meteor process - stat.berkeley.edualdous/pitman... · the \meteor hit" (mass redistribution...

$: METEOR PROCESS - stat.berkeley.edualdous/Pitman... · The \meteor hit" (mass redistribution event) occurs at a vertex when the corresponding Poisson process jumps. Krzysztof Burdzy$
METEOR PROCESS

Krzysztof BurdzyUniversity of Washington

Krzysztof Burdzy METEOR PROCESS

Collaborators and preprints

Joint work with Sara Billey, Soumik Pal and Bruce E. Sagan.

Math Arxiv:http://arxiv.org/abs/1308.2183http://arxiv.org/abs/1312.6865


Mass redistribution


Mass redistribution (2)

time

space


Related models

Chan and Pra lat (2012)Crane and Lalley (2013)Ferrari and Fontes (1998)Fey-den Boer, Meester, Quant and Redig (2008)Howitt and Warren (2009)


Model

G - simple connected graph (no loops, no multiple edges)V - vertex setMx

t - mass at x ∈ V at time tAssumption: Mx

0 ∈ [0,∞) for all x ∈ VMt = {Mx

t , x ∈ V }Nxt - Poisson process at x ∈ V

The Poisson processes are assumed to be independent.The “meteor hit” (mass redistribution event) occurs at a vertex when thecorresponding Poisson process jumps.


Existence of the process

THEOREM

If the graph has a bounded degree then the meteor process is well definedfor all t ≥ 0.


General graph - stationary distribution

Example. Suppose that G is a triangle. The following are possible massprocess transitions.(1, 2, 0)→ (0, 5/2, 1/2)

(1, π/2, 2− π/2)→ (1 + π/4, 0, 2− π/4)The state space is stratified.



Example. Suppose that G is a triangle. The following are possible massprocess transitions.(1, 2, 0)→ (0, 5/2, 1/2)(1, π/2, 2− π/2)→ (1 + π/4, 0, 2− π/4)

The state space is stratified.



Example. Suppose that G is a triangle. The following are possible massprocess transitions.(1, 2, 0)→ (0, 5/2, 1/2)(1, π/2, 2− π/2)→ (1 + π/4, 0, 2− π/4)The state space is stratified.


Stationary distribution - existence and uniqueness

THEOREM

Suppose that the graph is finite. The stationary distribution for theprocess Mt exists and is unique. The process Mt converges to thestationary distribution exponentially fast.

Proof (sketch). Consider two mass processes Mt and Mt on the samegraph, with different initial distributions but the same meteor hits.

t →∑

x∈V

∣∣∣Mxt − Mx

t

∣∣∣ is non-increasing.

Hairer, Mattingly and Scheutzow (2011)


Circular graphs

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Ck - circular graph with k verticesQk - stationary distribution for Mt on Ck


Circular graphs - moments of mass at a vertex

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THEOREM

EQkMx

0 = 1, x ∈ V , k ≥ 1

limk→∞

VarQkMx

0 = 1, x ∈ V

limk→∞

CovQk(Mx

0 ,Mx+10 ) = −1/2, x ∈ V

limk→∞

CovQk(Mx

0 ,My0 ) = 0, x 6↔ y


Circular graphs - correlation and independence

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limk→∞

CovQk(Mx

0 ,My0 ) = 0, x 6↔ y

If x 6↔ y then Mx0 and My

0 do not appear to be asymptoticallyindependent under Qk ’s.(Mx

0 )2 and My0 seem to be asymptotically correlated under Qk , if x 6↔ y .


From circular graphs to Z

Ck - circular graph with k verticesQk - stationary distribution for Mt on Ck

THEOREM

For every fixed n, the distributions of (M10 ,M

20 , . . . ,M

n0 ) under Qk

converge to a limit Q∞ as k →∞.

The theorem yields existence of a stationary distribution Q∞ for themeteor process on Z.

Similar results hold for meteor processes on Cdk and Zd .


Moments of mass at a vertex in Zd

THEOREM

EQ∞Mx0 = 1, x ∈ V

VarQ∞ Mx0 = 1, x ∈ V

CovQ∞(Mx0 ,M

y0 ) = − 1

2d, x ↔ y

CovQ∞(Mx0 ,M

y0 ) = 0, x 6↔ y


Mass fluctuations in intervals of Z

THEOREM

For every n,

EQ∞

∑1≤j≤n

M j0 = n,

VarQ∞

∑1≤j≤n

M j0 = 1.


Flow across the boundary

x x+1

F xt - net flow from x to x + 1 between times 0 and t

THEOREM

Under Q∞, for all x ∈ Z and t ≥ 0,

Var F xt ≤ 4.


Mass distribution at a vertex of Z

A simulation of Mx0 under Q∞.

0 1 2 3 4 5 6

5000

10 000

15 000

20 000


Mass distribution at a vertex of Z (2)

0 1 2 3 4 5 6

5000

10 000

15 000

20 000

PQ∞(Mx0 = 0) = 1/3

EQ∞Mx0 = 1, VarQ∞ Mx

0 = 1Is Q∞ a mixture of a gamma distribution and an atom at 0? No.

One can find an exact and rigorous value for EQk(Mx

0 )n for every n and k .We cannot find asymptotic formulas when k →∞.


Support of the stationary distribution

Assume that |V | = k , and∑

x∈V Mx0 = k .

Let S be the simplex consisting of all {Sx , x ∈ V } with Sx ≥ 0 for allx ∈ V and

∑x∈V Sx = k .

Let S∗ be the set of {Sx , x ∈ V } with Sx = 0 for at least one x ∈ V .

THEOREM

The (closed) support of the stationary distribution for Mt is equal to S∗.


WIMPs

DEFINITION

Suppose that M0 is given and k =∑

v∈V Mv0 .

For each j ≥ 1, let {Y jn , n ≥ 0} be a discrete time symmetric random walk

on G with the initial distribution P(Y j0 = x) = Mx

0 /k for x ∈ V . We

assume that conditional on M0, processes {Y jn , n ≥ 0}, j ≥ 1, are

independent.

Recall Poisson processes Nv and assume that they are independent of{Y j

n , n ≥ 0}, j ≥ 1. For every j ≥ 1, we define a continuous time Markovprocess {Z j

t , t ≥ 0} by requiring that the embedded discrete Markov chainfor Z j is Y j and Z j jumps at a time t if and only if Nv jumps at time t,where v = Z j

t−.


WIMPs

DEFINITION

Suppose that M0 is given and k =∑

v∈V Mv0 .

For each j ≥ 1, let {Y jn , n ≥ 0} be a discrete time symmetric random walk

on G with the initial distribution P(Y j0 = x) = Mx

0 /k for x ∈ V . We

assume that conditional on M0, processes {Y jn , n ≥ 0}, j ≥ 1, are

independent.Recall Poisson processes Nv and assume that they are independent of{Y j

n , n ≥ 0}, j ≥ 1. For every j ≥ 1, we define a continuous time Markovprocess {Z j

t , t ≥ 0} by requiring that the embedded discrete Markov chainfor Z j is Y j and Z j jumps at a time t if and only if Nv jumps at time t,where v = Z j

t−.


WIMPs and convergence rate

The rate of convergence to equilibrium for Mt cannot be faster than thatfor a simple random walk. Justification: Consider expected occupationmeasures.


Rate of convergence on tori

THEOREM

Consider the meteor process on a graph G = Cdn (the product of d copies

of the cycle Cn). Consider any distributions (possibly random) of mass

M0 and M0, and suppose that∑

x Mx0 =

∑x M

x0 = |V | = nd , a.s. There

exist constants c1, c2 and c3, not depending on G , such that ifn ≥ 1 ∨ c1

√d log d and t ≥ c2dn

2 then one can define a coupling of massprocesses Mt and Mt on a common probability space so that,

E

(∑x∈V|Mx

t − Mxt |

)≤ exp(−c3t/(dn2))|V |.


Earthworm

Earthworm = simple random walkRedistribution events occur at the sites visited by earthworm


Earthworms equidistribute soil

THEOREM

Fix d ≥ 1 and let Mnt be the empirical measure process for the earthworm

process on the graph G = Cdn . Assume that Mv

0 = 1/nd for v ∈ V (hence,∑v∈V Mv

0 = 1).

(i) For every n, the random measures Mnt converge weakly to a random

measure Mn∞, when t →∞.

(ii) For R ⊂ Rd and a ∈ R, let aR = {x ∈ Rd : x = ay for some y ∈ R}and Mn

∞(R) = Mn∞(nR). When n→∞, the random measures Mn

∞converge weakly to the random measure equal to, a.s., the uniformprobability measure on [0, 1]d .



THEOREM




0 = 1).(i) For every n, the random measures Mn

t converge weakly to a randommeasure Mn

∞, when t →∞.

(ii) For R ⊂ Rd and a ∈ R, let aR = {x ∈ Rd : x = ay for some y ∈ R}and Mn





THEOREM




0 = 1).(i) For every n, the random measures Mn

t converge weakly to a randommeasure Mn

∞, when t →∞.(ii) For R ⊂ Rd and a ∈ R, let aR = {x ∈ Rd : x = ay for some y ∈ R}and Mn




Craters in circular graphs

G = Ck

There is a crater at x at time t if Mxt = 0.

A crater exists at a site if and only if a meteor hit the site and there wereno more recent hits at adjacent sites.

Under the stationary distribution, the distribution of craters in Ck is thesame as the distribution of peaks in a random (uniform) permutation ofsize k .


Peaks in random permutations

It is possible to find a formula for the probability of a given peak set in arandom permutation.

THEOREM

P(crater at 1) = 1/3,

P(crater at 1 followed by exactly n non-craters) =n(n + 3)2n+1

(n + 4)!,

P(no craters at 1, 2, . . . , n) =2n+1

(n + 2)!.


Peaks in random permutations

THEOREM

P(crater at 1 followed by i non-craters, then a crater,

then exactly j non-craters)

=2i+j

(i + j + 5)!

[(i + j + 4)

(j

(i + j + 1

i − 1

)+ (j + 1)

(i + j + 1

i

)

+ (i + 1)

(i + j + 1

i + 1

)+ i

(i + j + 1

i + 2

)− 2(i + j + 1)

)+ ij

(i + j + 4

i + 2

)].


Craters repel each other

THEOREM

Consider the meteor process on a circular graph Ck in the stationaryregime. Let G be the family of adjacent craters, i.e., (i , j) ∈ G if an only ifthere are craters at i and j and there are no craters between i and j . Forr > 1, let

A1r =

{max(i ,j)∈G1 |i − j |min(i ,j)∈G1 |i − j |

≤ 1 + r

}.

Let H1n be the event that there are exactly n craters at time 0. For every

n ≥ 2, p < 1 and r > 1 there exists k1 <∞ such that for all k ≥ k1,P(A1

r | H1n) > p.

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Systems of non-crossing paths

time

space

Non-crossing continuous pathmodels:Harris (1965)Spitzer (1968) - shownDurr, Goldstein and Lebowitz (1985)Tagged particle in exclusion process:Arratia (1983)

Free path scaling: dX ≈ (dt)α

Non-crossing path scaling:dX ≈ (dt)α/2


Meteor process on Z

0

1

2

3

-3

-1

-2

-4

There is no time scale in this picture. The horizontal axis represents mass.


Meteor process on Z (2)

012345

-1-2-3-4

x y zv

The horizontal axis represents mass. The state of the meteor process attime t is represented as an RCLL function H ·t . For example, Hx

t = Hyt = 2,

Hvt = −4 and Hz

t = 3.


Jump of meteor process

012345

-1-2-3-4

x y zv


Jump of meteor process (2)

012345

-1-2-3-4

x y zv


Non-crossing paths

012345

-1-2-3-4

x y zv

If x ≤ y then Hxt ≤ Hy

t for all t ≥ 0.

THEOREM

Suppose that that the meteor process is in the stationary distribution Q.Then for every α < 2 there exists c <∞ such that for every x ∈ Z andt ≥ 0,

E |Hxt − Hx

0 |α ≤ c.


Systems of non-crossing paths

time

spaceNon-crossing continuous pathmodels:Harris (1965)Spitzer (1968) - shownDurr, Goldstein and Lebowitz (1985)Tagged particle in exclusion process:Arratia (1983)

Free path scaling: dX ≈ (dt)α

Non-crossing path scaling:dX ≈ (dt)α/2

Meteor modelFree path scaling: dX ≈ (dt)1/2

Non-crossing path scaling:dX ≈ (dt)0


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