METEOR PROCESS. Krzysztof Burdzy University of Washington

University of Washington

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

Mass redistribution

Mass redistribution (2) space time

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 set Mt x - mass at x V at time t Assumption: M0 x [0, ) for all x V M t = {Mt x, x V } Nt x - 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.

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

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)

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)

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.

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

Circular graphs 8 7 6 5 9 4 10 1 2 3 C k - circular graph with k vertices Q k - stationary distribution for M t on C k

Circular graphs - moments of mass at a vertex 8 7 6 5 9 10 1 2 3 4 THEOREM E Qk M x 0 = 1, x V, k 1

Circular graphs - moments of mass at a vertex 8 7 6 5 9 10 1 2 3 4 THEOREM E Qk M x 0 = 1, x V, k 1 lim Var Q k M0 x = 1, k x V

Circular graphs - moments of mass at a vertex 8 7 6 5 9 10 1 2 3 4 THEOREM E Qk M x 0 = 1, x V, k 1 lim Var Q k M0 x = 1, k x V lim Cov Q k (M0 x, M0 x+1 ) = 1/2, x V k

Circular graphs - moments of mass at a vertex 8 7 6 5 9 10 1 2 3 4 THEOREM E Qk M x 0 = 1, x V, k 1 lim Var Q k M0 x = 1, k x V lim Cov Q k (M0 x, M0 x+1 ) = 1/2, x V k lim k Cov Q k (M x 0, M y 0 ) = 0, x y

Circular graphs - correlation and independence 8 7 6 5 9 10 1 2 3 4 lim Cov Q k (M0 x, M y 0 ) = 0, k x y If x y then M0 x and My 0 do not appear to be asymptotically independent under Q k s. (M0 x)2 and M y 0 seem to be asymptotically correlated under Q k, if x y.

From circular graphs to Z C k - circular graph with k vertices Q k - stationary distribution for M t on C k THEOREM For every fixed n, the distributions of (M 1 0, M2 0,..., Mn 0 ) under Q k 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.

Moments of mass at a vertex in Z d THEOREM E Q M x 0 = 1, x V Var Q M x 0 = 1, x V Cov Q (M x 0, M y 0 ) = 1 2d, Cov Q (M x 0, M y 0 ) = 0, x y x y

Mass fluctuations in intervals of Z THEOREM For every n, E Q 1 j n M j 0 = n,

Mass fluctuations in intervals of Z THEOREM For every n, E Q Var Q 1 j n 1 j n M j 0 = n, M j 0 = 1.

Flow across the boundary x x+1 F x t - net flow from x to x + 1 between times 0 and t

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.

Mass distribution at a vertex of Z A simulation of M x 0 under Q. 20 000 15 000 10 000 5000

Mass distribution at a vertex of Z (2) 20 000 15 000 10 000 5000 P Q (M0 x = 0) = 1/3 E Q M0 x = 1, Var Q M0 x = 1 Is Q a mixture of a gamma distribution and an atom at 0? No.

Mass distribution at a vertex of Z (2) 20 000 15 000 10 000 5000 P Q (M0 x = 0) = 1/3 E Q M0 x = 1, Var Q M0 x = 1 Is Q a mixture of a gamma distribution and an atom at 0? No. One can find an exact and rigorous value for E Qk (M x 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 Mx 0 = k. Let S be the simplex consisting of all {S x, x V } with S x 0 for all x V and x V S x = k.

Support of the stationary distribution Assume that V = k, and x V Mx 0 = k. Let S be the simplex consisting of all {S x, x V } with S x 0 for all x V and x V S x = k. Let S be the set of {S x, x V } with S x = 0 for at least one x V.

WIMPs DEFINITION Suppose that M 0 is given and k = v V Mv 0. For each j 1, let {Yn, j n 0} be a discrete time symmetric random walk on G with the initial distribution P(Y j 0 = x) = Mx 0 /k for x V. We assume that conditional on M 0, processes {Yn, j n 0}, j 1, are independent. Recall Poisson processes N v and assume that they are independent of {Yn, j n 0}, j 1. For every j 1, we define a continuous time Markov process {Zt j, 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 N v jumps at time t, where v = Z j t.

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

Rate of convergence on tori THEOREM Consider the meteor process on a graph G = Cn d (the product of d copies of the cycle C n ). Consider any distributions (possibly random) of mass M 0 and M 0, and suppose that x Mx 0 = M x 0 x = V = nd, a.s. There exist constants c 1, c 2 and c 3, not depending on G, such that if n 1 c 1 d log d and t c2 dn 2 then one can define a coupling of mass processes M t and M t on a common probability space so that, ( ) E Mt x M t x exp( c 3 t/(dn 2 )) V. x V

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

Earthworms equidistribute soil THEOREM Fix d 1 and let M n t be the empirical measure process for the earthworm process on the graph G = C d n. Assume that M v 0 = 1/nd for v V (hence, v V Mv 0 = 1).

Earthworms equidistribute soil THEOREM Fix d 1 and let M n t be the empirical measure process for the earthworm process on the graph G = C d n. Assume that M v 0 = 1/nd for v V (hence, v V Mv 0 = 1). (i) For every n, the random measures M n t converge weakly to a random measure M n, when t. (ii) For R R d and a R, let ar = {x R d : x = ay for some y R} and M n (R) = M n (nr). When n, the random measures M n converge weakly to the random measure equal to, a.s., the uniform probability measure on [0, 1] d.

Craters in circular graphs G = C k There is a crater at x at time t if M x t = 0.

Craters in circular graphs G = C k There is a crater at x at time t if M x 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.

Craters in circular graphs G = C k There is a crater at x at time t if M x 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 C k is the same as the distribution of peaks in a random (uniform) permutation of size k.

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) = P(no craters at 1, 2,..., n) = 2n+1 (n + 2)!. n(n + 3)2n+1, (n + 4)!

Peaks in random permutations THEOREM P(crater at 1 followed by i non-craters, then a crater, then exactly j non-craters) [ ( 2 i+j ( i + j + 1 = (i + j + 4) j (i + j + 5)! i 1 ( ) ( i + j + 1 i + j + 1 + (i + 1) + i i + 1 i + 2 ) + (j + 1) ) 2(i + j + 1) ( ) i + j + 1 i ) ( ) ] i + j + 4 + ij. i + 2

Craters repel each other THEOREM Consider the meteor process on a circular graph C k 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 { } A 1 max(i,j) G1 i j r = min (i,j) G1 i j 1 + r. Let H 1 n be the event that there are exactly n craters at time 0. For every n 2, p < 1 and r > 1 there exists k 1 < such that for all k k 1, P(A 1 r H 1 n) > p. 8 7 6 5 9 4 10 1 2 3

Systems of non-crossing paths space time Non-crossing continuous path models: Harris (1965) Spitzer (1968) - shown Dürr, 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 3 2 1 0-1 -2-3 -4 There is no time scale in this picture. The horizontal axis represents mass.

Meteor process on Z (2) v 5 4 3 2 1 0-1 -2-3 -4 x y z The horizontal axis represents mass. The state of the meteor process at time t is represented as an RCLL function H t. For example, H x t = H y t = 2, H v t = 4 and H z t = 3.

Jump of meteor process v 5 4 3 2 1 0-1 -2-3 -4 x y z

Jump of meteor process (2) v 5 4 3 2 1 0-1 -2-3 -4 x y z

Non-crossing paths v 5 4 3 2 1 0-1 -2-3 -4 x y z If x y then H x t H y t for all t 0.

Non-crossing paths v 5 4 3 2 1 0-1 -2-3 -4 x y z If x y then H x t H y 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 H x t H x 0 α c.