Internal aggregation models on the comb

Transcription

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

2 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

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

4 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

5 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. 5 Wilfried Huss Internal aggregation models on the comb

6 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. 5 Wilfried Huss Internal aggregation models on the comb

7 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

8 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

9 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

10 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). 6 Wilfried Huss Internal aggregation models on the comb

11 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

27 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

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

29 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

30 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

31 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

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

33 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

34 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

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

36 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) > Wilfried Huss Internal aggregation models on the comb

37 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

38 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

39 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