METEOR PROCESS Krzysztof Burdzy University of Washington Krzysztof - - PowerPoint PPT Presentation

METEOR PROCESS

Krzysztof Burdzy University 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/1308.2183 http://arxiv.org/abs/1312.6865

Krzysztof Burdzy METEOR PROCESS

Mass redistribution

Krzysztof Burdzy METEOR PROCESS

Mass redistribution (2)

time space

Krzysztof Burdzy METEOR PROCESS

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)

Krzysztof Burdzy METEOR PROCESS

Model

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

t - mass at x ∈ V at time t

Assumption: Mx

0 ∈ [0, ∞) for all x ∈ V

Mt = {Mx

t , x ∈ V }

Nx

t - Poisson process at x ∈ V

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

Krzysztof Burdzy METEOR PROCESS

Existence of the process

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

Krzysztof Burdzy METEOR PROCESS

General graph - stationary distribution

• Example. Suppose that G is a triangle. The following are possible mass

process transitions. (1, 2, 0) → (0, 5/2, 1/2)

Krzysztof Burdzy METEOR PROCESS

General graph - stationary distribution

• Example. Suppose that G is a triangle. The following are possible mass

process transitions. (1, 2, 0) → (0, 5/2, 1/2) (1, π/2, 2 − π/2) → (1 + π/4, 0, 2 − π/4)

Krzysztof Burdzy METEOR PROCESS

General graph - stationary distribution

• Example. Suppose that G is a triangle. The following are possible mass

process 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.

Krzysztof Burdzy METEOR PROCESS

Stationary distribution - existence and uniqueness

THEOREM Suppose that the graph is finite. The stationary distribution for the process Mt exists and is unique. The process Mt converges to the stationary distribution exponentially fast. Proof (sketch). Consider two mass processes Mt and Mt on the same graph, with different initial distributions but the same meteor hits.

Krzysztof Burdzy METEOR PROCESS

Stationary distribution - existence and uniqueness

THEOREM Suppose that the graph is finite. The stationary distribution for the process Mt exists and is unique. The process Mt converges to the stationary distribution exponentially fast. Proof (sketch). Consider two mass processes Mt and Mt on the same graph, with different initial distributions but the same meteor hits. t →

x∈V

• Mx

t −

Mx

t

• is non-increasing.

Krzysztof Burdzy METEOR PROCESS

Stationary distribution - existence and uniqueness

THEOREM Suppose that the graph is finite. The stationary distribution for the process Mt exists and is unique. The process Mt converges to the stationary distribution exponentially fast. Proof (sketch). Consider two mass processes Mt and Mt on the same graph, with different initial distributions but the same meteor hits. t →

x∈V

• Mx

t −

Mx

t

• is non-increasing.

Hairer, Mattingly and Scheutzow (2011)

Krzysztof Burdzy METEOR PROCESS

Circular graphs

1 2 3 4 5 6 7 8 9 10

Ck - circular graph with k vertices Qk - stationary distribution for Mt on Ck

Krzysztof Burdzy METEOR PROCESS

Circular graphs - moments of mass at a vertex

1 2 3 4 5 6 7 8 9 10

THEOREM EQkMx

0 = 1,

x ∈ V , k ≥ 1

Krzysztof Burdzy METEOR PROCESS

Circular graphs - moments of mass at a vertex

1 2 3 4 5 6 7 8 9 10

THEOREM EQkMx

0 = 1,

x ∈ V , k ≥ 1 lim

k→∞ VarQk Mx 0 = 1,

x ∈ V

Krzysztof Burdzy METEOR PROCESS

Circular graphs - moments of mass at a vertex

1 2 3 4 5 6 7 8 9 10

THEOREM EQkMx

0 = 1,

x ∈ V , k ≥ 1 lim

k→∞ VarQk Mx 0 = 1,

x ∈ V lim

k→∞ CovQk(Mx 0 , Mx+1

) = −1/2, x ∈ V

Krzysztof Burdzy METEOR PROCESS

Circular graphs - moments of mass at a vertex

1 2 3 4 5 6 7 8 9 10

THEOREM EQkMx

0 = 1,

x ∈ V , k ≥ 1 lim

k→∞ VarQk Mx 0 = 1,

x ∈ V lim

k→∞ CovQk(Mx 0 , Mx+1

) = −1/2, x ∈ V lim

k→∞ CovQk(Mx 0 , My 0 ) = 0,

x ↔ y

Krzysztof Burdzy METEOR PROCESS

Circular graphs - correlation and independence

1 2 3 4 5 6 7 8 9 10

lim

k→∞ CovQk(Mx 0 , My 0 ) = 0,

x ↔ y If x ↔ y then Mx

0 and My 0 do not appear to be asymptotically

independent under Qk’s. (Mx

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

Krzysztof Burdzy METEOR PROCESS

From circular graphs to Z

Ck - circular graph with k vertices Qk - stationary distribution for Mt on Ck THEOREM For every fixed n, the distributions of (M1

0, M2 0, . . . , Mn 0 ) under Qk

converge to a limit Q∞ as k → ∞. The theorem yields existence of a stationary distribution Q∞ for the meteor process on Z. Similar results hold for meteor processes on C d

k and Zd.

Krzysztof Burdzy METEOR PROCESS

Moments of mass at a vertex in Zd

THEOREM EQ∞Mx

0 = 1,

x ∈ V VarQ∞ Mx

0 = 1,

x ∈ V CovQ∞(Mx

0 , My 0 ) = − 1

2d , x ↔ y CovQ∞(Mx

0 , My 0 ) = 0,

x ↔ y

Krzysztof Burdzy METEOR PROCESS

Mass fluctuations in intervals of Z

THEOREM For every n, EQ∞

• 1≤j≤n

Mj

0 = n,

Krzysztof Burdzy METEOR PROCESS

Mass fluctuations in intervals of Z

THEOREM For every n, EQ∞

• 1≤j≤n

Mj

0 = n,

VarQ∞

• 1≤j≤n

Mj

0 = 1.

Krzysztof Burdzy METEOR PROCESS

Flow across the boundary

x x+1

F x

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

Krzysztof Burdzy METEOR PROCESS

Flow across the boundary

x x+1

F x

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

THEOREM Under Q∞, for all x ∈ Z and t ≥ 0, Var F x

t ≤ 4.

Krzysztof Burdzy METEOR PROCESS

Mass distribution at a vertex of Z

A simulation of Mx

0 under Q∞.

1 2 3 4 5 6 5000 10000 15000 20000

Krzysztof Burdzy METEOR PROCESS

Mass distribution at a vertex of Z (2)

PQ∞(Mx

0 = 0) = 1/3

EQ∞Mx

0 = 1,

VarQ∞ Mx

0 = 1

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

Krzysztof Burdzy METEOR PROCESS

Mass distribution at a vertex of Z (2)

PQ∞(Mx

0 = 0) = 1/3

EQ∞Mx

0 = 1,

VarQ∞ Mx

0 = 1

Is 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 → ∞.

Krzysztof Burdzy METEOR PROCESS

Support of the stationary distribution

Assume that |V | = k, and

x∈V Mx 0 = k.

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

x∈V Sx = k.

Krzysztof Burdzy METEOR PROCESS

Support of the stationary distribution

Assume that |V | = k, and

x∈V Mx 0 = k.

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

x∈V Sx = k.

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

Krzysztof Burdzy METEOR PROCESS

Support of the stationary distribution

Assume that |V | = k, and

x∈V Mx 0 = k.

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

Krzysztof Burdzy METEOR PROCESS

WIMPs

DEFINITION Suppose that M0 is given and k =

v∈V Mv 0 .

For each j ≥ 1, let {Y j

n, n ≥ 0} be a discrete time symmetric random walk

• n G with the initial distribution P(Y j

0 = x) = Mx 0 /k for x ∈ V . We

assume that conditional on M0, processes {Y j

n, n ≥ 0}, j ≥ 1, are

independent.

Krzysztof Burdzy METEOR PROCESS

WIMPs

DEFINITION Suppose that M0 is given and k =

v∈V Mv 0 .

For each j ≥ 1, let {Y j

n, n ≥ 0} be a discrete time symmetric random walk

• n G with the initial distribution P(Y j

0 = x) = Mx 0 /k for x ∈ V . We

assume that conditional on M0, processes {Y j

n, 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 Markov

process {Z j

t , t ≥ 0} by requiring that the embedded discrete Markov chain

for 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−.

Krzysztof Burdzy METEOR PROCESS

WIMPs and convergence rate

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

Krzysztof Burdzy METEOR PROCESS

Rate of convergence on tori

THEOREM Consider the meteor process on a graph G = C d

n (the product of d copies

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

M0 and M0, and suppose that

x Mx 0 = x

Mx

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

exist constants c1, c2 and c3, not depending on G, such that if n ≥ 1 ∨ c1 √d log d and t ≥ c2dn2 then one can define a coupling of mass processes Mt and Mt on a common probability space so that, E

• x∈V

|Mx

t −

Mx

t |

• ≤ exp(−c3t/(dn2))|V |.

Krzysztof Burdzy METEOR PROCESS

Earthworm

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

Krzysztof Burdzy METEOR PROCESS

Earthworms equidistribute soil

THEOREM Fix d ≥ 1 and let Mn

t be the empirical measure process for the earthworm

process on the graph G = C d

n . Assume that Mv 0 = 1/nd for v ∈ V (hence,

• v∈V Mv

0 = 1).

Krzysztof Burdzy METEOR PROCESS

Earthworms equidistribute soil

THEOREM Fix d ≥ 1 and let Mn

t be the empirical measure process for the earthworm

process on the graph G = C d

n . Assume that Mv 0 = 1/nd for v ∈ V (hence,

• v∈V Mv

0 = 1).

(i) For every n, the random measures Mn

t converge weakly to a random

measure Mn

∞, when t → ∞.

Krzysztof Burdzy METEOR PROCESS

Earthworms equidistribute soil

THEOREM Fix d ≥ 1 and let Mn

t be the empirical measure process for the earthworm

process on the graph G = C d

n . Assume that Mv 0 = 1/nd for v ∈ V (hence,

• v∈V Mv

0 = 1).

(i) For every n, the random measures Mn

t 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 uniform probability measure on [0, 1]d.

Krzysztof Burdzy METEOR PROCESS

Craters in circular graphs

G = Ck There is a crater at x at time t if Mx

t = 0.

Krzysztof Burdzy METEOR PROCESS

Craters in circular graphs

G = Ck There is a crater at x at time t if Mx

t = 0.

A crater exists at a site if and only if a meteor hit the site and there were no more recent hits at adjacent sites.

Krzysztof Burdzy METEOR PROCESS

Craters in circular graphs

G = Ck There is a crater at x at time t if Mx

t = 0.

A crater exists at a site if and only if a meteor hit the site and there were no more recent hits at adjacent sites. Under the stationary distribution, the distribution of craters in Ck is the same as the distribution of peaks in a random (uniform) permutation of size k.

Krzysztof Burdzy METEOR PROCESS

Peaks in random permutations

It is possible to find a formula for the probability of a given peak set in a random 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)!.

Krzysztof Burdzy METEOR PROCESS

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

• .

Krzysztof Burdzy METEOR PROCESS

Craters repel each other

THEOREM Consider the meteor process on a circular graph Ck in the stationary

• regime. Let G be the family of adjacent craters, i.e., (i, j) ∈ G if an only if

there are craters at i and j and there are no craters between i and j. For r > 1, let A1

r =

max(i,j)∈G1 |i − j| min(i,j)∈G1 |i − j| ≤ 1 + r

• .

Let H1

n 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 | H1 n) > p. 1 2 3 4 5 6 7 8 9 10

Krzysztof Burdzy METEOR PROCESS

Systems of non-crossing paths

time space

Non-crossing continuous path models: Harris (1965) Spitzer (1968) - shown D¨ urr, 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

Krzysztof Burdzy METEOR PROCESS

Meteor process on Z

1 2 3

• 3
• 1
• 2
• 4

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

Krzysztof Burdzy METEOR PROCESS

Meteor process on Z (2)

1 2 3 4 5

• 1
• 2
• 3
• 4

x y z v

The horizontal axis represents mass. The state of the meteor process at time t is represented as an RCLL function H·

• t. For example, Hx

t = Hy t = 2,

Hv

t = −4 and Hz t = 3.

Krzysztof Burdzy METEOR PROCESS

Jump of meteor process

1 2 3 4 5

• 1
• 2
• 3
• 4

x y z v

Krzysztof Burdzy METEOR PROCESS

Jump of meteor process (2)

1 2 3 4 5

• 1
• 2
• 3
• 4

x y z v

Krzysztof Burdzy METEOR PROCESS

Non-crossing paths

1 2 3 4 5

• 1
• 2
• 3
• 4

x y z v

If x ≤ y then Hx

t ≤ Hy t for all t ≥ 0.

Krzysztof Burdzy METEOR PROCESS

Non-crossing paths

1 2 3 4 5

• 1
• 2
• 3
• 4

x y z v

If x ≤ y then Hx

t ≤ 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 and t ≥ 0, E|Hx

t − Hx 0 |α ≤ c.

Krzysztof Burdzy METEOR PROCESS

Systems of non-crossing paths

time space

Non-crossing continuous path models: Harris (1965) Spitzer (1968) - shown D¨ urr, 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

Krzysztof Burdzy METEOR PROCESS

Systems of non-crossing paths

time space

Non-crossing continuous path models: Harris (1965) Spitzer (1968) - shown D¨ urr, 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 model Free path scaling: dX ≈ (dt)1/2 Non-crossing path scaling: dX ≈ (dt)0

Krzysztof Burdzy METEOR PROCESS