polynomial chaos and scaling limits of disordered...

White noise Continuum partition functions The continuum DPRE Pinning models

Polynomial Chaos andScaling Limits of Disordered Systems

2. Continuum model and free energy estimates

Francesco Caravenna

Universita degli Studi di Milano-Bicocca

YEP XIII, Eurandom ∼ March 7-11, 2016

Francesco Caravenna Scaling Limits of Disordered Systems March 7-11, 2016 1 / 43


Overview

In the previous lecture we saw how to build continuum partition functions

Zωδ

d−−−→δ→0

ZW (scaling limits of discrete partition functions)

In this lecture we present two interesting applications of ZW

I Scaling limit of the full probability measure Pωδd−−−→

δ→0PW

constructing a continuum version of the disordered system

We will focus on the DPRE [Alberts, Khanin, Quastel 2014b]drawing inspiration from the Pinning [C., Sun, Zygouras 2016]

I Sharp asymptotics on the discrete model, in terms of free energy andcritical curve

For this we will focus on Pinning models (rather than DPRE)



Outline

1. White noise and Wiener chaos

2. Continuum partition functions

3. The continuum DPRE

4. Pinning models



Outline




4. Pinning models



White noise (1 dim.)

We are familiar with (1-dim.) Brownian motion B = (B(t))t≥0

We are interested in its derivative “W (t) := ddtB(t)” called white noise

Think of W as a stochastic process W = (W (·)) indexed by

Intervals I = [a, b] 7−→ W (I ) = B(b)− B(a) ∼ N (0, b − a)

Borel sets A ∈ B(R) 7−→ W (A) =

∫

R1A(t) dB(t) ∼ N (0, |A|)

W is a Gaussian process with

E[W (A)] = 0 Cov[W (A),W (B)] = |A ∩ B|



White noise

White noise on Rd

It is a Gaussian process W = (W (A))A∈B(Rd ) with

E[W (A)] = 0 Cov[W (A),W (B)] = |A ∩ B|

I ∀ (An)n∈N disjoint =⇒ W

( ⋃

n∈N

An

)a.s.=∑

n∈N

W (An)

Almost a random signed measure on Rd . . . (but not quite!)

We can define single stochastic integrals W (f ) :=∫f (x)W (dx)

E[W (f )] = 0 E[W (f )2] = ‖f ‖2L2(Rd )



Multiple stochastic integrals

We can define

W⊗k(g) =

∫

(Rd )kg(x1, . . . , xk)W (dx1) · · ·W (dxk)

For d = 1 we can restrict x1 < x2 < . . . < xk iterated Ito integrals

For symmetric functions we have

E[W⊗k(g)] = 0 E[W⊗k(g)2] = k! ‖g‖2L2((Rd )k )

Cov[W⊗k(f ),W⊗k′(g)] = 0 ∀k 6= k ′

Wiener chaos expansion

Any r.v. X ∈ L2(ΩW ) measurable w.r.t. σ(W ) can be written as

X =∞∑

k=0

1

k!W⊗k(fk) with fk ∈ L2

sym((Rd)k)



Discrete sums and stochastic integrals

Consider a lattice Tδ ⊆ Rd whose cells have volume vδ → 0

Take i.i.d. random variables (X z)z∈Tδ with zero mean and unit variance

Consider the “stochastic Riemann sum” (multi-linear polynomial)

Ψδ :=∑

(z1,...,zk )∈(Tδ)k

zi 6=zi ∀i 6=j

f (z1, . . . , zk)X z1 X z2 · · ·X zk

where f ∈ L2(Rd) is (say) continuous.

(√vδ)k Ψδ

d−−−→δ→0

∫

(Rd )kg(z1, . . . , zk)W (dz1) · · ·W (dzk)

(Check the variance!)



Outline




4. Pinning models



Continuum partition function for DPRE

1d rescaled RW Sδt :=√δSt/δ lives on Tδ =

([0, 1] ∩ δN0

)×√δZ

Zωδ = Eref

[exp

(Hω)]

= Eref

[exp

(N∑

n=1

(βω(n,Sn) − λ(β)

))]

= 1 +∑

(t,x)∈Tδ

Pref(Sδt = x)X t,x

+1

2

∑

(t,x)6=(t′,x′)∈Tδ

Pref(Sδt = x ,Sδt′ = x ′)X t,x X t′,x′ + . . .

Recall the LLT: Pref(Sn = x) ∼ 1√ng(

x√n

)with g(z) = e−

z2

2√2π

Pref(Sδt = x) = Pref(S tδ

= x√δ

) ∼√δ gt(x) gt(x) =

e−x2

2t√2πt

Replacing X t,x = e(βω(t,x)−λ(β)) − 1 ≈ βY t,x with Y t,x i.i.d. N (0, 1)



Continuum partition function for DPRE

ZωN = 1 + β

√δ∑

(t,x)∈Tδ

gt(x)Y t,x

+1

2(β√δ)2

∑

(t,x)6=(t′,x′)∈Tδ

gt(x) gt′−t(x′ − x)Y t,x Y t′,x′ + . . .

Cells in Tδ have volume vδ = δ√δ = δ

32 “Stochastic Riemann sums”

converge to stochastic integrals if β√δ ≈ √vδ

β ∼ β δ 14 =

β

N14

ZωN

d−−−→δ→0

ZW = 1 + β

∫

[0,1]×Rgt(x)W (dtdx)

+β2

2

∫

([0,1]×R)2

gt(x) gt′−t(x′ − x)W (dtdx)W (dt ′dx ′)

+ . . .



Constrained partition functions

We have constructed ZW = “free” partition function on [0, 1]× RRW paths starting at (0, 0) with no constraint on right endpoint

ZW = ZW((0, 0), (1, ?)

)E[ZW

]= 1

Consider now constrained partition functions: for (s, y), (t, x) ∈ [0, 1]× R

Discrete: Zωδ

((s, y), (t, x)

)= Eref

[exp

(Hω)1Sδt =x

∣∣∣Sδs = y]

Divided by√δ, they converge to a continuum limit:

ZW((s, y), (t, x)

)E[ZW

((s, y), (t, x)

)]= gt−s(x − y)

This is a function of white noise in the stripe W ([s, t]× R)

Four-parameter random process ZW((s, y), (t, x)

) regularity?



Key properties

Key properties

For a.e. realization of W the following properties hold:

I Continuity: ZW ((s, y), (t, x)) is jointly continuous in (s, y , t, x)(on the domain s < t)

I Positivity: ZW ((s, y), (t, x)) > 0 for all (s, y , t, x) satisfying s < t

I Semigroup (Chapman-Kolmogorov): for all s < r < t and x , y ∈ R

ZW ((s, y), (t, x)) =

∫

RZW ((s, y), (r , z))ZW ((r , z), (t, x)) dz

(Inherited from discrete partition functions: drawing!)

How to prove these properties?



The 1d Stochastic Heat Equation

The four-parameter field ZW ((s, y), (t, x)) solves the 1d SHE∂tZW = 1

2 ∆xZW + βW ZW

limt↓s ZW ((s, y), (t, x)) = δ(y − x)

Checked directly from Wiener chaos expansion (mild solution)

It is known that solutions to the SHE satisfy the properties above

Alternative approach (to check, OK for pinning [C., Sun, Zygouras 2016])

I Prove continuity by Kolmogorov criterion, showing that

ZW ((s, y), (t, x))

gt−s(x − y)is continuous also for t = s

I Use continuity to prove semigroup for all times

I Use continuity to deduce positivity for close times, then bootstrap toarbitrary times using semigroup



Outline




4. Pinning models



A naive approach

Consider DPRE in d = 1 (random walk + disorder)

Pω(S) ∝ e∑N

n=1 β ω(n,Sn) Pref(S)

Can we define its continuum analogue (BM + disorder)? Naively

PW (dB) ∝ e∫ 1

0βW (t,Bt) dt Pref(dB)

Pref = law of BM W (t, x) = white noise on R2 (space-time)

I∫ 1

0W (t,Bt) dt ill-defined. Regularization?

NO! The problem is more subtle (and interesting!)



Partition functions and f.d.d.

Start from discrete: distribution of DPRE at two times 0 < t < t ′ < 1

Pωδ (Sδt = x ,Sδt′ = x ′) =Zωδ

((0, 0), (t, x)

)Zωδ

((t, x), (t ′, x ′)

)Zωδ

((t ′, x ′), (1, ?)

)

Zωδ

((0, 0), (1, ?)

)

(drawing!) Analogous formula for any finite number of times

Idea: Replace Zωδ ZW to define the law of continuum DPRE

Recall: to define a process (Xt)t∈[0,1] it is enough (Kolmogorov) to assignfinite-dimensional distributions (f.d.d.)

µt1,...,tk (A1, . . . ,Ak) “ = P(Xt1 ∈ A1, . . . ,Xtk ∈ Ak) ”

that are consistent

µt1,...,tj ,...,tk (A1, . . . ,R, . . . ,Ak) = µt1,...,tj−1,tj+1,...,tk (A1, . . . ,Aj−1,Aj+1, . . . ,Ak)



The continuum 1d DPRE

I Fix β ∈ (0,∞) (on which ZW depend) [recall that β ∼ βδ14 ]

I Fix space-time white noise W on [0, 1]× R and a realization ofcontinuum partition functions ZW satisfying the key properties(continuity, strict positivity, semigroup)

The Continuum DPRE is the process (Xt)t∈[0,1] with f.d.d.

PW (Xt ∈ dx ,Xt′ ∈ dx ′)

dx dx ′

:=ZW

((0, 0), (t, x)

)ZW

((t, x), (t ′, x ′)

)ZW

((t ′, x ′), (1, ?)

)

ZW((0, 0), (1, ?)

)

I Well-defined by strict positivity of ZW

I Consistent by semigroup property



Relation with Wiener measure

The law of the continuum DPRE is a random probability

PW (X ∈ · ) (quenched law)

for the process X = (Xt)t∈[0,1] [ Probab. kernel S ′(R)→ R[0,1] ]

Define a new law P (mutually absolutely continuous) for disorder W by

dPdP

(W ) = ZW((0, 0), (1, ?)

)

Key Lemma

Pann(X ∈ ·) :=

∫

S′(R)

PW (X ∈ · ) P(dW ) = P(BM ∈ · )

Proof. The factor ZW in P cancels the denominator in the f.d.d. for PW

Since E[ZW

((s, y), (t, x)

)]= gt−s(x − y) one gets f.d.d. of BM



Absolute continuity properties

Theorem

∀A ⊆ R[0,1] : P(BM ∈ A) = 1 ⇒ PW (X ∈ A) = 1 for P-a.e. W

Any given a.s. property of BM is an a.s. property of continuum DPRE,for a.e. realization of the disorder W

Corollary

PW (X has Holder paths with exp. 12−) = 1 for P-a.e. W

We can thus realize PW as a law on C ([0, 1],R), for P-a.e. W

(More precisely: PW admits a modification with Holder paths)

One is tempted to conclude that PW is absolutely continuous w.r.t.Wiener measure, for P-a.e. W . . .

NO ! “ ∀A ” and “ for P-a.e. W ” cannot be exchanged!



Singularity properties

Theorem

The law PW is singular w.r.t. Wiener measure, for P-a.e. W .

for P-a.e. W ∃ A = AW ⊆ C ([0, 1],R) :

PW (X ∈ A) = 1 vs. P(BM ∈ A) = 0

Unlike discrete DPRE, there is no continuum Hamiltonian

PW (X ∈ · ) 6∝ eHW ( · )P(BM ∈ · )

Absolute continuity is lost in the scaling limit

In a sense, the laws PW are just barely not absolutely continuous w.r.t.Wiener measure (“stochastically absolutely continuous”)



Proof of singularity

Let (Xt)t∈[0,1] be the canonical process on C ([0, 1],R) [ Xt(f ) = f (t) ]

Let Fn := σ(Xtni: tni = i

2n , 0 ≤ i ≤ 2n) be the dyadic filtration

Fix a typical realization of W . Setting Pref = Wiener measure

RWn (X ) :=

dPW |Fn

dPref |Fn

(X )

The process (RWn )n∈N is a martingale w.r.t. Pref (exercise!)

Since RWn ≥ 0, the martingale converges: RW

na.s.−−−→

n→∞RW∞

I PW Pref if and only if Eref [RW∞ ] = 1 (the martingale is UI)

I PW is singular w.r.t. Pref if and only if RW∞ = 0




It suffices to show that RWn (X ) −−−→

n→∞0 in P⊗ Pref -probability

Fractional moment

For Pref -a.e. X EE[(RWn (X )

)γ] −−−→n→∞

0 for some γ ∈ (0, 1)

RWn (X ) =

1

ZW((0, 0), (1, ?)

)2n−1∏

i=0

ZW((tni ,Xtni

), (tni+1,Xtni+1))

g 12n

(Xtni+1− Xtni

)

I Switch from E to equivalent law E to cancel the denominator

I For fixed X , the ZW((tni ,Xtni

), (tni+1,Xtni+1))

’s are independent

We need to exploit translation and scale invariance of their laws




Lemma 1 (Translation and scale invariance)

If we set ∆ni :=

Xtni+1− Xtni√

tni+1 − tniwe have

ZWβ

((tni ,Xtni

), (tni+1,Xtni+1))

g 12n

(Xtni+1− Xtni

)d=

ZWβ

2n/4

((0, 0), (1,∆n

i ))

g1(∆ni )

Lemma 2 (Expansion)

For z ∈ R and ε ∈ [0, 1] (say)

ZWε

((0, 0), (1, z)

)

g1(z)= 1 + εX z + ε2 Y ε,z

E[X z ] = 0 E[X ε,z ] = 0 E[X 2

z

]≤ C E

[Y 2ε,z

]≤ C unif. in ε, z




By Taylor expansion, for fixed γ ∈ (0, 1)

E

[(ZWε

((0, 0), (1, z)

)

g1(z)

)γ]= E

[(1 + εX z + ε2Y ε,z

)γ]

= 1 + γεE[Xz ] + ε2E[Yε,z ]

+

γ(γ − 1)

2

ε2E[(X x)2] + . . .

+ . . .

= 1 − c ε2 ≤ e−c ε2

(?) First order terms vanish (?) γ(γ − 1) < 0 (?) For some c > 0

Estimate is uniform over z ∈ R We can set z = ∆ni and ε = 1

2n/4

E[(RWn (X )

)γ]=

2n−1∏

i=0

E

[(ZWε

((0, 0), (1,∆n

i ))

g1(∆ni )

)γ]≤ e−c ε

2 2n

= e−c 2n/2

which vanishes as n→∞



Proof of Lemma 1

Introducing the dependence on β

ZWβ

((s, y), (t, x)

) d= ZW

β

((0, 0), (t − s, x − y)

)

ZWβ

((0, 0), (t, x)

) d=

1√tZW

βt14

((0, 0),

(1,

x√t

))

transl. invariance + diffusive rescaling (prefactor, new β) (drawing!)

ZW((0, 0), (t, x)

)= gt(x) + β

∫

[0,t]×Rgs(z) gt−s(x − z)W (dsdz) + . . .

=1√tg1( x√

t) +

1√t

(β t

34√t

)∫

[0,t]×Rg s

t( z√

t) g1− s

t( x−z√

t)W (dsdz)

t34

+ . . .

= OK !



Convergence of discrete DPRE

I Pωδ = law of discrete DPRE (recall that Sδt :=√δSt/δ)

“Rescaled RW Sδ moving in an i.i.d. environment ω”

I PW = law of continuum DPRE

“BM moving in a white noise environment W ”

Both Pωδ and PW are random probability laws on E := C ([0, 1],R)i.e. RVs (defined on different probab. spaces) taking values in M1(E )

Does Pωδ converge in distribution toward PW as δ → 0 ?

∀ ψ ∈ Cb(M1(E )→ R) : E[ψ(Pωδ )

]−−−→δ→0

E[ψ(PW )

]

The answer is positive. . . almost surely ;-)

Statement for Pinning model proved in [C., Sun, Zygouras 2016]

Details need to be checked for DPRE (stronger assumptions on RW ?)



Universality

The convergence of Pωδ toward PW is an instance of universality

There are many discrete DPRE:

I any RW S (zero mean, finite variance + technical assumptions)

I any (i.i.d.) disorder ω (finite exponential moments)

In the continuum (δ → 0) and weak disorder (β → 0) regime, all thesemicroscopic models Pωδ give rise to a unique macroscopic model PW

Tomorrow we will see how the continuum model PW can tellquantitative information on discrete models Pωδ (free energy estimates)



Convergence

How to prove convergence in distribution Pωδd−−−→

δ→0PW ?

Prove a.s. convergence through a suitable coupling of (ω,W )

Assume we have convergence in distribution of discrete partition functionsto continuum ones, in the space of continuum functions of (s, y), (t, x)

Zωδ

((s, y), (t, x)

) d−−−→δ→0

ZW((s, y), (t, x)

)

By Skorokhod representation theorem, there is a coupling of (ω,W )under which this convergence holds a.s.

Fix such a coupling: for a.e. (ω,W ) the f.d.d. of Pωδ converge weakly tothose of PW . It only remains to prove tightness of Pωδ (·).



Outline




4. Pinning models



Ingredients: renewal process & disorder

0 = τ0 τ1 τ2 τ3 τ4 τ5 τ6

Discrete renewal process τ = 0 = τ0 < τ1 < τ2 < . . . ⊆ N0

Gaps (τi+1 − τi )i≥0 are i.i.d. with polynomial-tail distribution:

Pref(τ1 = n) ∼ cKn1+α

, cK > 0, α ∈ (0, 1)

τ = n ∈ N0 : Sn = 0 zero level set of a Markov chain S = (Sn)n≥0

Disorder ω = (ωi )i∈N: i.i.d. real random variables with law P

λ(β) := log E[eβω1 ] <∞ E[ω1] = 0 Var[ω1] = 1



Bessel random walks

For α ∈ (0, 1) the α-Bessel random walk is defined as follows:

0prob. 1

2

xprob. 1

2

(1 + cα

x

)Sn

prob. 12

prob. 12

(1− cα

x

) cα := 12 − α

I (α = 12 ) no drift (cα = 0) simple random walk

I (α < 12 ) drift away from the origin (cα > 0)

I (α > 12 ) drift toward the origin (cα < 0)



Disordered pinning model

0 N

Pinning model rewards ωn > 0 penalties ωn < 0

N ∈ N (system size) β ≥ 0, h ∈ R (disorder strength, bias)

The pinning model

Gibbs change of measure PωN = PωN,β,h of the renewal distribution Pref

dPωNdPref

(τ) :=1

ZωNexp

(N∑

n=1

(βωn + h − λ(β))1n∈τ 1Sn=0

)



The phase transition

How are the typical paths τ of the pinning model PωN?

Contact number CN :=∣∣τ ∩ (0,N]

∣∣ =∑N

n=1 1n∈τ =∑N

n=1 1Sn=0

Theorem (phase transition)

∃ continuous, non decreasing, deterministic critical curve hc(β):

I Localized regime: for h > hc(β) one has CN ≈ N

∃µ = µβ,h > 0 : PωN

(∣∣∣∣CNN− µ

∣∣∣∣ > ε

)−−−−→N→∞

0 ω–a.s.

I Deocalized regime: for h < hc(β) one has CN = O(logN)

∃A = Aβ,h > 0 : PωN

( CNlogN

> A

)−−−−→N→∞

0 ω–a.s.



Estimates on the critical curve

For β = 0 (homogeneous pinning, no disorder) one has hc(0) = 0

What is the behavior of hc(β) for β > 0 small ?

Theorem ( P(τ1 = n) ∼ cKn1+α )

I (α < 12 ) disorder is irrelevant: hc(β) = 0 for β > 0 small

[Alexander] [Toninelli] [Lacoin] [Cheliotis, den Hollander]

I (α ≥ 12 ) disorder is relevant: hc(β) > 0 for all β > 0

I (α > 1) hc(β) ∼ C β2 with explicit C = α1+α

12E(τ1)

[Berger, C., Poisat, Sun, Zygouras]

I ( 12< α < 1) C1 β

2α2α−1 ≤ hc(β) ≤ C2 β

2α2α−1 hc(β) ∼ C β

2α2α−1

using continuum part. funct.![Derrida, Giacomin, Lacoin, Toninelli] [Alexander, Zygouras] [C., Torri, Toninelli]

I (α = 12) hc(β) = e

− c+o(1)

β2 [Giacomin, Lacoin, Toninelli] [Berger, Lacoin]



Discrete free energy and critical curve

Partition function ZωN := E

[eHN (τ)

]= E

[e∑N

n=1(h+βωn−Λ(β))1n∈τ]

Consider first the regime of N →∞ with fixed β, h

I Free energy F(β, h) := limN→∞

1N log Zω

N ≥ 0 P(dω)-a.s.

ZωN ≥ E

[eHN (τ) 1τ∩(0,N]=∅

]= P(τ ∩ (0,N] = ∅) ∼ (const.)

Nα

I Critical curve hc(β) = suph ∈ R : F(β, h) = 0 non analiticity!

(convexity)∂F(β, h)

∂h= lim

N→∞EωN

[CNN

] > 0 if h > hc(β)

= 0 if h < hc(β)

F(β, h) and hc(β) depend on the law of τ and ω

Universality as β, h→ 0 ? YES, connected to continuum model



Continuum partition functions

Build continuum partition functions for Pinning Model with α ∈ ( 12 , 1)

(disorder relevant) following “usual” approach [C, Sun, Zygouras 2015+]

We need to rescale

β = βN =β

Nα−1/2h = hN =

h

Nα

One has ZωN

d−−−−→N→∞

ZW with

ZW := 1 + C

∫

0<t<1

dW β,ht

t1−α + C 2

∫

0<t<t′<1

dW β,ht dW β,h

t′

t1−α(t ′ − t)1−α + . . .

where W β,ht := βW t + h t and C = α sin(απ)

π cK



Continuum free energy

In analogy with the discrete model, define

Continuum free energy F(β, h) := limt→∞

1

tlogZW

β,h(0, t) a.s.

(existence and self-averaging need some work)

Again F(β, h) ≥ 0 and define

Continuum critical curve Hc(β) := suph ∈ R : F(β, h) = 0

Scaling relations

∀c > 0 : ZWβ,h

(c t)d= ZW

cα−12 β,cαh

(t) (Wiener chaos exp.)

F(cα−

12 β, cαh

)= c F(β, h) Hc(β) = Hc(1) β

2α2α−1



Interchanging the limits

Can we relate continuum free energy to the discrete one?

By construction of continuum partition functions

ZWβ,h

(t)d= lim

N→∞ZωβN ,hN (Nt)

Assuming uniform integrability of log Zω (OK)

F(β, h) = limt→∞

1

tE[

logZWβ,h

(t)]

= limt→∞

1

tlim

N→∞E[

log ZωβN ,hN (Nt)

]

Assuming we can interchange the limits N →∞ and t →∞

F(β, h) = limN→∞

N limt→∞

1

Nt tE[

log ZωβN ,hN (Nt)

]= lim

N→∞N F(βN , hN)

Setting δ = 1N for clarity, we arrive at. . .




Conjecture

F(β, h) = limδ→0

F(βδα−

12 , hδα

)

δ

Theorem [C., Toninelli, Torri 2015]

For all β > 0, h ∈ R and η > 0 there is δ0 > 0 such that ∀δ < δ0

F(β, h − η) ≤ F(βδα−

12 , hδα

)

δ≤ F(β, h + η)

This implies Conj. and hc(β) ∼ Hc(β) ∼ Hc(1)β2α

2α−1

For any discrete Pinning model with α ∈ ( 12 , 1), the free energy F (β, h)

and the critical curve hc(β) have a universal shape in the regime β, h→ 0




Very delicate result. How to prove it?

I Assume that there is a continuum Hamiltonian:

Zω = E[eHωNt

]ZW = E

[eH

Wt]

I Couple HωNt and HW

t on the same probability space in such a waythat the difference ∆N,t := Hω

Nt −HWt is “small”

I Deduce that

E[

log Zω]≤ E

[logZW

]+ log EE

[e∆N,t

]

and show that the last term is “negligible”

Problem: there is no continuum Hamiltonian!

Solution: perform coarse-graining and define an “effective” Hamiltonian



The DPRE case

What about the DPRE?

We can still define discrete F(β) and continuum F(β) free energy

Since F(β) ∼ F(1)β4 we can hope that

F(β) ∼ F(1)β4 as β → 0

provided the “interchanging of limits” is justified

N. Torri is currently working on this problem. A finer coarse-graining isneeded, together with sharper estimates on continuum partition functions



References

I T. Alberts, K. Khanin, J. QuastelThe intermediate disorder regime for directed polymers in dimension1 + 1Ann. Probab. 42 (2014), 1212–1256

I T. Alberts, K. Khanin, J. QuastelThe Continuum Directed Random PolymerJ. Stat. Phys. 154 (2014), 305–326

I F. Caravenna, R. Sun, N. ZygourasPolynomial chaos and scaling limits of disordered systemsJ. Eur. Math. Soc. (JEMS), to appear

I F. Caravenna, R. Sun, N. ZygourasThe continuum disordered pinning modelProbab. Theory Related Fields 164 (2016), 17-59.


polynomial chaos and scaling limits of disordered...

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