the kpz fixed pointdremenik/kpzfixedpointnotes.pdfpresentation of the main results, stripping the...

COURSE NOTES ON THE KPZ FIXED POINT

DANIEL REMENIK

Abstract These notes are intended as a supplement to a minicourse on the KPZ fixed pointfirst given at the CIRM Research School ldquoRandom Structures in Statistical Mechanics andMathematical Physicsrdquo in March 2017 Much of the material is taken directly from the paper[MQR17] by Matetski Quastel and the author the aim here is to simplify and shorten thepresentation of the main results stripping the arguments of many technical details addingsome details in other parts of the derivation and focusing mostly on the key ideas leading tothis development

1 The KPZ fixed point

The aim of these notes is to present the recent development [MQR17] where the KPZ fixedpoint a scaling invariant Markov process taking values in real valued functions which looklocally like Brownian motion was constructed as limit of the totally asymmetric exclusionlsquorsquoprocess The construction leads to an explicit Fredholm determinant formula for its transitionprobabilities and to proofs of its main properties We begin with some brief motivation

11 The KPZ equation and universality class The Kardar-Parisi-Zhang (KPZ) univer-sality class is a broad collection of one-dimensional asymmetric randomly forced systemsincluding stochastic interface growth on a one-dimensional substrate directed polymer chainsin a random potential driven lattice gas models reaction-diffusion models in two-dimensionalrandom media and randomly forced Hamilton-Jacobi equations It can be loosely characterizedby having local dynamics a smoothing mechanism slope dependent growth rate (lateralgrowth) and space-time random forcing with rapid decay of correlations

Models in this class present an unusual size and distribution of fluctuations Thinking interms of the growth of a random one-dimensional interface KPZ models exhibit interfacesmoving at a non-zero velocity proportional to time t with fluctuations of size t13 anddecorrelating at a spatial scale of t23 The distribution of the fluctuations depends on thegeometry of the specific problem being studied and it has been found that in many interestingcases they are related to objects coming from random matrix theory (RMT) For backgroundon different aspects of the study of this class and an account of some of the main developmentsin the subject in the last fifteen years see the reviews [Qua11 Cor12 QR14 BP14 QS15]

The model which gives its name to this class is the Kardar-Parisi-Zhang equation [KPZ86]

parttH = λ(partxH)2 + νpart2xH +

radicDξ (11)

a canonical continuum equation for the evolution of a randomly growing one-dimensionalinterface It was predicted in [HH85 KPZ86] that the 123 rescaled solution

Hε(t x) = ε12H(εminus32t εminus1x) (12)

should have a non-trivial limiting behavior The (conjectural) limit under this scaling isthe so-called KPZ fixed point which is believed to be the universal limit for models on theKPZ universality class under the KPZ (123) scaling and should contain all the fluctuationbehaviour seen in this class As a (conjectural) object it was introduced non-rigorouslyin [CQR15] to which we point (together with [MQR17]) for further background One canunderstand many (although by no means all) of the main advances in the field as attempts to

Revision January 29 2019

1

COURSE NOTES ON THE KPZ FIXED POINT 2

understand some aspects of this limiting object (mostly restricted to a very special family ofparticular initial conditions)

12 TASEP and the KPZ fixed point

121 TASEP Our starting point in the construction of the KPZ fixed point will be anotherbasic model in the KPZ universality class the totally asymmetric exclusion process (TASEP)We will introduce this model in more detail in Section 3 so for now describe it slightlyinformally

Definition 11 (TASEP height function) The height function associated to TASEP is acontinuous time Markov process

(ht(x)

)xisinZ taking values in the space of simple random walk

paths by which we mean continuous curves obtained by linearly interpolating the graph of afunction from Z to Z which moves up or down by one in each step We think of this curve asan interface growing (downwards) in time according to the following dynamics each localmaximum (and) turns into a local minimum (or) at rate one that is if ht(z) = ht(z plusmn 1) + 1 thenht(z) 7rarr ht(z)minus 2 at rate 1 (see Figure 1)

122 The KPZ fixed point For each ε gt 0 the 123 rescaled TASEP height function is1

hε(tx) = ε12[h2εminus32t(2ε

minus1x) + εminus32t] (13)

The initial data corresponds to just rescaling h0 diffusively hε(0x) = ε12h0(2εminus1x) (the extra

2 is just a choice of normalization) and we allow h0 = h(ε)0 itself to depend on ε and assume

that the limith0 = lim

εrarr0hε(0 middot) (14)

exists (in a sense to be specified later)

Definition 12 (The KPZ fixed point) We define the KPZ fixed point as the limit

h(tx h0) = limεrarr0

hε(tx) (15)

We will often omit h0 from the notation when it is clear from the context

Of course one of the main points which we need to settle is that the limit (15) exists in asuitable sense this will in fact be part of the content of our main result

As we already mentioned the KPZ fixed point h(tx h0) should be the 123 scaling limitof all models in the KPZ universality class This is known as the strong KPZ universalityconjecture and could arguably be posed as the definition of the KPZ universality class Inthese lectures (and in [MQR17]) we only deal with TASEP but our method works for severalvariants of TASEP (which also have a representation through biorthogonal ensembles such asPushASEP as well as discrete time TASEPs and PNGs) this is the content of [MQR+]

123 Earlier work TASEP has been one of the most heavily studied models in the KPZ classand much effort has been devoted to the study of the distributional limit (15) for a few veryspecial choices of initial data h0

Example 13 (Stepnarrow wedge) The most basic case is the step initial data h0(x) =minus|x| (the name step comes from the derivative of h0) in which case it is known that hε(1x)converges to the Airy2 process minus a parabola A2(x)minus x2 [PS02 Joh03] which has one-point marginals given by the Tracy-Widom GUE distribution [TW94] In this case we have

1There is a difference between the scalings chosen in (13) and in the first arXiv version of [MQR17] wherethe analogous definition was done with t replaced by t2 on the right hand side The present choice is betterbecause it makes the usual Airy processes arise at time t = 1 at the fixed point level (instead of t = 2) andbecause it matches a natural choice of parameters for (11) it is the scaling used in later revised versions ofthat paper This difference translates into some constants being off by a factor of 2 when comparing the twoversions but it does not affect the arguments in any other way


rate 1

Figure 1 The height function associated to TASEP A particle moving to theright corresponds to a local maximum of the height function moving down toturn into a local minimum (see also Defn 31)

that h(0 middot) goes to d0 where du is the narrow wedge at u which is the function du(u) = 0du(x) = minusinfin for x 6= u So in view of our definition (15) we have

h(1x d0) + x2 = A2(x) (16)

Example 14 (Flat) Another basic case is that of flat initial data h0(x) = 1 for even xh0(x) = 0 for odd x interpolated lineraly in between Here we have convergence to the Airy1

process A1(x) [Sas05 BFPS07] which has one-point marginals given by the Tracy-WidomGOE distribution [TW96] and so we have

h(1x 0) = A1(x) (17)

Example 15 (StationaryBrownian) The other basic case corresponds to starting withh0 sampled as a double-sided simple symmetric random walk path (independently of theevolution of the height function) which yields h(1x B) with B a double-sided Brownianmotion with diffusion coefficient 2 this is known as the Airystat process [BFP10] The namestationary comes from the fact that TASEP is invariant under the product measure initialcondition associated to this choice of h0 this translates into the fact that h(1x B)minus h(1 0 B)is itself a double-sided Brownian motion (this Brownian motion is however not independent ofh(1 0 B) whose law is known as the Baik-Rains distribution [BR01])

What has made these cases tractable is the fact that exact formulas were available forTASEP which were derived using very specific properties of these initial conditions The samemethods also yield formulas for the three mixed cases corresponding to putting one of theseinitial conditions on one side of the origin and another one on the other

13 State space and topology The state space in which we will show that (15) holds andwhere (14) will be assumed to hold (in distribution) will be the following

Definition 16 (UC functions) We define UC as the space of upper semicontinuousfunctions h Rrarr [minusinfininfin) with h(x) le C(1 + |x|) for some C ltinfin

We will endow this space with the topology of local UC convergence This is the naturaltopology for lateral growth and will allow us to compute h(tx h0) in all cases of interest2In order to define this topology recall that h is upper semicontinuous (UC) if and only if itshypograph

hypo(h) = (xy) y le h(x)is closed in [minusinfininfin)times R Slightly informally local UC convergence can be defined as follows

Definition 17 (Local UC convergence) We say that(hε)εsube UC converges locally in

UC to h isin UC if there is a C gt 0 such that hε(x) le C(1 + |x|) for all ε gt 0 and for everyM ge 1 there is a δ = δ(εM) gt 0 going to 0 as εrarr 0 such that the hypographs HεM and HMof hε and h restricted to [minusMM ] are δ-close in the sense that

cup(tx)isinHεMBδ((tx)) sube HM and cup(tx)isinHM

Bδ((tx)) sube HεM

2Actually the bound h(x) le C(1 + |x|) which we are imposing here and in [MQR17] on UC functions is notas general as possible but makes the arguments a bit simpler and it suffices for most cases of interest (see also[MQR17 Foot 9])


See [MQR17 Sec 31] for more details We will also use an analogous space LC made oflower semicontinuous functions

Definition 18 (LC functions and local convergence) We let

LC =g minusg isin UC

We endow this space with the topology of local LC convergence which is defined analogouslyto local UC convergence now in terms of epigraphs

epi(g) = (xy) y ge g(x) (18)

Explicitly gε converges locally in LC to g if and only if minusgε minusrarr minusg locally in UC

14 Operators In order to state our main result we need to introduce several operatorswhich will appear in the explicit Fredholm determinant formula for the fixed point

Definition 19 (Projections) For a fixed vector a isin Rm and indices n1 lt lt nm weintroduce the functions

χa(nj x) = 1xgtaj χa(nj x) = 1xleaj

which we also regard as multiplication operators acting on the spaces L2(t1 tmtimesR) and`2(n1 nm times Z) We will use the same notation if a is a scalar writing

χa(x) = 1minus χa(x) = 1xgta

We think of χa as a projection from L2(R) to L2((ainfin)) (and analogously in the vector case)

Our basic building block is the following (almost) group of operators

Definition 110 We define

Stx = expxpart2 + t3 part

3 x t isin R2 x lt 0 t = 0

which satisfySsxSty = Ss+tx+y (19)

as long as all subscripts avoid x lt 0 t = 0

We can think of these as unbounded operators with domain Cinfin0 (R)

Remark 111 In these notes we will not worry about precise statements (or proofs) justifyingthe convergence of kernels as needed in each step So for example at this particular pointwe will not worry about making the domains and analytical properties of the Stx operatorsprecise

But it is not even clear whether the operators make any sense for x lt 0 t 6= 0 (notice thatfor t = 0 they clearly donrsquot) The fact that they do can be checked using the following explicitrepresentation for the operators Stx acts by convolution

Stxf(z) =

int infinminusinfin

dy Stx(z y)f(y) =

int infinminusinfin

dy Stx(z minus y)f(y)

with convolution kernel Stx(z y) = Stx(z minus y) given by

Stx(z) =1

2πi

int〈dw e

t3w3+xw2minuszw = tminus13e

2x3

3t2minus zx

t Ai(minustminus13z + tminus43x2) (110)

for t gt 0 together with

Stx = (Sminustx)lowast or Stx(z y) = Stx(z minus y) = Sminustx(y minus z) (111)

for t lt 0 where 〈 is the positively oriented contour going in straight lines from eminusiπ3infin to

eiπ3infin through 0 and Ai is the Airy function Ai(z) = 12πi

int〈 dwe

13w3minuszw


Remark 112 The fact that these operators make sense is a generalization of the fact that

eminusxpart2Ai is well defined for all x isin R which has appeared been used earlier in the field (see eg

[BFPS07 QR13 QR16]) and corresponds simply to taking x = 1 in the above definition

From (111) we get directly the identity

(Stx)lowastStminusx = I

We get to the definition of the most important operator which will appear in our formulaIn order to introduce it we first take h isin UC and define local ldquoHitrdquo and ldquoNo hitrdquo operators by

PNo hit h`1`2

(u1 u2)du2 = PB(`1)=u1(B(y) gt h(y) on [`1 `2] B(`2) isin du2)

and PHit h`1`2

= I minus PNo hit h`1`2

where B is a Brownian motion with diffusion coefficient 2 The

Brownian scattering transform of h is then the t gt 0 dependent operator on L2(R) given by

Khypo(h)t = lim

`1rarrminusinfin`2rarrinfin

(St`1)lowastPHit h`1`2

Stminus`2 (112)

One should think then of Khypo(h)t as a sort of asymptotic transformed ldquotransition densityrdquo

for B in the whole line killed if it doesnrsquot hit hypo(h) Of course it is far from obvious thatthis makes any sense or in other words that the above limit exists This was first proved in[QR16] (for t = 1 which makes no difference and under an additional regularity assumption)and it turns out not to be hard to write a relatively explicit expression for the limit We sketchnext how this is done

Observe first that

PNo hit h`1`2

(u1 u2) = p`1u1(`2 u2)p`1u1`2u2(no hit) (113)

where p`1u1(`2 u2) is the transition density for Brownian motion to be at u2 at time `2 giventhat it started at u1 at time `1 and where the second factor is the probability for a Brownianbridge with the same endpoints not to hit hypo(h) which we can write for x isin [`1 `2] asint infin

h(x)p`1u1`2u2(B(x) isin dz)p0zxminus`1u1(no hit hminusx on [0xminus `1])

times p0z`2minusxu2(no hit h+x on [0 `2 minus x])

Now p`1u1(`2 u2)p`1u1`2u2(B(x) isin dz) = p0z(xminus `1 u1)p0z(`2 minus x u2)dz so (113) becomesint infinh(x)

dz p0z(no hit hminusx on [0xminus `1]xminus `1 u1)p0z(no hit h+x on [0 `2 minus x] `2 minus x u2)

where the notation is meant to indicate that the factors in the integrand are now densitiesAgain we can write p0z(no hit h on [0 `] ` u) as p0z(` u)minus p0z(hit h on [0 `] ` u) which is

just e`part2(z u)minus

int `0 pz(τ isin ds)e(`minuss)part2(B(τ ) u) so we have

PNo hit h`1`2

(u1 u2) =

int infinh(x)

dz

(e(xminus`1)part2(z u1)minus

int xminus`1

0pz(τ

minus isin ds)e(xminus`1minuss)part2(B(τminus) u1)

)times(e(`2minusx)part2(z u2)minus

int `2minusx

0pz(τ

+ isin ds)e(`2minusxminuss)part2(B(τ+) u2)

)where τ+ and τminus are respectively the hitting times by B of

h+x (y) = h(x + y) and hminusx (y) = h(xminus y)


If we now apply St`1 on the left and Stminus`2 on the right then all the `1rsquos and `2rsquos in heatkernels cancel and we get

St`1PNo hit h`1`2

Stminus`2(u1 u2) =

int infinh(x)

dz

(Stx(z u1)minus

int xminus`1

0pz(τ

minus isin ds)Stxminuss(B(τminus) u1)

)times(

Stminusxminuss(z u2)minusint `2minusx

0pz(τ

+ isin ds)Stminusxminuss(B(τ+) u2)

) (114)

The limit as `1 rarr minusinfin and `2 rarrinfin is now easy to compute (at least formally) In order towrite it down we introduce the following

Definition 113 (Hit operators) For h isin UC we define the operator

Shypo(g)tx (v u) = EB(0)=v

[Stxminusτ (B(τ ) u)1τltinfin

]=

int infin0

PB(0)=v(τ isin ds)Stxminuss(B(τ ) u)

where B(x) is a Brownian motion with diffusion coefficient 2 and τ is the hitting time of thehypograph of h|[0infin) Note that trivially

Shypo(h)tx (v u) = Stx(v u) for v le h(0) (115)

If g isin LC there is a similar operator Sepi(g)tx with the same definition except that now τ is the

hitting time of the epigraph of g and Sepi(g)tx (v u) = Stx(v u) for v ge g(0)

As above one can think of Shypo(h)tx (v u) as a sort of asymptotic transformed ldquotransition

densityrdquo for the Brownian motion B to go from v to u hitting the hypograph of h (note that his not necessarily continuous so hitting h is not the same as hitting hypo(h) B(τ ) le h(τ )and in general the equality need not hold) To see what we mean here write

Shypo(h)tx = lim

TrarrinfinShypo(h)TStxminusT with Shypo(h)T(v u) = EB(0)=v

[S0Tminusτ (B(τ ) u)1τleT

](116)

and note that Shypo(h)T(v u) is nothing but the transition probability for B to go from v attime 0 to u at time T hitting hypo(h) in [0T]

We are ready now to state the formal definition of Khypo(h)t (following from (112) and

(114))

Definition 114 (Brownian scattering transform) We define the Brownian scatteringtranform of h isin UC as

Khypo(h)t = Iminus (Stx minus S

hypo(hminusx )tx )lowastχh(x)(Stminusx minus S

hypo(h+x )tminusx ) (117)

where x isin R Note that the projection χh(x) can be removed from (117) thanks to (115)

There is another operator which uses g isin LC and hits ldquofrom belowrdquo

Kepi(g)t = Iminus (Stx minus S

epi(gminusx )tx )lowastχg(x)(Stminusx minus S

epi(g+x )tminusx ) (118)

The full proof involves showing that the integrals in (114) indeed converge in the limit as`1 rarr minusinfin and `2 rarr infin The proof appears in [QR16] (where it is actually shown that theconvergence in (112) happens in trace norm)

Remark 115 Note from the above computation that the point x at which h is split in (117)

plays absolutely no role the operator Khypo(h)t actually does not depend on the choice of

splitting point x This is of course clear one if one focuses on (112) but it is not at all obvious

at the level of (117) This fact is useful in explicit computations of Khypo(h)t (since it allows

us to split h at any convenient point) and will also be useful in several steps in the derivation


Remark 116 For t = 1 I minus Kepi(g)minust coincides with the Brownian scattering operator

introduced in [QR16] One of the aims of that paper was the derivation under an assumptionwhich is widely believed to hold but which currently escapes rigorous treatment (namely thatsolution of the KPZ equation with narrow wedge initial data converges to the Airy2 process)of explicit formulas for the one-point marginals of the limit of the rescaled solution of the KPZequation (12) These formulas coincide with those which follow from the KPZ fixed pointformulas to be given below which provides further (if not rigorous) confirmation of the strongKPZ universality conjecture

15 Existence and formulas for the KPZ fixed point We are ready to state the resultwhich states the existence of the KPZ fixed point h(tx) as the limit of the 123 rescaledTASEP height function (15) and characterizes its distribution for each t ge 0

Theorem 117 (KPZ fixed point) Fix h0 isin UC Let h(ε)0 be initial data for the TASEP

height function such that the corresponding rescaled height functions hε0 (13) converge to h0

locally in UC as εrarr 0 Then the limit (15) for h(tx) exists (in distribution) locally in UCand its finite dimensional distributions are given as follows for x1 lt x2 lt middot middot middot lt xm

Ph0(h(tx1) le a1 h(txm) le am)

= det(Iminus χaK

hypo(h0)text χa

)L2(x1xmtimesR)

(119)

= det(IminusK

hypo(h0)txm

+ Khypo(h0)txm

e(x1minusxm)part2χa1e(x2minusx1)part2χa2 middot middot middot e(xmminusxmminus1)part2χam

)L2(R)

(120)

where

Khypo(h0)text (xi middot xj middot) = minuse(xjminusxi)part

21xiltxj + eminusxipart

2K

hypo(h0)t exjpart

2(121)

and

Khypo(h0)tx = eminusxpart

2K

hypo(h0)t expart

2(122)

with Khypo(h0)t the kernel defined in (118)

The determinants appearing in the theorem are Fredholm determinants which may bedefined as follows for K an integral operator acting on L2(X dmicro) with integral kernel K(x y)

det(I minusK)L2(Xdmicro) =sumnge0

(minus1)n

n

intXn

dmicro(x1) middot middot middot dmicro(xn) det[K(xi xj)]nij=1

For background on Fredholm determinants we refer to [Sim05] or [QR14 Sec 2]

Remark 118 The fact that eminusxpart2K

hypo(h)t expart

2in (122) makes sense is not entirely obvious

but follows from the fact that Khypo(h)t equals

(Stx)lowastχh(x)Stminusx + (Shypo(hminusx )tx )lowastχh(x)Stminusx + (Stx)lowastχh(x)S

hypo(h+x )tminusx minus (S

hypo(hminusx )tx )lowastχh(x)S

hypo(h+x )tminusx

together with the group property (19) and the definition of the hit operators

Remark 119 The fact that the Fredholm determinant in (119) is finite is a consequence of

the fact that there is a (multiplication) operator M such that MχaKhypo(h0)text χaM

minus1 is trace

class A similar statement holds for (120) (these facts are proved in [MQR17 Appdx A])However as we already mentioned in these lectures we will omit all these issues

A straightforward consequence of the definition of the KPZ fixed point as the limit of the123 rescaled TASEP height function is that it is invariant in distribution under the same123 rescaling


Proposition 120 (123 scaling invariance) Let α gt 0 and hα0 (x) = αminus1h0(α2x) Then

αh(αminus3t αminus2x hα0 )dist= h(tx h0)

This property is what justifies the name of the process The KPZ fixed point is conjectured tobe the unique non-trivial field which is fixed in distribution under the 123 rescaling and whichis local in a certain sense and satisfies some symmetry properties (i) and (ii) in Proposition28 below (see [MQR17 Rem 412])

The kernel in (119) is usually referred to as an extended kernel (note that the Fredholmdeterminant is being computed on the ldquoextended L2 spacerdquo L2(x1 xmtimesR)) The kernelappearing after the second hypo operator in (120) is sometimes referred to as a path integralkernel [BCR15] and should be thought of as a discrete pre-asymptotic version of the Brownianscattering operator (117) (on a finite interval) The extended kernel version of the formula isthe way in which formulas of this type have most frequently been written in the literatureand in many situations it is easier to handle than its alternative The path integral kernelversion on the other hand will play a crucial role later in the proof of the regularity of theKPZ fixed point which as we will see is in turn crucial in proving that it satisfies the Markovproperty Another nice consequence of the path integral version is that it leads to an explicitformula for the probability that the fixed point lies below a given LC function This type ofldquocontinuum statisticsrdquo formulas have been previously derived for Airy and related processes[CQR13 QR13 BCR15 NR17a NR17b]

Theorem 121 (Continuum statistics formula for the KPZ fixed point) Let h0 isin UCand g isin LC Then for every t gt 0

P(h(tx h0) le g(x) x isin R) = det(IminusK

hypo(h0)t2 K

epi(g)minust2

)L2(R)

(123)

2 Properties of the KPZ fixed point

In this section we present the main properties of the KPZ fixed point as well as a sketch ofthe proof of some of them

21 Markov property The sets Ag = h isin UC h(x) le g(x) x isin R g isin LC form agenerating family for the Borel sets B(UC) Hence from (123) we can define the fixed pointtransition probabilities ph0(t Ag)

Lemma 21 For fixed h0 isin UC and t gt 0 the measure ph0(t middot) defined above is a probabilitymeasure on UC

Sketch of the proof It is clear from the construction that Ph0(h(txi) le ai i = 1 n) isnon-decreasing in each ai and is in [0 1] We need to show then that this quantity goes to 1as all airsquos go to infinity and to 0 if any ai goes to 0 The first one is standard and relies onthe inequality |det(IminusK)minus 1| le K1eK1+1 (with middot 1 denoting trace norm) and the factthat the kernel inside the determinant in (119) obviously vanishes in the limit as all airsquos goto infinity The second limit is in general very hard to show for a formula given in terms ofa Fredholm determinant but it turns out to be rather easy in our case because the abovemultipoint probability is trivially bounded by Ph0(h(txi) le ai) and then one can use theskew time reversal symmetry of the fixed point (Prop 28(ii)) to compare this to the one-pointmarginals of the Airy2 process See [MQR17 Sec 35] for the details

Theorem 22 The KPZ fixed point(h(t middot)

)tgt0

is a (Feller) Markov process taking values inUC

The proof is based on the fact that h(tx) is the limit of hε(tx) which is Markovian Toderive from this the Markov property of the limit requires some compactness which in ourcase is provided by Thm 24 below


22 Regularity and local Brownian behavior Up to this point we only know that thefixed point is in UC but due to the smoothing mechanism inherent to models in the KPZclass one should expect h(t middot) to at least be continuous for each fixed t gt 0 The next resultshows that in fact

For every M gt 0 h(t middot)∣∣[minusMM ]

is Holder-β for any β lt 12 with probability 1

The statement of the theorem we include is actually stronger since it shows that this regu-larity holds uniformly (in ε gt 0) for the approximating hε(t middot)rsquos which yields the compactnessneeded to get the Markov property

Definition 23 (Local Holder norm) For a continuous function h R minusrarr [minusinfininfin) β gt 0and M gt 0 we define local Holder β norm of h as

hβ[minusMM ] = supx1 6=x2isin[minusMM ]

|h(x2)minus h(x1)||x2 minus x1|β

A function for which these norms are finite for all M gt 0 is said to be locally Holder β

We note that the topology on UC when restricted to the space Cγ of continuous functions hwith |h(x)| le γ(1 + |x|) is the topology of uniform convergence on compact sets and that UCis a Polish space in which the subspaces of Cγ made of locally Holder β functions are compactin UC (this is what connects the regularity of the fixed point with the tightness needed forproving the proof of the Markov)

Theorem 24 (Space regularity) Fix t gt 0 h0 isin UC and initial data h(ε)0 for the TASEP

height function such that hε0 minusrarr h0 locally in UC as ε rarr 0 Then for each β isin (0 12) andM ltinfin

limArarrinfin

limsupεrarr0

P(hε(t)β[minusMM ] ge A) = limArarrinfin

P(hβ[minusMM ] ge A) = 0

The proof proceeds through an application of the Kolmogorov continuity theorem whichreduces regularity to two-point functions It depends heavily on the representation (120)for the two-point function in terms of path integral kernels and is based on the argumentsintroduced in [QR13] for the Airy1 and Airy2 processes We skip the details (see [MQR17Appdx 42])

From the 123 scaling invariance of the fixed point (Prop 120 below) one expects that theHolder 1

2minus regularity in space translates into Holder 13minus regularity in time This is indeed the

case

Theorem 25 (Time regularity) Fix x0 isin R and initial data h0 isin UC For t gt 0 h(tx0)is locally Holder α in t for any α lt 13

The proof uses the variational formula for the fixed point we sketch it in Section 25

Now we turn to the Brownian behavior in space of the fixed point paths

Theorem 26 (Local Brownian behavior) For any initial condition h0 isin UC and anyt gt 0 the process h(tx) is locally Brownian in x in the sense that for each y isin R the finitedimensional distributions of

b+ε (x) = εminus12(h(ty + εx)minus h(ty)) and bminusε (x) = εminus12(h(ty minus εx)minus h(ty))

converge as ε 0 to Brownian motions with diffusion coefficient 2

Very brief sketch of the proof The proof is based again on the arguments of [QR13] One uses(120) and Brownian scale invariance to show that

P(h(t εx1) le u +

radicεa1 h(t εxn) le u +

radicεan

∣∣ h(t 0) = u)

= E(1B(xi)leaii=1n φ

εxa(uB(xn))

)


for some explicit function φεxa(ub) The Brownian motion appears from the product ofheat kernels in (120) while φεxa contains the dependence on everything else in the formula

(including the Fredholm determinant structure and h0 through the hypo operator Khypo(h0)t )

Then one shows that φεxa(ub) goes to 1 in a suitable sense as εrarr 0

By the 123 scaling invariance of the fixed point Prop 120 we have h(tx h0)dist=

t13h(1 tminus23x tminus13h0(t23x)) which suggests that the local Brownian behaviour of the fixedpoint should yield ergodicity In fact an argument very similar to the one sketched aboveyields

Theorem 27 (Ergodicity) For any (possibly random) initial condition h0 isin UC such

that for some ρ isin R ε12(h0(εminus1x)minus ρεminus1x) is convergent in distribution in UC the finitedimensional distributions of the process

h(tx h0)minus h(t 0 h0)minus ρx

converge as trarrinfin to those of a double-sided Brownian motion B with diffusion coefficient 2

A very similar result was first stated and proved (by a method based on coupling) in Pimentel[Pim17]

23 Symmetries The KPZ fixed point satisfies a number of properties which follow directlyfrom the construction

Proposition 28 (Symmetries of h)

(i) (Skew time reversal) P(h(tx g) le minusf(x)

)= P

(h(tx f) le minusg(x)

) f g isin UC

(ii) (Shift invariance) h(tx + u h0(xminus u))dist= h(tx h0)

(iii) (Reflection invariance) h(tminusx h0(minusx))dist= h(tx h0)

(iv) (Affine invariance) h(tx f(x) + a + cx)dist= h(tx + 1

2ct f) + a + cx + 14c

2t

(v) (Preservation of max) h(tx f1 or f2)dist= h(tx f1) or h(tx f2)

Implicit in (v) is a certain coupling which allows one to run the fixed point started with twodifferent initial conditions This coupling is inherited naturally from the basic coupling forTASEP see Rem 215

Proof (ii) and (iii) follow directly from the fact that the TASEP height function satisfiesexactly the same properties

(i) is slightly more delicate but it also follows easily from the analog property for TASEPwhich is perhaps most easily seen by using the graphical representation for TASEP see eg[Lig85] to couple TASEP starting with height profile g at time 0 and running to time t withanother copy of TASEP started now with height profile minusf at time t and running backwards totime 0 Exercise prove the skew time reversal property directly from the continuum statisticsformula (123)

(v) also follows from the exact same property for TASEP which can be proved by showingthat if h1(t x) and h2(t x) are two copies of TASEP coupled again using the graphicalrepresentation and such that h1(0 x) ge h2(0 x) for all x then h1(t x) ge h2(t x) for all x andall t

(iv) can be also be proved from TASEP but it is easier to use the variational formulaprovided in Thm 216 so we postpone the proof until Section 25

24 Airy processes We show here how to recover (at time t = 1) the known Airy1 Airy2

and Airy2rarr1 processes from Thm 117 and the definition of the Brownian scattering transformThe first two actually follow from the third (which interpolates between them) we will presenta quick derivation for them and then do the third case in a bit more detail


Definition 29 The UC function du(u) = 0 du(x) = minusinfin for x 6= u is known as a narrowwedge at u (we already used this function in (16))

Example 210 (Airy2 process) Narrow wedge initial data leads to the Airy2 process[PS02 Joh03] (this was stated above in (16))

h(1x du) + (xminus u)2 = A2(x)

Obtaining this from the KPZ fixed point formula is trivial First we may take for simplicityu = 0 since the general case follows by shift invariance and the stationarity of the Airy2

process The only way in which a Brownian motion can hit the hypograph of d0 is to pass

below 0 at time 0 so Phit d0[minusLL] = eLpart

2χ0e

Lpart2 Then (StminusL)lowastPhit d0[minusLL]StminusL = (St0)lowastχ0(St0) so

setting t = 1 we get (note here we donrsquot even have to compute a limit)

Khypo(d0)1 (u v) = eminus

13part3χ0e

13part3(u v) =

int infin0

dλAi(uminus λ)Ai(v minus λ) = KAi(u v)

the Airy kernel which is exactly the kernel appearing in the formulas for the Airy2 process

Example 211 (Airy1 process) Flat initial data h0 equiv 0 leads to the Airy1 process [Sas05BFPS07] (this was stated above in (17))

h(1x 0) = A1(x)

This one needs a bit more work but the basic idea is simple By the reflection principle if u v gt 0

then Phit 0[minusLL](u v) = e2Lpart2(u v) = eLpart

2eLpart

2(u v) where f(x) = f(minusx) (so A(u v) =

A(uminusv)) If we pretend like this formula works for all u v then we get (StminusL)lowastPhit 0[minusLL]StminusL =

(St0)lowast(St0) It can be shown (see [QR16 Prop 37]) that the difference between this andwhat one gets if one uses the correct formula for all u v goes to 0 as Lrarrinfin so setting t = 1we get

Khypo(0)1 (u v) = eminus

13part3e

13part3(u v) =

int infinminusinfin

dλAi(u+ λ)Ai(v minus λ) = 2minus13 Ai((2minus13(u+ v))

(the last one is a known identity for the Airy function) which is exactly the kernel appearingin the formulas for the Airy1 process

Example 212 (Airy2rarr1 process) Wedge or half-flat initial data hh-f(x) = minusinfin for x lt 0hh-f(x) = 0 for x ge 0 leads to the Airy2rarr1 process [BFS08b]

h(1x hh-f) + x21xlt0 = A2rarr1(x)

This one is a combination of the above two here we sketch how it can be derived directly froma formula like (117) where we need to take hminus0 equiv minusinfin and h+

0 (x) equiv 0

It is straightforward to check that Shypo(hminus0 )t0 equiv 0 On the other hand and similarly to the

last example an application of the reflection principle based on (116) (see [QR16 Prop 37]again) yields that for v ge 0

Shypo(h+0 )t0 (v u) =

int infin0

Pv(τ0 isin dy)Stminusy(0 u) = St0(minusv u)

which gives (with f(x) = f(minusx) as above)

Khypo(h0)t = Iminus (St0)lowastχ0[St0 minus St0] = (St0)lowast(I + )χ0St0

This yields (using (121) and (122))

Khypo(hh-f)text (xi middot xj middot) = minuse(xjminusxi)part

21xiltxj + S0minusxi(St0)lowast(I + )χ0St0S0xi

= minuse(xjminusxi)part21xiltxj + (Stminusxi)

lowast(I + )χ0Stxi


Choosing t = 1 and using (110) yields that the second term on the right hand side equalsint 0

minusinfindλ eminus2x3

i 3minusxi(uminusλ)Ai(uminus λ+ x2i )e

2x3j3+xj(vminusλ)Ai(v minus λ+ x2

j )

+

int 0


i 3minusxi(u+λ)Ai(u+ λ+ x2i )e


j )

which after a simple conjugation gives the kernel for the Airy2rarr1 process [BFS08b App A]

25 Variational formulas and the Airy sheet

Definition 213 (Airy sheet)

A(xy) = h(1y dx) + (xminus y)2

is called the Airy sheet [CQR15] Fixing either one of the variables it is an Airy2 process inthe other We also write

A(xy) = A(xy)minus (xminus y)2

Thanks to the symmetry properties of the KPZ fixed point 28 the Airy sheet is stationary

(A(x0 + xy0 + y)dist= A(xy) for each fixed x0y0) and symmetric (A(xy)

dist= A(yx))

Remark 214 It is natural to wonder whether the fixed point formulas at our disposaldetermine the joint probabilities P(A(xiyi) le ai i = 1 M) for the Airy sheet Un-fortunately this is not the case In fact the most we can compute using our formulas is

P(A(xy) le f(x) + g(y) xy isin R) = det(IminusK

hypo(minusg)1 K

epi(f)minus1

) Suppose we want to compute

the two-point distribution for the Airy sheet P(A(xiyi) le ai i = 1 2) from this We would

need to choose f and g taking two non-infinite values which yields a formula for P(A(xiyj) lef(xi) + g(yj) i j = 1 2) and thus we need to take f(x1) + g(y1) = a1 f(x2) + g(y2) = a2 andf(x1) + g(y2) = f(x2) + g(y1) = L with Lrarrinfin But f(xi) + g(yj) i j = 1 2 only spans a3-dimensional linear subspace of R4 so this is not possible

Remark 215 The KPZ fixed point inherits from TASEP a canonical coupling between theprocesses started with different initial data (using the same ldquonoiserdquo) It is this the propertythat allows us to construct the two-parameter Airy sheet An annoying difficulty however isthat we cannot prove that this process is unique (see also the last remark) More precisely theconstruction of the Airy sheet in [MQR17] is based on using tightness of the coupled processesat the TASEP level and taking subsequential limits Since at the present time we have noway to assure that the limit points are unique what we are really doing is calling any suchsubsequential limit an Airy sheet While one should keep this important caveat in mind westress that the lack of uniqueness does not affect any of the statements we will make below

The preservation of max property allows us to write an important variational formula forthe KPZ fixed point in terms of the Airy sheet which was conjectured in [CQR15]

Theorem 216 (Airy sheet variational formula)

h(tx h0)dist= sup

y

t13A(tminus23x tminus23y) + h0(y)

(21)

In particular any such Airy sheet satisfies the semi-group property If A1 and A2 areindependent copies and t1 + t2 = t are all positive then

supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)

dist= t13A1(tminus23x tminus23y)

Proof Let hn0 be a sequence of initial conditions taking finite values hn0 (yni ) at yni i = 1 knand minusinfin everywhere else which converges to h0 in UC as nrarrinfin By repeated application of


Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )

and taking nrarrinfin yields

h(tx h0)dist= sup

y

h(tx dy) + h0(y)

The result now follows from scaling invariance Prop 120

One of the interests in this variational formula is that it leads to proofs of properties of thefixed point as we already mentioned in earlier sections

Proof of affine invariance Prop 28(iv) The fact that the fixed point is invariant undertranslations of the initial data is straightforward so we may assume a = 0 By Thm 216 andthe stationarity of the Airy sheet we have

h(tx h0 + cx)dist= sup

y

t13A(tminus23x tminus23y)minus tminus1(xminus y)2 + h0(y) + cy

= sup

y

t13A(tminus23x tminus23(y + ct2))minus tminus1(xminus y)2 + h0(y + ct2) + cx + c2t4

dist= sup

y

t13A(tminus23x tminus23y) + h0(y + ct2) + cx + c2t4

dist= h(tx h0(x + ct2)) + cx + c2t4

Sketch of the proof of time regularity Thm 25 Fix α lt 13 and choose β lt 12 so thatβ(2 minus β) = α By the Markov property and Thm 24 it is enough to assume that h0 islocally Holder β and check the Holder α regularity at time 0 By space regularity of theAiry2 process (proved in [QR13] but which also follows from Thm 24) there is an R ltinfinas such that |A2(x)| le R(1 + |x|β) and making R larger if necessary we may also assume|h0(x)minush0(x0)| le R(|xminusx0|β+|xminusx0|) From the variational formula (21) |h(tx0)minush(0x0)|is then bounded by

supxisinR

(R(|xminus x0|β + |xminus x0|+ t13 + t(1minus2β)3|x|β)minus 1

t (x0 minus x)2)

The supremum is attained roughly at xminus x0 = tminusη with η such that |xminus x0|β sim 1t (x0 minus x)2

Then η = 1(2minus β) and the supremum is bounded by a constant multiple of tβ(2minusβ) = tα asdesired

3 Full solution for TASEP

Our goal is to obtain a full solution for TASEP (by which we mean an explicit formula forits finite dimensional distributions) which is suitable for asymptotics in the form needed in(15) We begin with a more precise description of the process

31 The model

Definition 31 (TASEP) The totally asymmetric simple exclusion process consists ofparticles at positions middot middot middot lt Xt(2) lt Xt(1) lt Xt(0) lt Xt(minus1) lt Xt(minus2) lt middot middot middot on Zcupminusinfininfinperforming totally asymmetric nearest neighbour random walks with exclusion each particleindependently attempts jumps to the neighbouring site to the right at rate 1 the jump beingallowed only if that site is unoccupied (see [Lig85] for the non-trivial fact that the process withan infinite number of particles makes sense)


We follow the standard practice of ordering particles from right to left for right-finite datathe rightmost particle is labelled 1 Let

Xminus1t (u) = mink isin Z Xt(k) le u

denote the label of the rightmost particle which sits to the left of or at u at time t TheTASEP height function associated to Xt is given for z isin Z by

ht(z) = minus2(Xminus1t (z minus 1)minusXminus1

0 (minus1))minus z (31)

which fixes h0(0) = 0 This is the height function which we already introduced in Section 12(exercise check directly that the dynamics induced on ht by the particle dynamics coincideexactly with the dynamics specified there) see also Fig 1 We will find formulas for the finitedimensional distributions of Xt (31) will allow us to use those formulas to compute the limit(15)

32 The master equation and Schutzrsquos formula3 The first step is to solve the masterequation for TASEP in order to obtain a formula for its transition probabilities We will workonly in the case where TASEP starts with a finite number N ge 2 of particles

Define the Weyl chamber ΩN = (x1 xN ) isin ZN x1 gt middot middot middot gt xN and for x y isin ΩN

consider the transition probabilities

P(N)t (x y) = P(Xt = x|X0 = y)

Then P(N)t satisfies the master equation (or Kolmogorov forward equation)4

d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)

where the infinitesimal generator of the process L(N) is given for F ΩN minusrarr R by

L(N)F (x) =Nsumk=1

1xkminus1minusxkgt1nablakF (x)

with x0 =infin and nablakF (x) = F (x1 xk + 1 xN )minus F (x)

The master equation for TASEP was solved by Schutz [Sch97] using the Bethe ansatz[Bet31] The first step is to rewrite (32) as an equation without the exclusion constraint(that is without the the factor 1xkminus1minusxkgt1 in LN ) together with suitable boundary conditions

Explicitly if for fixed y isin ΩN the function u(N)t ZN minusrarr R solves

d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)

(where nablalowastkF (x) = F (x1 xk minus 1 xN )minus F (x)) with the boundary conditions

nablalowastku(N)t (x) = 0 when xk = xk+1 + 1 (34)

then when restricted to the Weyl chamber u(N)t coincides with P

(N)t that is

P(N)t (x y) = u

(N)t (x) forallx isin ΩN (35)

The effect of the boundary constraint (34) on the free evolution (33) is simply to impose thata configuration (x1 xk xk+1 xN ) can be reached from (x1 xk minus 1 xk+1 xN )only if xk minus 1 gt xk+1 which corresponds to the exclusion rule in TASEP (exercise check that(35) holds under (33)(34))

3The reader may choose to skip directly to Section 34 this and the next sections are not strictly needed inthe derivation of the fixed point

4The adjoint in (32) comes from the fact that in order to keep the notation from the literature we have

written P(N)t (x y) for the probability of reaching x from y in t steps instead of the other way around


Theorem 32 (Schutzrsquos formula [Sch97])

P(N)t (x y) = det(Fiminusj(t xN+1minusi minus yN+1minusj))1leijleN (36)

with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn

wxminusn+1et(wminus1) (37)

where Γ01 is any simple loop oriented anticlockwise which includes w = 0 and w = 1

The argument of [Sch97] shows how this remarkable solution can be derived and the methodturns out to work for other similar processes (see for instance [BFP07 BF08 BFS08a])However once one has the explicit solution it is not hard to check that it satisfies (33)-(34)so this is the proof we present

Proof of Thm 32 The proof is based only on the following identities which follow directlyfrom (37)

parttFn(t x) = nablalowastFn(t x) Fn(t x) = minusnablaFn+1(t x) (38)

where nablaF (x) = F (x+ 1)minus F (x) nablalowastF (x) = F (xminus 1)minus F (x)

Define the column vectors

Hi(t x) =(Fiminus1(t xminus yN ) middot middot middot FiminusN (t xminus y1)

)T

Then denoting by u(N)t (x) the right-hand side of (36) we can write

parttu(N)t (x) =

Nsumk=1

det[ parttHk(t xN+1minusk)

]=

Nsumk=1

det[ nablalowastHk(t xN+1minusk t)

]=

Nsumk=1

nablalowastk det[Fiminusj(t xN+1minusi minus yN+1minusj)

]

where we used the multilinearity of the determinant This gives the first equation in (33)

To get the boundary conditions (34) take xkminus1 = xk + 1 and use again the multilinearityof the determinant to get

nablalowastk det[H1(t xN ) HN (t x1)

]= det

[ HNminusk(t xk+1)nablalowastHN+1minusk(t xk)

]= 0

because nablalowastHN+1minusk(t xk) = minusnablaHN+1minusk(t xk minus 1) = minusnablaHN+1minusk(t xk+1) = HNminusk(t xk+1)

To get the initial condition we write first n ge 0

Fminusn(0 x) =(minus1)n

2πi

∮Γ0

dw(1minus w)n

wx+n+1

which in particular implies that Fminusn(0 x) = 0 for x lt minusn and x gt 0 and F0(0 x) = 1x=0In the case xN lt yN since y isin ΩN we have xN le yN minus 1 le yNminus1 minus 2 le middot middot middot le yN+1minusj minus jand thus F1minusj(0 xN minus yN+1minusj) = 0 so the determinant in (36) vanishes because the matrixcontains a row of zeros If xN ge yN then we have xk gt yN for all k = 1 N minus 1 and allentries of the first column in the matrix from (36) vanish except for the first one which equals1xN=yN Repeating this argument for xNminus1 xNminus2 and so on we deduce that the matrix isupper-triangular with indicator functions 1XNminusi+1=yNminusi+1 in the diagonal which yields theclaimed formula

33 The non-intersecting line ensemble representation Schutzrsquos formula itself is notwell-suited for asymptotics but the remarkable fact that it is given by a determinant opensup an avenue for its treatment In a breakthrough by Sasamoto [Sas05] which was pursuedand extended rigorously in [BFPS07] he realized that the finite dimensional distributions forTASEP can be expressed in terms of a ldquosignedrdquo non-intersecting line ensemble as follows LetGTN be the set of Gelfand-Tsetlin patterns x given as

GTN =xji isin Z 1 le i le j le N xj+1

i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4

Figure 2 Visualization of a Gelfand-Tsetlin pattern in GT4 The bottom leftdiagonal is distributed as TASEP with initial condition y (at time t) under the(signed) measure WN given in (39)

(see Figure 2) For y isin ΩN define the (signed) weight

WN (x y) =( Nprodn=2

det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x

Ni+1 minus yNminusj)

)0leijltN (39)

with φ(x1 x2) = 1x1gtx2 WN defines a signed measure on GTN We think of it as describing

the evolution of((xji )i=1j

)j=1N

(of course this makes no sense since the measure is not

positive it is meant only as intuition) We are thinking now of j as time but it has nothing todo with the ldquorealrdquo TASEP time which is just the parameter t in the formula The statementcan then be expressed as follows

Theorem 33 ([BFPS07]) For x y isin ΩN

P(N)t (x y) =

sumxisinGTNxi1=xi i=1N

WN (x y)

In other words TASEP at time t can be identified with the evolution of the first particle in

the Gelfand-Tsetlin pattern (x11 x

N1 ) (that is the leftmost diagonal in Figure 2) and P

(N)t

can be obtained as a marginal of WN The proof of this result is based on applying the secondequality of (38) and the multilinearity of the determinant repeatedly on Schutzrsquos formula andthen employing a clever symmetrization argument see [BFPS07] for the details

34 Biorthogonal ensembles The key point is that from the form of (39) [Sas05 BFPS07]could recognize that the correlation functions associated to the ldquorandomrdquo Gelfand-Tsetlinpattern x should be determinantal and that one could then apply a suitable version of theEynard-Mehta theorem [EM98] (see [BFPS07 Lem 34]) to obtain a Fredholm determinantformula for the finite dimensional distributions of TASEP This step is crucial and far fromtrivial but we refer to [BFPS07] for the details and just describe the result here

Definition 34 Consider a decreasing sequence of integers(X0(k)

)kge1

(the TASEP initial

condition) Given a fixed n isin Zgt0 we define two families of functions on the integers(Ψnk

)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k

2xminusX0(nminusk)wx+k+1minusX0(nminusk)et(wminus1) (310)

where Γ0 is any simple loop anticlockwise oriented and contained in D which includes w = 0but does not include w = 1 while the functions Φn

k(x) k = 0 nminus 1 are defined implicitlyby

(1) The biorthogonality relationsum

xisinZ Ψnk(x)Φn

` (x) = 1k=`

(2) 2minusxΦnk(x) is a polynomial of degree at most nminus 1 in x

Additionally we define the following stochastic matrix


Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m

Moreover Q and Qm are invertible

Qminus1(x y) = 2 middot 1x=yminus1 minus 1x=y Qminusm(x y) = (minus1)yminusx+m2yminusx(

m

y minus x

) (311)

A crucial identity is that for all mn isin Zgt0

QnminusmΨnnminusk = Ψm

mminusk (312)

Theorem 36 (Biorthogonal ensemble formula for TASEP [BFPS07]) Suppose thatTASEP starts with particles labeled 1 2 (so that in particular there is a rightmost particle)and let 1 le n1 lt n2 lt middot middot middot lt nm Then for t gt 0 we have

P(Xt(nj) gt aj j = 1 m) = det(I minus χaKtχa)`2(n1nmtimesZ) (313)

where

Kt(ni xinj xj) = minusQnjminusni(xi xj)1niltnj +

njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)

with the Ψnk rsquos and the Φn

k rsquos given as in Definition 34

Remark 37

1 We are assuming in the theorem that X0(j) lt infin for all j ge 1 particles at minusinfin areallowed

2 The [BFPS07] result is stated only for initial conditions with finitely many particlesbut the extension to right-finite (infinite) initial conditions is straightforward becausegiven fixed indices n1 lt n2 lt middot middot middot lt nm the distribution of

(Xt(n1) Xt(nm)

)does

not depend on the initial positions of the particles with indices beyond nm

3 In (314) we have conjugated the kernel Kt from [BFPS07] by 2x for convenience Theadditional X0(nminus k) in the power of 2 in the Ψn

k rsquos is there also for convenience and is

allowed because it just means that the Φnk rsquos have to be multiplied by 2X0(nminusk)

Our goal is to find the Φnk rsquos explicitly for any initial data These functions had been

computed previously only for very special initial conditions (such as step X0(i) = minusi forall i ge 1and half-periodic X0(i) = minus2i forall i ge 1 which lead respectively to the Airy2 and Airy2rarr1

processes)

35 Reversibility and the TASEP path integral kernel The solution of the biorthog-onalization problem is based on two ingredients The first one is the following skew timereversibility property satisfied by TASEP

Pf (ht(x) le g(x) x isin Z) = Pminusg(ht(x) le minusf(x) x isin Z) (315)

the subscript indicating the initial data In other words the height function evolving backwardsin time is indistinguishable from minus the height function as can be proved directly fromthe definition of its evolution A good way to prove (315) is through the connection betweenTASEP and last passage percolation (LPP) with exponential weights In this model we considera family w(i j)ijisinZ of independent exponential random variables with parameter 1 and definethe point-to-point last passage time L((m1 n1) (m2 n2)) = maxπisinΠ(m1n1)rarr(m2n2)

sumiw(πi)

where the max is taken over the set of all paths connecting (m1 n1) to (m2 n2) which take steps


x1x2xn a

Figure 3 The time reversibility property of TASEP starting TASEP withparticles at (x1 xn) and computing the probability that Xt(n) gt a is thesame (by the exclusion property of the dynamics) as asking that Xt(1) gta+ 1 Xt(2) gt a+ 2 Xt(n) gt a+ n and this turns out to be the same asstarting TASEP at (a+ 1 a+ n) running it backwards and computing theprobability that Xt(i) le xn+1minusi for each i

in the southeast or southwest directions (that is steps of the form (mn)rarr (mminus 1 nplusmn 1))5 ifthis set is empty we take the max to be minusinfin Next given an infinite space-like curve γ (by whichwe mean that γ is the graph of a TASEP initial condition) we define the curve-to-point lastpassage time L(γ (mn)) = max(mprimenprime)isinγ L((mprime nprime) (mn)) and similarly the point-to-curvelast passage time L((mn) γ) = max(mprimenprime)isinγ L((mn) (mprime nprime)) Let ρ(mn) = (mminusn) bethe reflection about the horizontal axis The reversibility property at the level of LPP is then

just the straightforward fact that(L(γ (mn))

)(mn)isinZ2

dist=(L(ρ(mn) ρ(γ)

)(mn)isinZ2 The

connection between LPP and TASEP on the other hand is the following if h0 is a given TASEPinitial condition and γ0 is its graph then

(x y) y ge ht(x)

=

(mn) L(γ0 (mn)) le t

indistribution (this means that a down flip in the TASEP height function at time t correspondsto a adding the corresponding site to the set of sites which are reachable by time t by anLPP path starting at γ0) Through this correspondence (315) is just a direct translation toTASEP of the LPP reversibility property

The time reversibility property (315) for the height function translates (through (31)) intothe following statement for the particle system for any x1 gt x2 gt middot middot middot gt xn and any a isin Z

P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)

(Xt(i) le xn+1minusi i = 1 n

)

where the subscript denotes the initial positions of the particles and the superscript ldquoreversedrdquoindicates that the process is run backwards (ie with jumps to the left) see Figure 3 But theright hand side can be written in terms of the process running forwards giving (using alsoshift invariance)

P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)

(Xt(i) ge aminus xn+1minusi i = 1 n

)

And this corresponds exactly to computing the multipoint distribution of TASEP with stepinitial data X0(i) = minusi which was already known (it is actually the starting point in thecomputation of the limit (16)) So we have at least a formula for the one-point distribution ofTASEP with general initial condition and we can use to try guess a formula for the Φn

k rsquos

The key to making a guess is provided by our second ingredient Formula (313) is what wecalled an extended kernel formula after Thm 121 There is aso a path integral kernel formulawhich is given as follows

P(Xt(nj) gt aj j = 1 M)

= det(I minusK(nm)

t (I minusQn1minusnmχa1Qn2minusn1χa2 middot middot middotQnmminusnmminus1χam)

)`2(Z)

(316)

where

K(n)t = Kt(n middotn middot)

5In the usual description of LPP this picture is rotated by 135 degrees with paths taking upright stepsthe present description is more convenient in our setting


This formula follows from the framework of [BCR15] (or rather a minor variation of it see[MQR17 Appdx D2]) applied to (313) a formula of this type was first obtained by [PS02]for the Airy2 process

In order to see how this helps observe that the kernel Q is the transition probability of arandom walk with geometric steps (which is why we conjugated the [BFPS07] kernel by 2x)and thus χa1Q

n2minusn1χa2 middot middot middotQnmminusnmminus1χam(x y) is the probability that this walk goes from xto y in nm minus n1 steps staying above a1 at time n1 above a2 at time n2 etc This tells us thatthe Φn

k rsquos should be related to the problem of hitting the curved prescribed by the airsquos by therandom walk Based on this intuition we obtained in [MQR17] a formula for Φn

k in terms ofthe solution of certain boundary value problem for a backwards heat equation involving QThis is the content of the next result

36 Explicit biorthogonalization

Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)

2xminusywxminusy+1= eminust

txminusy

2xminusy(xminus y)1xgey (317)

Rt is invertible with

Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)

2xminusywxminusy+1= et

(minust)xminusy


Note that by Cauchyrsquos residue theorem we have Ψn0 = RtδX0(n) where δy(x) = 1x=y so from

(312) we have

Ψnk = RtQ

minuskδX0(nminusk) (319)

Theorem 39 Fix 0 le k lt n and consider particles at X0(1) gt X0(2) gt middot middot middot gt X0(n) Lethnk(` z) be the unique solution to the initialndashboundary value problem for the backwards heatequation

(Qlowast)minus1hnk(` z) = hnk(`+ 1 z) ` lt k z isin Z (320a)

hnk(k z) = 2zminusX0(nminusk) z isin Z (320b)

hnk(`X0(nminus `)) = 0 ` lt k (320c)

ThenΦnk(z) = (Rlowastt )

minus1hnk(0 middot)(z) =sumyisinZ

hnk(0 y)Rminus1t (y z)

Here Qlowast(x y) = Q(y x) is the kernel of the adjoint of Q (and likewise for Rlowastt )

Remark 310 It is not true that Qlowasthnk(`+ 1 z) = hnk(` z) In fact in general Qlowasthnk(k z) isdivergent

Before the proof we need

Lemma 311 With the definitions in the above theorem 2minusxhnk(0 x) is a polynomial of degreeat most k

Proof We proceed by induction By (320b) 2minusxhnk(k x) is a polynomial of degree 0 Assume

now that hnk(` x) = 2minusxhnk(` x) is a polynomial of degree at most k minus ` for some 0 lt ` le kBy (320a) and (311) we have

hnk(` y) = 2minusy(Qlowast)minus1hnk(`minus 1 y) = hnk(`minus 1 y minus 1)minus hnk(`minus 1 y) (321)

Taking x ge X0(nminus `+ 1) and summing (321) gives hnk(`minus 1 x) = minussumx

y=X0(nminus`+1)+1 hnk(` y)

thanks to (320c) which by the inductive hypothesis is a polynomial of degree at most kminus `+ 1

in x Similarly taking x lt X0(n minus ` + 1) we get hnk(` minus 1 x) =sumX0(nminus`+1)

y=x+1 hnk(` y) whichagain is a polynomial of degree at most k minus `+ 1 The two polynomials are the same as canbe checked for instance using Faulhaberrsquos formula and thus the claim follows


Proof of Thm 39 Note first that the dimension of ker(Qlowast)minus1 is 1 and it consists of the

function 2z This allows us to march forwards from the initial condition hnk(k z) = 2zminusX0(nminusk)

uniquely solving the boundary value problem hnk(`X0(nminus k)) = 0 at each step and thus getthe existence and uniqueness (exercise fill in the details)

We need to show that the proposed Φnk rsquos satisfy conditions (1) and (2) of Definition 34 We

check the biorthogonality first Using (319) we getsumzisinZ

Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ

Rt(z z1)Qminus`(z1 X0(nminus `))hnk(0 z2)Rminus1t (z2 z)

=sumzisinZ

Qminus`(zX0(nminus `))hnk(0 z) = (Qlowast)minus`hnk(0 X0(nminus `))

(exercise justify the application of Fubini here) We want to show that the last expression equals1k=` Consider first ` le k Then we may use the boundary condition hnk(`X0(nminus `)) = 1k=`which is both (320b) and (320c) to get

(Qlowast)minus`hnk(0 X0(nminus `)) = hnk(`X0(nminus `)) = 1k=`

Next for ` gt k we use (320a) and 2z isin ker (Qlowast)minus1

(Qlowast)minus`hnk(0 X0(nminus `)) = (Qlowast)minus(`minuskminus1)(Qlowast)minus1hnk(kX0(nminus `)) = 0

Next we show that 2minusxΦnk(x) is a polynomial of degree at most k in x By (317) we have

2minusxΦnk(x) =

sumyge0

eminustty

y2minus(x+y)hnk(0 x+ y)

This sum is absolutely convergent so we may compute the (k + 1)-th derivate in x of theright hand side by taking the derivative inside the sum and use the fact that 2minuszhnk(0 z) is a

polynomial of degree at most k to deduce that dk+1

dxk+1 [2minusxΦnk(x)] = 0 as desired

37 Representation of the kernel below the curve as a transition probability withhitting

Definition 312 (Geometric random walks) We denote by Blowastm a random walk withtransition probabilities given by Qlowast (that is Blowastm has Geom[1

2 ] jumps strictly to the right) Wealso introduce for 0 le ` le k le nminus 1 the stopping times

τ `n = minm isin ` nminus 1 Blowastm gt X0(nminusm)

with the convention that min empty =infin

Similarly we denote by Bm a random walk with transition probabilities given by Q (that isBm has Geom[1

2 ] jumps strictly to the left) and introduce the stopping time

τ = minm ge 0 Bm gt X0(m+ 1) (322)

Lemma 313 For z le X0(nminus `) we have

hnk(` z) = PBlowast`minus1=z

(τ `n = k

)

Proof It is easy to see that hnk(` z) satisfies (320b) and (320c) On the other hand forz le X0(n minus ` minus 1) it also satisfies (320a) and it is given by 2z times a polynomial in z ofdegree at most nminus 1 Exercise conclude the proof by using Lemma 311

Definition 314 We write

G0n(z1 z2) =

nminus1sumk=0

Qnminusk(z1 X0(nminus k))hnk(0 z2)


with hnk the solution of (320) so that

K(n)t = RtQ

minusnG0nRminus1t

Lemma 315 For z2 le X0(n)

G0n(z1 z2) = PBlowastminus1=z2

(τ0n lt n Blowastnminus1 = z1

)(323)

(which is the probability for the walk starting at z2 at time minus1 to end up at z1 after n stepshaving passed above the curve

(X0(nminusm)

)m=0nminus1

in between)

Proof From the memoryless property of the geometric distribution we have for all z le X0(nminusk)that

PBlowastminus1=z

(τ0n = k Blowastk = y

)= 2X0(nminusk)minusyPBlowastminus1=z

(τ0n = k

) (324)

and as a consequence we get for z2 le X0(n)

G0n(z1 z2) =nminus1sumk=0

PBlowastminus1=z2

(τ0n = k

)(Qlowast)nminusk(X0(nminus k) z1)

=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2

(τ0n = k Blowastk = z

)(Qlowast)nminuskminus1(z z1)

= PBlowastminus1=z2


)

38 Polynomial extension above the curve To finish our derivation of our TASEPsolution we need to extend the result of Lemma 315 to all values of z2 We will do this byextending our formula polynomally

Definition 316 (Polynomial extension of Qn) Let

Q(n)(y1 y2) =1

2πi

∮Γ0

dw(1 + w)y1minusy2minus1

2y1minusTy2wn=

(y1 minus y2 minus 1)nminus1

2y1minusy2(nminus 1) (325)

where (x)k = x(xminus 1) middot middot middot (xminus k + 1) for k gt 0 and (x)0 = 1 is the Pochhammer symbol For

each fixed y1 2minusy2Qn(y1 y2) extends in y2 to a polynomial 2minusy2Q(n)(y1 y2) of degree nminus 1We have

Q(n)(y1 y2) = Qn(y1 y2) for y1 minus y2 ge 1 (326)

and

Qminus1Q(n) = Q(n)Qminus1 = Q(nminus1) for n gt 1 but Qminus1Q(1) = Q(1)Qminus1 = 0

Remark 317 Q(n)Q(m) is divergent so the Q(n) are no longer a group like the Qn

Lemma 318 For all z1 z2 isin Z we have for τ as in Definition 312

G0n(z1 z2) = EB0=z1

[Q(nminusτ)(Bτ z2)1τltn

] (327)

Proof From Lemma 311 and its definition 2minusz2G0n(z1 z2) is a polynomial in z2 for everyfixed z1 and thus it is enough to check that the right hand side of (327) equals 2z2 times apolynomial in z2 and it coincides with (323) for all z2 le X0(n)

From (323) we have for z2 le X0(n)


(τ0n le nminus 1 Blowastnminus1 = z1

)= PB0=z1

(τ le nminus 1 Bn = z2

)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z

)Qnminusk(z z2) = EB0=z1

[Qnminusτ

(Bτ z2

)1τltn

] (328)


Note that we reversed the direction of the walk in this formula Crucially on the right handside z2 appears as an argument inside the expectation and not as the inital condition So if wereplace the Qnminusτ in the expectation by Q(nminusτ) we obtain a formula in the form of 2z2 times apolynomial in z2 To finish the proof we need to check that the resulting formula (327) coincideswith the right hand side of (328) below the curve which can be checked easily using the last

equality in (328) all we need to check is that χX0(k+1)Q(nminusk)χX0(n) = χX0(k+1)Q

nminuskχX0(n)which follows from X0(k+ 1)minusX0(n) gt nminuskminus1 and (326) (see [MQR17] for the details)

39 Main formula for TASEP We need to introduce the discrete version of the operatorsin Definitions 110 and 113 In the first case we need two discrete versions

Definition 319 We let

Stminusn(z1 z2) = (et2RtQminusn)lowast(z1 z2) =

1

2πi

∮Γ0

dw(1minus w)n

2z2minusz1wn+1+z2minusz1 et(wminus12) (329)

Stn(z1 z2) = eminust2Q(n)Rminus1t (z1 z2) =

1

2πi

∮Γ0

dw(1minus w)z2minusz1+nminus1

2z1minusz2wnet(12minusw) (330)

(where we used (310) and (319) for the first one and (318) and (325) together with a residuecomputation for the second one)

Definition 320 (Discrete hit operator)

Sepi(X0)tn (z1 z2) = EB0=z1

[Stnminusτ (Bτ z2)1τltn

] (331)

Note that τ (defined in (322)) is the hitting time of the strict epigraph of the curve(X0(k +

1))k=0nminus1

by the random walk Bk the strict epigraph of(g(m)

)mge0

is the set6

epi(g) =

(m y) m ge 0 y gt g(m)

The following formula now follows directly from the above arguments and the last twodefinitions

Theorem 321 (TASEP formula for right-finite initial data) Assume that initiallywe have X0(j) =infin for all j le 0 X0(1) ltinfin Then for 1 le n1 lt n2 lt middot middot middot lt nm and t ge 0

P(Xt(nj) gt aj j = 1 m) = det(I minus χaKtχa)`2(n1nmtimesZ)

whereKt(ni middotnj middot) = minusQnjminusni1niltnj + (Stminusni)

lowastSepi(X0)tnj

(332)

The path integral version (316) also holds

Remark 322 Note that by definition Sepi(X0)tnj

= Stnj (y z) for y gt X0(1) so (332) can

also be written as

Kt(ni middotnj middot) = minusQnjminusni1niltnj + (Stminusni)lowastχX0(1)Stnj + (Stminusni)

lowastχX0(1)Sepi(X0)tnj

(333)

Example 323 (Step initial data) Consider TASEP with step initial data X0(i) = minusifor i ge 1 If we start the random walk in (331) from B0 = z1 below the curve ie z1 le minus1

then the random walk clearly never hits the epigraph Hence χX0(1)Sepi(X0)tn equiv 0 and the last

term in (333) vanishes For the second term in (333) we have from (329) and (330)

(Stminusni)lowastχX0(1)Stnj (z1 z2) =

1

(2πi)2

∮Γ0

dw

∮Γ0

dv(1minus w)ni(1minus v)nj+z2

2z1minusz2wni+z1+1vnj

et(w+vminus1)

1minus v minus w

which is exactly the formula previously derived in the literature (modulo the conjugation2z2minusz1 see eg [Fer15 Eq 82])

6There is a slight abuse of notation here since for a function g over R we defined epi(g) in (18) as the(non-strict) epigraph of g (ie with a non-strict inequality in the definition) the meaning of epi however canalways be disntiguished from the context


Example 324 (Periodic initial data) Consider now TASEP with periodic initial dataX0(i) = 2i i isin Z One can obtain a formula for the kernel in this case by approximationconsidering first the finite periodic initial data X0(i) = 2(N minus i) for i = 1 2N Thecomputation is more involved than in the previous example but it can be carried out explicitly(see [MQR17]) and leads to

K(n)t (z1 z2) = minus 1

2πi

∮1+Γ0

dvvz2+2m

2z1minusz2(1minus v)z1+2m+1et(1minus2v)

which coincides with the kernel derived (modulo the conjugation 2z2minusz1 and after a simplechange of variables) in [BFPS07 Thm 22]

4 From TASEP to the KPZ fixed point

We will only sketch this part In particular for simplicity we will mostly only address thecase of one-sided initial data for the fixed point which means

h0(x) = minusinfin forall x gt 0

This corresponds to right-finite TASEP initial data as in Thm 321 From (13) (15) and(31) and because the Xε

0(k) are in reverse order we have that hε(0x) converges to h0(x) inUC if and only if

ε12(Xε

0(εminus1x) + 2εminus1xminus 1)minusminusminusrarrεrarr0

minush0(minusx)

in LC and it can be proved that any h0 isin UC can be approximated by TASEP initial data inthis way

41 Scaling and asymptotics Our goal is to take such a sequence of initial data Xε0 and

compute

Ph0(h(txi) le ai i = 1 M)

= limεrarr0

PX0

(X2εminus32t(

12εminus32tminus εminus1xi minus 1

2εminus12ai + 1) gt 2εminus1xi minus 2 i = 1 M

)(the equality comes again from (13) (15) and (31)) We therefore want to consider Thm321 with

t = 2εminus32t ni = 12εminus32tminus εminus1xi minus 1

2εminus12ai + 1 (41)

(and with ai = 2εminus1xi minus 2)

Lemma 41 Under the scaling (41) and assuming that ε12(Xε

0(εminus1x) + 2εminus1xminus 1)minusrarr

minush0(minusx) as εrarr 0 in LC if we set zi = 2εminus1xi + εminus12(ui + ai)minus 2 and yprime = εminus12v then wehave as ε 0

Sεminustxi(v ui) = εminus12Stminusni(y

prime zi) minusrarr Sminustxi(v ui) (42)

Sεminustminusxj(v uj) = εminus12Stnj (y

prime zj) minusrarr Sminustminusxj (v uj) (43)

Sεepi(minushminus0 )minustminusxj

(v uj) = εminus12Sepi(X0)tnj

(yprime zj) minusrarr Sepi(minushminus0 )minustminusxj

(v uj) (44)

pointwise where hminus0 (x) = h0(minusx) for x ge 0

The first two limits follow from standard arguments (although in [MQR17] we need detailed

estimates which are very involved) The main point is that the limit of Sεepi(minushminus0 )minustminusxj

can be

computed naturally essentially one uses the asymptotics of Sεminustminusxjtogether with the fact

that under our scaling the random walk B goes to the Brownian motion B


Sketch of the proof of Lemma 41 From (329)

Stminusni(zi y) =1

2πi

∮Γ0

eεminus32F (3)+εminus1F (2)+εminus12F (1)+F (0)

dw

whereF (3) = t

[(2w minus 1) + 1

2 log(1minusww )] F (2) = minusxi log 4w(1minus w)

F (1) = (ui minus v minus 12ai) log 2w minus 1

2ai log 2(1minus w) F (0) = log 8w2(45)

The leading term has a double critical point at w = 12 so we introduce the change of variables

w 7rarr 12(1minus ε12w) which leads to

εminus32F (3) asymp t3 w

3 εminus1F (2) asymp xiw2 εminus12F (1) asymp minus(ui minus v)w

We also have F (0) asymp log(2) which cancels the prefactor 12 coming from the change of variablesIn view of (110) this gives (42) (43) follows in the same way now using (330)

Now define the scaled walk Bε(x) = ε12(Bεminus1x + 2εminus1xminus 1

)for x isin εZge0 interpolated

linearly in between and let τ ε be the hitting time by Bε of epi(minushε(0 middot)minus) By Donskerrsquosinvariance principle [Bil99] Bε converges locally uniformly in distribution to a Brownianmotion B(x) with diffusion coefficient 2 and therefore the hitting time τ ε converges to τ aswell (330) and (43) now show that modulo explicit estimates (44) should hold

Theorem 42 (One-sided fixed point formulas) Let h0 isin UC with h0(x) = minusinfin forx gt 0 Then given x1 lt x2 lt middot middot middot lt xM and a1 aM isin R we have for h(tx) given as in(15)

Ph0(h(tx1) le a1 h(txM ) le aM )

= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM

e(x1minusxM )part2χa1e(x2minusx1)part2χa2 middot middot middot e(xMminusxMminus1)part2χaM

)L2(R)

(47)

where Khypo(h0)tx was defined in Definition 114

Proof We have ni lt nj for small ε if and only if xj lt xi and in this case we have under ourscaling

εminus12Qnjminusni(zi zj) minusrarr e(ximinusxj)part2(ui uj)

as ε 0 From this and the above lemma we obtain the following limiting formula

Ph0(h(tx1) le a1 h(txM ) le aM ) = det(Iminus χaKlimχa)L2(x1xMtimesR)

with

Klim(xi ui xj uj) = minuse(ximinusxj)part2(ui uj)1xigtxj + (Sminustxi)lowastS

epi(minushminus0 )minustminusxj

(ui uj)

We may turn the above projections χminusa into χa by changing variables ui 7minusrarr minusui in the kernelIf we additionally replace the Fredholm determinant of the kernel by that of its adjoint to get

det(Iminus χaKlimχa

)with Klim(ui uj) = Klim(xj minusuj ximinusui)

Now

Sminustx(minusu v) = (Stx)lowast(minusv u) and Sepi(minushminus0 )minustx (vminusu) = (S

hypo(hminus0 )tx )lowast(uminusv)

so

Klim = minuse(xjminusxi)part21xiltxj + (S

hypo(hminus0 )tminusxi

)lowastStxj = Khypo(h0)text

This gives the extended kernel formula 46 The path integral version 47 follows from theframework of [BCR15]


42 From one-sided to two-sided formulas In the last result we assumed h0(x) = minusinfinfor x gt 0 Obtaining from this a formula for general h0 isin UC involves two separate arguments

1 For h0 isin UC let hL0 (x) = h0(x)1xleL minus infin middot 1xgtL Then we need to compute the limitof (46)(47) as L rarr infin with initial data hL0 To this end we use shift invariance (whichfollows directly from that of TASEP) to translate this into a problem involving a (shifted)initial condition which is minusinfin on the positive axis The kernels appearing in the Fredholmdeterminants now involve some shifts by L but in view of Rem 115 it is possible to rewritethem in such a way that they look exactly like those in (46)(47) but with h0 replaced by hL0

and so what one needs is to show that Shypo(hL0 )tx minusrarr S

hypo(h0)tx See [MQR17 Sec 34] for the

details

2 We need to show that the limit we get in the previous point is in fact the same as thelimit we would get for TASEP with an initial height profile hε0 going to h0 in UC as ε rarr 0

This amounts to considering a truncated version hεL0 of hε0 which goes to hL0 taking ε rarr 0and Lrarrinfin and then showing that the limits can be interchanged This can be justified by

showing that the error we incur at the level of TASEP by replacing hε0 by hεL0 can be boundedby something that goes to 0 with L rarr infin uniformly in ε This is the content of the finitespeed of propagation result in [MQR17 Lem 34] which can be stated roughly as follows

Lemma 43 Consider TASEP initial data Xε0

(the proof of this result turns out to be very delicate and depends on precise estimates onthe rescaled kernels appearing in Lemma 41)

These arguments lead in [MQR17 Sec 34] to Thm 117 above

43 Continuum limit Finally to get the continuum statistics formula in Thm 121 we needto compute the continuum limit in the airsquos of the path integral formula (120) on the full lineBy this we mean that we take g isin UC let x1 xM be a fine mesh of [minusRR] take M rarrinfinwith ai = g(xi) to obtain a continuum statistics formula and finally take R rarr infin To this

end one notes that with these choices e(x1minusxM )part2 becomes eminus2Rpart2 (which makes sense when

applied after Khypo(h0)t ) while

χa1e(x2minusx1)part2χa2 middot middot middot e(xMminusxMminus1)part2χaM (u1 u2)

minusminusminusrarrεrarr0

PB(`1)=u1

(B(s) le g(s) forall s isin [`1 `2] B(`2) isin du2

)du2

that is the transition probability for B not to hit epi(g) This establishes the connection with

the hit operator Kepi(g)minust For the details and in particular the computation of the R rarr infin

limit (which is essentially contained already in [QR16]) see [MQR17 Sec 42]

Acknowledgements The author is grateful to the organizers of the CIRM Research SchoolldquoRandom Structures in Statistical Mechanics and Mathematical Physicsrdquo in Marseille in March2017 where these lectures where first presented He is also grateful for the invitation to givethis minicourse at CIMAT in Guanajuato in 2018 at the ldquoRandom Physical Systemsrdquo meetingin Patagonia Chile in 2018 and at the ICTS program ldquoUniversality in random structuresinterfaces matrices sandpilesrdquo in Bangalore in 2019 which lead to improvements in theselecture notes The author was supported by Conicyt Basal-CMM by Programa IniciativaCientıfica Milenio grant number NC120062 through Nucleus Millenium Stochastic Models ofComplex and Disordered Systems and by Fondecyt Grant 1160174

References

[BFP10] J Baik P L Ferrari and S Peche Limit process of stationary TASEP near thecharacteristic line Comm Pure Appl Math 638 (2010) pp 1017ndash1070


[BR01] J Baik and E M Rains Symmetrized random permutations In Random matrixmodels and their applications Vol 40 Math Sci Res Inst Publ CambridgeCambridge Univ Press 2001 pp 1ndash19

[Bet31] H A Bethe On the theory of metals i eigenvalues and eigenfunctions of a linearchain of atoms Zeits Phys 74 (1931) pp 205ndash226

[Bil99] P Billingsley Convergence of probability measures Second Wiley Series in Proba-bility and Statistics Probability and Statistics A Wiley-Interscience PublicationJohn Wiley amp Sons Inc New York 1999 pp x+277

[BCR15] A Borodin I Corwin and D Remenik Multiplicative functionals on ensembles ofnon-intersecting paths Ann Inst H Poincare Probab Statist 511 (2015) pp 28ndash58

[BF08] A Borodin and P L Ferrari Large time asymptotics of growth models on space-likepaths I PushASEP Electron J Probab 13 (2008) no 50 1380ndash1418

[BFP07] A Borodin P L Ferrari and M Prahofer Fluctuations in the discrete TASEPwith periodic initial configurations and the Airy1 process Int Math Res PapIMRP (2007) Art ID rpm002 47

[BFPS07] A Borodin P L Ferrari M Prahofer and T Sasamoto Fluctuation propertiesof the TASEP with periodic initial configuration J Stat Phys 1295-6 (2007)pp 1055ndash1080

[BFS08a] A Borodin P L Ferrari and T Sasamoto Large time asymptotics of growthmodels on space-like paths II PNG and parallel TASEP Comm Math Phys 2832(2008) pp 417ndash449

[BFS08b] A Borodin P L Ferrari and T Sasamoto Transition between Airy1 and Airy2

processes and TASEP fluctuations Comm Pure Appl Math 6111 (2008) pp 1603ndash1629

[BP14] A Borodin and L Petrov Integrable probability from representation theory tomacdonald processes Probab Surveys 11 (2014) pp 1ndash58

[Cor12] I Corwin The Kardar-Parisi-Zhang equation and universality class Random Ma-trices Theory Appl 1 (2012)

[CQR13] I Corwin J Quastel and D Remenik Continuum statistics of the Airy2 processComm Math Phys 3172 (2013) pp 347ndash362

[CQR15] I Corwin J Quastel and D Remenik Renormalization fixed point of the KPZuniversality class J Stat Phys 1604 (2015) pp 815ndash834

[EM98] B Eynard and M L Mehta Matrices coupled in a chain I Eigenvalue correlationsJ Phys A 3119 (1998) pp 4449ndash4456

[Fer15] P L Ferrari Dimers and orthogonal polynomials connections with random matricesIn Dimer models and random tilings Vol 45 Panor Syntheses Soc Math FranceParis 2015 pp 47ndash79

[HH85] D A Huse and C L Henley Pinning and roughening of domain walls in Isingsystems due to random impurities Phys Rev Lett 5425 (1985) pp 2708ndash2711

[Joh03] K Johansson Discrete polynuclear growth and determinantal processes CommMath Phys 2421-2 (2003) pp 277ndash329

[KPZ86] M Kardar G Parisi and Y-C Zhang Dynamical scaling of growing interfacesPhys Rev Lett 569 (1986) pp 889ndash892

[Lig85] T M Liggett Interacting particle systems Vol 276 Grundlehren der Mathematis-chen Wissenschaften [Fundamental Principles of Mathematical Sciences] New YorkSpringer-Verlag 1985 pp xv+488

[MQR+] K Matetski J Quastel and D Remenik In preparation[MQR17] K Matetski J Quastel and D Remenik The KPZ fixed point 2017 arXiv

170100018v1[mathPR][NR17a] G B Nguyen and D Remenik Extreme statistics of non-intersecting Brownian

paths Electron J Probab 22 (2017) Paper No 102 40


[NR17b] G B Nguyen and D Remenik Non-intersecting Brownian bridges and the Laguerreorthogonal ensemble Ann Inst Henri Poincare Probab Stat 534 (2017) pp 2005ndash2029

[Pim17] L Pimentel Ergodicity of the KPZ fixed point 2017 arXiv 170806006[mathPR][PS02] M Prahofer and H Spohn Scale invariance of the PNG droplet and the Airy

process J Stat Phys 1085-6 (2002) pp 1071ndash1106[Qua11] J Quastel The Kardar-Parisi-Zhang equation In Current developments in mathe-

matics 2011 Int Press Somerville MA 2011[QR14] J Quastel and D Remenik Airy processes and variational problems In Topics in

Percolative and Disordered Systems Ed by A Ramırez G Ben Arous P A FerrariC Newman V Sidoravicius and M E Vares Vol 69 Springer Proceedings inMathematics amp Statistics 2014 pp 121ndash171

[QR16] J Quastel and D Remenik How flat is flat in random interface growth To appearin Trans Amer Math Soc 2016 arXiv 160609228[mathPR]

[QR13] J Quastel and D Remenik Local behavior and hitting probabilities of the Airy1

process Probability Theory and Related Fields 1573-4 (2013) pp 605ndash634[QS15] J Quastel and H Spohn The one-dimensional KPZ equation and its universality

class J Stat Phys 1604 (2015) pp 965ndash984[Sas05] T Sasamoto Spatial correlations of the 1D KPZ surface on a flat substrate Journal

of Physics A Mathematical and General 3833 (2005) p L549[Sch97] G M Schutz Exact solution of the master equation for the asymmetric exclusion

process J Statist Phys 881-2 (1997) pp 427ndash445[Sim05] B Simon Trace ideals and their applications Second Vol 120 Mathematical

Surveys and Monographs American Mathematical Society 2005 pp viii+150[TW94] C A Tracy and H Widom Level-spacing distributions and the Airy kernel Comm

Math Phys 1591 (1994) pp 151ndash174[TW96] C A Tracy and H Widom On orthogonal and symplectic matrix ensembles Comm

Math Phys 1773 (1996) pp 727ndash754

Departamento de Ingenierıa Matematica and Centro de Modelamiento Matematico Universidadde Chile Av Beauchef 851 Torre Norte Piso 5 Santiago Chile

Email address dremenikdimuchilecl


11 The KPZ equation and universality class


13 State space and topology

14 Operators

15 Existence and formulas for the KPZ fixed point


21 Markov property

22 Regularity and local Brownian behavior

23 Symmetries

24 Airy processes



31 The model

32 The master equation and Schuumltzs formulaThe reader may choose to skip directly to Section 34 this and the next sections are not strictly needed in the derivation of the fixed point

33 The non-intersecting line ensemble representation

34 Biorthogonal ensembles

35 Reversibility and the TASEP path integral kernel


37 Representation of the kernel below the curve as a transition probability with hitting

38 Polynomial extension above the curve

39 Main formula for TASEP


41 Scaling and asymptotics

42 From one-sided to two-sided formulas

43 Continuum limit

References


understand some aspects of this limiting object (mostly restricted to a very special family ofparticular initial conditions)


121 TASEP Our starting point in the construction of the KPZ fixed point will be anotherbasic model in the KPZ universality class the totally asymmetric exclusion process (TASEP)We will introduce this model in more detail in Section 3 so for now describe it slightlyinformally

Definition 11 (TASEP height function) The height function associated to TASEP is acontinuous time Markov process

(ht(x)

)xisinZ taking values in the space of simple random walk

paths by which we mean continuous curves obtained by linearly interpolating the graph of afunction from Z to Z which moves up or down by one in each step We think of this curve asan interface growing (downwards) in time according to the following dynamics each localmaximum (and) turns into a local minimum (or) at rate one that is if ht(z) = ht(z plusmn 1) + 1 thenht(z) 7rarr ht(z)minus 2 at rate 1 (see Figure 1)

122 The KPZ fixed point For each ε gt 0 the 123 rescaled TASEP height function is1

hε(tx) = ε12[h2εminus32t(2ε

minus1x) + εminus32t] (13)

The initial data corresponds to just rescaling h0 diffusively hε(0x) = ε12h0(2εminus1x) (the extra

2 is just a choice of normalization) and we allow h0 = h(ε)0 itself to depend on ε and assume

that the limith0 = lim

εrarr0hε(0 middot) (14)

exists (in a sense to be specified later)

Definition 12 (The KPZ fixed point) We define the KPZ fixed point as the limit

h(tx h0) = limεrarr0

hε(tx) (15)

We will often omit h0 from the notation when it is clear from the context

Of course one of the main points which we need to settle is that the limit (15) exists in asuitable sense this will in fact be part of the content of our main result

As we already mentioned the KPZ fixed point h(tx h0) should be the 123 scaling limitof all models in the KPZ universality class This is known as the strong KPZ universalityconjecture and could arguably be posed as the definition of the KPZ universality class Inthese lectures (and in [MQR17]) we only deal with TASEP but our method works for severalvariants of TASEP (which also have a representation through biorthogonal ensembles such asPushASEP as well as discrete time TASEPs and PNGs) this is the content of [MQR+]

123 Earlier work TASEP has been one of the most heavily studied models in the KPZ classand much effort has been devoted to the study of the distributional limit (15) for a few veryspecial choices of initial data h0

Example 13 (Stepnarrow wedge) The most basic case is the step initial data h0(x) =minus|x| (the name step comes from the derivative of h0) in which case it is known that hε(1x)converges to the Airy2 process minus a parabola A2(x)minus x2 [PS02 Joh03] which has one-point marginals given by the Tracy-Widom GUE distribution [TW94] In this case we have

1There is a difference between the scalings chosen in (13) and in the first arXiv version of [MQR17] wherethe analogous definition was done with t replaced by t2 on the right hand side The present choice is betterbecause it makes the usual Airy processes arise at time t = 1 at the fixed point level (instead of t = 2) andbecause it matches a natural choice of parameters for (11) it is the scaling used in later revised versions ofthat paper This difference translates into some constants being off by a factor of 2 when comparing the twoversions but it does not affect the arguments in any other way


rate 1



h(1x d0) + x2 = A2(x) (16)



h(1x 0) = A1(x) (17)










Bδ((tx)) sube HεM







epi(g) = (xy) y ge g(x) (18)

















Stxf(z) =

int infinminusinfin

dy Stx(z y)f(y) =

int infinminusinfin



Stx(z) =1

2πi

int〈dw e


2x3

3t2minus zx






int〈 dwe

13w3minuszw








PNo hit h`1`2


and PHit h`1`2




Khypo(h)t = lim



Stminus`2 (112)



Observe first that

PNo hit h`1`2

(u1 u2) = p`1u1(`2 u2)p`1u1`2u2(no hit) (113)









PNo hit h`1`2

(u1 u2) =

int infinh(x)

dz


int xminus`1

0pz(τ



int `2minusx

0pz(τ






St`1PNo hit h`1`2

Stminus`2(u1 u2) =

int infinh(x)

dz

(Stx(z u1)minus

int xminus`1

0pz(τ


)times(


0pz(τ


) (114)





]=

int infin0








Shypo(h)tx = lim



](116)



(114))
























= det(Iminus χaK

hypo(h0)text χa

)L2(x1xmtimesR)

(119)

= det(IminusK

hypo(h0)txm

+ Khypo(h0)txm


)L2(R)

(120)

where



2K

hypo(h0)t exjpart

2(121)

and


2K

hypo(h0)t expart

2(122)




(minus1)n

n

intXn




hypo(h)t expart






hypo(h+x )tminusx




minus1 is trace










hypo(h0)t2 K

epi(g)minust2

)L2(R)

(123)







)tgt0















limArarrinfin

limsupεrarr0






case








P(h(t εx1) le u +


radicεan

∣∣ h(t 0) = u)


εxa(uB(xn))

)















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References


rate 1



h(1x d0) + x2 = A2(x) (16)



h(1x 0) = A1(x) (17)










Bδ((tx)) sube HεM







epi(g) = (xy) y ge g(x) (18)

















Stxf(z) =

int infinminusinfin

dy Stx(z y)f(y) =

int infinminusinfin



Stx(z) =1

2πi

int〈dw e


2x3

3t2minus zx






int〈 dwe

13w3minuszw








PNo hit h`1`2


and PHit h`1`2




Khypo(h)t = lim



Stminus`2 (112)



Observe first that

PNo hit h`1`2

(u1 u2) = p`1u1(`2 u2)p`1u1`2u2(no hit) (113)









PNo hit h`1`2

(u1 u2) =

int infinh(x)

dz


int xminus`1

0pz(τ



int `2minusx

0pz(τ






St`1PNo hit h`1`2

Stminus`2(u1 u2) =

int infinh(x)

dz

(Stx(z u1)minus

int xminus`1

0pz(τ


)times(


0pz(τ


) (114)





]=

int infin0








Shypo(h)tx = lim



](116)



(114))
























= det(Iminus χaK

hypo(h0)text χa

)L2(x1xmtimesR)

(119)

= det(IminusK

hypo(h0)txm

+ Khypo(h0)txm


)L2(R)

(120)

where



2K

hypo(h0)t exjpart

2(121)

and


2K

hypo(h0)t expart

2(122)




(minus1)n

n

intXn




hypo(h)t expart






hypo(h+x )tminusx




minus1 is trace










hypo(h0)t2 K

epi(g)minust2

)L2(R)

(123)







)tgt0















limArarrinfin

limsupεrarr0






case








P(h(t εx1) le u +


radicεan

∣∣ h(t 0) = u)


εxa(uB(xn))

)















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References






epi(g) = (xy) y ge g(x) (18)

















Stxf(z) =

int infinminusinfin

dy Stx(z y)f(y) =

int infinminusinfin



Stx(z) =1

2πi

int〈dw e


2x3

3t2minus zx






int〈 dwe

13w3minuszw








PNo hit h`1`2


and PHit h`1`2




Khypo(h)t = lim



Stminus`2 (112)



Observe first that

PNo hit h`1`2

(u1 u2) = p`1u1(`2 u2)p`1u1`2u2(no hit) (113)









PNo hit h`1`2

(u1 u2) =

int infinh(x)

dz


int xminus`1

0pz(τ



int `2minusx

0pz(τ






St`1PNo hit h`1`2

Stminus`2(u1 u2) =

int infinh(x)

dz

(Stx(z u1)minus

int xminus`1

0pz(τ


)times(


0pz(τ


) (114)





]=

int infin0








Shypo(h)tx = lim



](116)



(114))
























= det(Iminus χaK

hypo(h0)text χa

)L2(x1xmtimesR)

(119)

= det(IminusK

hypo(h0)txm

+ Khypo(h0)txm


)L2(R)

(120)

where



2K

hypo(h0)t exjpart

2(121)

and


2K

hypo(h0)t expart

2(122)




(minus1)n

n

intXn




hypo(h)t expart






hypo(h+x )tminusx




minus1 is trace










hypo(h0)t2 K

epi(g)minust2

)L2(R)

(123)







)tgt0















limArarrinfin

limsupεrarr0






case








P(h(t εx1) le u +


radicεan

∣∣ h(t 0) = u)


εxa(uB(xn))

)















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References








PNo hit h`1`2


and PHit h`1`2




Khypo(h)t = lim



Stminus`2 (112)



Observe first that

PNo hit h`1`2

(u1 u2) = p`1u1(`2 u2)p`1u1`2u2(no hit) (113)









PNo hit h`1`2

(u1 u2) =

int infinh(x)

dz


int xminus`1

0pz(τ



int `2minusx

0pz(τ






St`1PNo hit h`1`2

Stminus`2(u1 u2) =

int infinh(x)

dz

(Stx(z u1)minus

int xminus`1

0pz(τ


)times(


0pz(τ


) (114)





]=

int infin0








Shypo(h)tx = lim



](116)



(114))
























= det(Iminus χaK

hypo(h0)text χa

)L2(x1xmtimesR)

(119)

= det(IminusK

hypo(h0)txm

+ Khypo(h0)txm


)L2(R)

(120)

where



2K

hypo(h0)t exjpart

2(121)

and


2K

hypo(h0)t expart

2(122)




(minus1)n

n

intXn




hypo(h)t expart






hypo(h+x )tminusx




minus1 is trace










hypo(h0)t2 K

epi(g)minust2

)L2(R)

(123)







)tgt0















limArarrinfin

limsupεrarr0






case








P(h(t εx1) le u +


radicεan

∣∣ h(t 0) = u)


εxa(uB(xn))

)















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References



St`1PNo hit h`1`2

Stminus`2(u1 u2) =

int infinh(x)

dz

(Stx(z u1)minus

int xminus`1

0pz(τ


)times(


0pz(τ


) (114)





]=

int infin0








Shypo(h)tx = lim



](116)



(114))
























= det(Iminus χaK

hypo(h0)text χa

)L2(x1xmtimesR)

(119)

= det(IminusK

hypo(h0)txm

+ Khypo(h0)txm


)L2(R)

(120)

where



2K

hypo(h0)t exjpart

2(121)

and


2K

hypo(h0)t expart

2(122)




(minus1)n

n

intXn




hypo(h)t expart






hypo(h+x )tminusx




minus1 is trace










hypo(h0)t2 K

epi(g)minust2

)L2(R)

(123)







)tgt0















limArarrinfin

limsupεrarr0






case








P(h(t εx1) le u +


radicεan

∣∣ h(t 0) = u)


εxa(uB(xn))

)















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References









= det(Iminus χaK

hypo(h0)text χa

)L2(x1xmtimesR)

(119)

= det(IminusK

hypo(h0)txm

+ Khypo(h0)txm


)L2(R)

(120)

where



2K

hypo(h0)t exjpart

2(121)

and


2K

hypo(h0)t expart

2(122)




(minus1)n

n

intXn




hypo(h)t expart






hypo(h+x )tminusx




minus1 is trace










hypo(h0)t2 K

epi(g)minust2

)L2(R)

(123)







)tgt0















limArarrinfin

limsupεrarr0






case








P(h(t εx1) le u +


radicεan

∣∣ h(t 0) = u)


εxa(uB(xn))

)















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References








hypo(h0)t2 K

epi(g)minust2

)L2(R)

(123)







)tgt0















limArarrinfin

limsupεrarr0






case








P(h(t εx1) le u +


radicεan

∣∣ h(t 0) = u)


εxa(uB(xn))

)















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References













limArarrinfin

limsupεrarr0






case








P(h(t εx1) le u +


radicεan

∣∣ h(t 0) = u)


εxa(uB(xn))

)















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References















)= P


) f g isin UC




2ct f) + a + cx + 14c

2t
















2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References








2χ0e




13part3χ0e

13part3(u v) =

int infin0




h(1x 0) = A1(x)



2eLpart





13part3e

13part3(u v) =

int infinminusinfin






0 (x) equiv 0




int infin0








lowast(I + )χ0Stxi






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References






j )

+

int 0




j )









dist= A(yx))



hypo(minusg)1 K

epi(f)minus1







h(tx h0)dist= sup

y


(21)


supz

t

131 A

1(tminus231 x t

minus231 z) + t

132 A

2(tminus232 z t

minus232 y)




Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References


Prop 28(ivv) we get

h(tx hn0 )dist= sup

i=1kn

h(tx dyn

i) + hn0 (yni )


h(tx h0)dist= sup

y

h(tx dy) + h0(y)





y


= sup

y


dist= sup

y




supxisinR


t (x0 minus x)2)





31 The model












P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References











P(N)t (x y) = P(Xt = x|X0 = y)


d

dtP

(N)t (x y) =

(L(N)

)lowastP

(N)t (x y) P

(N)0 (x y) = 1y=x (32)


L(N)F (x) =Nsumk=1





d

dtu

(N)t =

Nsumk=1

nablalowastku(N)t u

(N)0 (x) = 1y=x (33)




(N)t that is

P(N)t (x y) = u









with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References




with

Fn(t x) =(minus1)n

2πi

∮Γ01

dw(1minus w)minusn









)T


parttu(N)t (x) =

Nsumk=1


]=

Nsumk=1


]=

Nsumk=1


]




]= det


]= 0




2πi

∮Γ0

dw(1minus w)n

wx+n+1




i lt xji le xj+1i+1


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References


lt lex1

1

lt lex2

1

lt lex3

1

x41

lt lex2

2

lt lex3

2

x42

lt lex3

3

x43 x4

4




det(φ(xnminus1

i xnj ))

1leijlen

)det(Fminusj(t x


)0leijltN (39)



)j=1N




P(N)t (x y) =


WN (x y)




(N)t




)kge1

(the TASEP initial


)klenminus1

and(Φnk

)k=0nminus1

as follows

Ψnk(x) =

1

2πi

∮Γ0

dw(1minus w)k





xisinZ Ψnk(x)Φn

` (x) = 1k=`




Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References


Definition 35 Let

Q(x y) =1

2xminusy1xgty

We have for m ge 1

Qm(x y) =1

2xminusy

(xminus y minus 1

mminus 1

)1xgey+m



m

y minus x

) (311)



mminusk (312)



where


njsumk=1

Ψniniminusk(xi)Φ

nj

njminusk(xj) (314)



Remark 37



(Xt(n1) Xt(nm)

)does







processes)




sumiw(πi)



x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References


x1x2xn a




)(mn)isinZ2


)(mn)isinZ2 The


(x y) y ge ht(x)

=




P(x1xn)

(Xt(n) gt a

)= Preversed

(a+1a+n)


)


P(x1xn)

(Xt(n) gt a

)= P(minusnminus1)


)


k rsquos



= det(I minusK(nm)


)`2(Z)

(316)

where










Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References








Definition 38 Let

Rt(x y) =1

2πi

∮Γ0

dwet(wminus1)


txminusy



Rminus1t (x y) =

1

2πi

∮Γ0

dwet(1minusw)


(minust)xminusy



(312) we have

Ψnk = RtQ





hnk(`X0(nminus `)) = 0 ` lt k (320c)






















Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References







Ψn` (z)Φn

k(z) =sum

z1z2isinZ

sumzisinZ


=sumzisinZ







2minusxΦnk(x) =

sumyge0

eminustty












τ = minm ge 0 Bm gt X0(m+ 1) (322)



(τ `n = k

)



G0n(z1 z2) =

nminus1sumk=0




K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References



K(n)t = RtQ

minusnG0nRminus1t




)(323)


(X0(nminusm)

)m=0nminus1

in between)


PBlowastminus1=z



(τ0n = k

) (324)



PBlowastminus1=z2

(τ0n = k


=nminus1sumk=0

sumzgtX0(nminusk)

PBlowastminus1=z2



= PBlowastminus1=z2


)



Q(n)(y1 y2) =1

2πi

∮Γ0


2y1minusTy2wn=






and




G0n(z1 z2) = EB0=z1


] (327)





)= PB0=z1


)=

nminus1sumk=0

sumzgtX0(k+1)

PB0=z1

(τ = k Bk = z


[Qnminusτ

(Bτ z2

)1τltn

] (328)








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References








1

2πi

∮Γ0

dw(1minus w)n



1

2πi

∮Γ0







] (331)


1))k=0nminus1


)mge0

is the set6

epi(g) =






lowastSepi(X0)tnj

(332)




also be written as



(333)





1

(2πi)2

∮Γ0

dw

∮Γ0



et(w+vminus1)

1minus v minus w






2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References




2πi

∮1+Γ0

dvvz2+2m








ε12(Xε


minush0(minusx)



compute


= limεrarr0

PX0

(X2εminus32t(

















(v uj) (44)




can be





Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References



Stminusni(zi y) =1

2πi

∮Γ0


dw

whereF (3) = t

[(2w minus 1) + 1














= det(Iminus χaK

hypo(h0)text χa

)L2(x1xMtimesR)

(46)

= det(IminusK

hypo(h0)txM

+ Khypo(h0)txM


)L2(R)

(47)






with



(ui uj)




Now



so










details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References






details












PB(`1)=u1


)du2





References















































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References














































14 Operators



21 Markov property


23 Symmetries

24 Airy processes



31 The model












43 Continuum limit

References

the kpz fixed pointdremenik/kpzfixedpointnotes.pdfpresentation of the main results, stripping the...

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