Internal aggregation models on the comb

Internal aggregation models on the comb Wilfried Huss joint work with Ecaterina Sava Cornell Probability Summer School 21. July 2011

Internal Diffusion Limited Aggregation Internal Diffusion Limited Aggregation G infinite Graph; { X n (t) } i.i.d. random walks on G. n 1 IDLA is a random process of subsets A(n) of G. Start with n particles at the origin o G. Each particle performs a random walk, until it visits an unoccupied vertex. A(n) = {set of occupied vertices} is called IDLA cluster. Definition Let A(1) = { o } and recursively σ n = inf { t 0 : X n (t) A(n 1) } P [ A(n) = A(n 1) {z} ] [ = P o X n (σ n ) = z ]. 2 Wilfried Huss Internal aggregation models on the comb

Example Internal Diffusion Limited Aggregation G = Z 2 and X n (t) simple random walks 3 Wilfried Huss Internal aggregation models on the comb

IDLA on lattices Internal Diffusion Limited Aggregation Theorem (Lawler, Bramson & Griffeath, 1992) Let A(n) be the IDLA cluster for simple random walk on Z d. Then for all ε > 0 with P [ B n(1 ε) A(ω d n d ) B n(1+ε), n n ε ] = 1, B r = { x Z d : x < r }, and ω d the volume of the unit ball in R d. 4 Wilfried Huss Internal aggregation models on the comb

IDLA on other state spaces Internal Diffusion Limited Aggregation Green function: G(x, y) = E x [ #{t 0 : X(t) = y} ]. Levelsets of the Green function: { x G : G(o, x) N }. Theorem The levelsets of the Green function are the limiting shape for IDLA with probability 1, for: (Lawler, Bramson & Griffeath, 1992) SRW on Z d. (Blachère, 2002) symmetric random walks on Z d. (Blachère & Brofferio, 2006) symmetric random walks on finitely generated groups with exponential growth. 5 Wilfried Huss Internal aggregation models on the comb

IDLA on other state spaces Internal Diffusion Limited Aggregation Green function: G(x, y) = E x [ #{t 0 : X(t) = y} ]. Levelsets of the Green function: { x G : G(o, x) N }. Theorem The levelsets of the Green function are the limiting shape for IDLA with probability 1, for: (Lawler, Bramson & Griffeath, 1992) SRW on Z d. (Blachère, 2002) symmetric random walks on Z d. (Blachère & Brofferio, 2006) symmetric random walks on finitely generated groups with exponential growth. (H., 2008) strongly reversible, uniformly irreducible random walks on non-amenable graphs. 5 Wilfried Huss Internal aggregation models on the comb

IDLA on other state spaces Internal Diffusion Limited Aggregation Green function: G(x, y) = E x [ #{t 0 : X(t) = y} ]. Levelsets of the Green function: { x G : G(o, x) N }. Theorem The levelsets of the Green function are the limiting shape for IDLA with probability 1, for: (Lawler, Bramson & Griffeath, 1992) SRW on Z d. (Blachère, 2002) symmetric random walks on Z d. (Blachère & Brofferio, 2006) symmetric random walks on finitely generated groups with exponential growth. (H., 2008) strongly reversible, uniformly irreducible random walks on non-amenable graphs. (Shellef, 2010) SRW on percolation cluster of Z d (only inner bound). 5 Wilfried Huss Internal aggregation models on the comb

The Rotor-Router is a deterministic analogue of random walk. For each vertex v G, we define a cyclical ordering of its neighbours. At each vertex we have a rotor (an arrow pointing to one of the neighbours). Transition rule If a particle is at a vertex v: 1. It first changes the rotor at v to point to its next neighbour, 2. then it moves into the direction of the rotor. 6 Wilfried Huss Internal aggregation models on the comb

Example G = Z 2, clockwise rotor sequence 7 Wilfried Huss Internal aggregation models on the comb

Rotor-Router Aggregation Definition Let R(1) = { o } and R(n) = R(n 1) { z n }, for n > 1, where z n is the vertex where the n-th rotor-router walk exits the set R(n 1). R(n) is called rotor-router cluster. 8 Wilfried Huss Internal aggregation models on the comb

Example G = Z 2 9 Wilfried Huss Internal aggregation models on the comb

Rotor-Router Aggregation Theorem (Levine & Peres, 2007) Let R(n) be the rotor-router cluster in Z d, then for all initial rotor configurations B r c log r R(#B r ) B r(1+c r 1/d log r), with B r = { x Z d : x < r }, and constants c, c only depending on the dimension d. 10 Wilfried Huss Internal aggregation models on the comb

Internal growth models on the comb Internal growth models on the comb The comb C 2 is the spanning tree of Z 2 obtained by deleting all horizontal edges of except for the x-axis. o 11 Wilfried Huss Internal aggregation models on the comb

Internal growth models on the comb Internal growth models on the comb The comb C 2 is the spanning tree of Z 2 obtained by deleting all horizontal edges of except for the x-axis. Question: Behaviour of IDLA cluster A(n), for n particles starting at the origin o? Question: Behaviour of rotor-router cluster R(n)? o 11 Wilfried Huss Internal aggregation models on the comb

Internal growth models on the comb IDLA Rotor-Router 12 Wilfried Huss Internal aggregation models on the comb

Internal growth models on the comb Inner bound for IDLA Theorem (H. & Sava) Let A(n) be the IDLA cluster of n particles starting at the origin o, for the simple random walk on the comb C 2. Then, for all ε > 0 with and l = 1 2 B n = P [ B n(1 ε) A(n), for all n n ε ] = 1, { (x, y) C 2 : ( 3 ) 1/3, ( 2 k = 3 ) 2/3. 2 x k + ( ) } y 1/2 n 1/3, l 13 Wilfried Huss Internal aggregation models on the comb

Internal growth models on the comb Inner bound for router-router aggregation Theorem (H. & Sava) Let R(n) be the rotor-router cluster of n particles on the comb C 2. Then, for n n 0 and independently of the initial rotor configuration and the choice of rotor sequence, we have the following inner bound B n R(n), where { B n = (x, y) C 2 : x kn 1/3 c 1 n 1/6, ( y l n 1/3 x ) 2 + c2 x c 3 n 1/3}, k k = ( ) 3 2/3 ( 2 and l = 1 3 ) 1/3 2 2 and c1, c 2 and c 3 are constants. 14 Wilfried Huss Internal aggregation models on the comb

Internal growth models on the comb Inner bound for router-router aggregation 1000 Particles 15 Wilfried Huss Internal aggregation models on the comb

Proof idea Internal growth models on the comb Proof idea Comparison with a growth model based on continuous masses. Sequence µ k : G [0, ) (Sandpiles). We call x unstable at time k, if µ k (x) > 1. Transition rule (µ k µ k+1 ) Choose an unstable vertex x. The sandpile x keeps a mass of 1. The remaining mass is distributed equally among the neighbours of x. 1, if x = y µ k+1 (y) = µ k (y) + 1 d(x) (µ k(x) 1), if y x µ k (y), otherwise. 16 Wilfried Huss Internal aggregation models on the comb

Internal growth models on the comb Proof idea The limit µ = lim k µ k is well defined. Definition (Divisible Sandpile cluster) with µ 0 = n δ o. S(n) = { x G : µ(x) = 1 } S(n) can be obtained as the solution of a discrete obstacle problem. 17 Wilfried Huss Internal aggregation models on the comb

Internal growth models on the comb Proof idea The limit µ = lim k µ k is well defined. Definition (Divisible Sandpile cluster) with µ 0 = n δ o. S(n) = { x G : µ(x) = 1 } S(n) can be obtained as the solution of a discrete obstacle problem. Note: S(n) is believed to be the limiting set of IDLA and Rotor-Router Aggregation for general state spaces. 17 Wilfried Huss Internal aggregation models on the comb