The Limiting Shape of Internal DLA with Multiple Sources

Transcription

1 The Limiting Shape of Internal DLA with Multiple Sources January 30, 2008 (Mostly) Joint work with Yuval Peres

2 Finite set of points x 1,...,x k Z d. Start with m particles at each site x i. Each particle performs simple random walk in Z d until reaching an unoccupied site. Get a random set of km occupied sites in Z d. The distribution of this random set does not depend on the order of the walks (Diaconis-Fulton 91).

3 100 point sources arranged on a grid in Z 2. Sources are at the points (50i,50j) for 0 i,j 9. Each source started with 2200 particles.

4 50 point sources arranged at random in a box in Z 2. The sources are iid uniform in the box [0,500] 2. Each source started with 3000 particles.

5 Questions Fix sources x 1,...,x k R d. Run internal DLA on 1 n Zd with n d particles per source. As the lattice spacing goes to zero, is there a scaling limit? If so, can we describe the limiting shape? Lawler-Bramson-Griffeath 92 studied the case k = 1: For a single source, the limiting shape is a ball in R d. Not clear how to define dynamics in R d.

6 Limiting Shape Fix points x 1,...,x k R d and λ 1,...,λ k > 0. Let I n be the random set of occupied sites for internal DLA in the lattice 1 n Zd, if λ i n d particles start at each site x i :: (closest lattice site to x i ). Theorem (L.-Peres) There exists a deterministic domain D R d such that I n D as n in the following sense: for any ε > 0, with probability one D :: ε I n D ε:: for all sufficiently large n, where D ε,d ε are the inner and outer ε-neighborhoods of D.

7 Overlapping Internal DLA Clusters Idea: First let the particles at each source x i perform internal DLA ignoring the particles from the other sources. Get k overlapping internal DLA clusters, each of which is close to a ball. Hard part: How does the shape change when the particles in the overlaps continue walking until they reach unoccupied sites?

8 Two-source internal DLA cluster built from overlapping single-source clusters.

9 Diaconis-Fulton Addition Finite sets A,B Z d. In our application, A and B will be overlapping internal DLA clusters from two different sources. Write A B = {y 1,...,y k }. To form A + B, let C 0 = A B and C j = C j 1 {z j } where z j is the endpoint of a simple random walk started at y j and stopped on exiting C j 1. Define A + B = C k. Abeilan property: the law of A + B does not depend on the ordering of y 1,...,y k.

10 Diaconis-Fulton sum of two squares in Z 2 overlapping in a smaller square.

11 Divisible Sandpile Given A,B Z d, start with mass 2 on each site in A B; and mass 1 on each site in A B A B. At each time step, choose x Z d with mass m(x) > 1, and distribute the excess mass m(x) 1 equally among the 2d neighbors of x. As t, get a limiting region A B Z d of sites with mass 1. Sites in (A B) have fractional mass. Sites outside have zero mass. Abelian property: A B does not depend on the choices.

12 Divisible sandpile sum of two squares in Z 2 overlapping in a smaller square.

13 Diaconis-Fulton sum Divisible sandpile sum

14 Odometer Function u(x) = total mass emitted from x. Discrete Laplacian: u(x) = 1 2d u(y) u(x) y x = mass received mass emitted = 1 1 A (x) 1 B (x), x A B. Boundary condition: u = 0 on (A B). Need additional information to determine the domain A B.

15 Free Boundary Problem Unknown function u, unknown domain D = {u > 0}. u 0 Alternative formulation: u 1 1 A 1 B u( u A + 1 B ) = 0. u = 1 1 A 1 B on D; u = u = 0 on D.

16 Least Superharmonic Majorant Given A,B Z d, let γ(x) = x 2 g(x,y) g(x,y), y A y B where g is the Green s function for simple random walk g(x,y) = E x #{k X k = y}. Let s(x) = inf{φ(x) φ is superharmonic on Z d and φ γ}. Claim: odometer = s γ.

17 Proof of the claim Claim: odometer = s γ. Let m(x) = amount of mass present at x in the final state. Then u = m 1 A 1 B 1 1 A 1 B. Since γ = 1 A + 1 B 1 the sum u + γ is superharmonic, so u + γ s. Reverse inequality: s γ u is superharmonic on A B and nonnegative outside A B, hence nonnegative inside as well.

18 Defining the Scaling Limit A,B R d bounded open sets such that A, B have measure zero Let D = A B {s > γ} where Z Z γ(x) = x 2 g(x,y)dy g(x,y)dy A B and s(x) = inf{φ(x) φ is continuous, superharmonic, and φ γ} is the least superharmonic majorant of γ. Odometer: u = s γ.

19 Obstacle for two overlapping disks A and B: Z Z γ(x) = x 2 g(x,y)dy g(x,y)dy A B Obstacle for two point sources x 1 and x 2 : γ(x) = x 2 g(x,x 1 ) g(x,x 2 )

20 The domain D = {s > γ} for two overlapping disks in R 2. The boundary D is given by the algebraic curve ( x 2 + y 2) 2 2r 2 ( x 2 + y 2) 2(x 2 y 2 ) = 0.

21 Main Result Let A,B R d be bounded open sets such that A, B have measure zero. Lattice spacing δ n 0. Theorem (L.-Peres) With probability one where D n,r n,i n D as n, Dn, R n, I n are the smash sums of A δ n Z d and B δ n Z d, computed using divisible sandpile, rotor-router, and Diaconis-Fulton dynamics, respectively. D = A B {s > γ}. Convergence is in the sense of ε-neighborhoods: for all ε > 0 D :: ε D n,r n,i n D ε:: for all sufficiently large n.

22 Internal DLA Divisible Sandpile Rotor-Router Model

23 Steps of the Proof convergence of densities convergence of obstacles convergence of odometer functions convergence of domains.

24 Multiple Point Sources Fix centers x 1,...,x k R d and λ 1,...,λ k > 0. Theorem (L.-Peres) With probability one D n,r n,i n D as n, where Dn, R n, I n are the domains of occupied sites δ n Z d, if λ i δ d n particles start at each site x i :: and perform divisible sandpile, rotor-router, and Diaconis-Fulton dynamics, respectively. D is the smash sum of the balls B(x i,r i ), where λ i = ω d ri d. Follows from the main result and the case of a single point source.

25 A Quadrature Identity If h is harmonic on δ n Z d, then M t = h(xt j ) j is a martingale for internal DLA, where (X j t ) t 0 is the random walk performed by the j-th particle. Optional stopping: E h(x) = EM T = M 0 = x I n k i=1 λ i δn d h(x i ). Therefore if I n D, we expect the limiting domain D R d to satisfy Z k h(x)dx = λ i h(x i ). D i=1 for all harmonic functions h on D.

26 Quadrature Domains Given x 1,...x k R d and λ 1,...,λ k > 0. D R d is called a quadrature domain for the data (x i,λ i ) if Z D h(x)dx k i=1 λ i h(x i ). for all superharmonic functions h on D. (Aharonov-Shapiro 76, Gustafsson, Sakai,...) The smash sum B 1... B k is such a domain, where B i is the ball of volume λ i centered at x i. The boundary of B 1... B k lies on an algebraic curve of degree 2k.

27 ZZ D h(x,y)dx dy = h( 1,0) + h(1,0)

28 Inverse Problems Subtraction: Given domains B A R d, can we find a domain C such that A = B C? Trivial solution C = A \ B. Is there a simply connected solution? Division: Can we find D such that A = D D?

29 Erosion Algorithm Given B A Z d, start with mass 1 at each site in B. Sites on the boundary of A begin eroded. Sites in the interior of A begin uneroded. At each time step, each uneroded site distributes its mass equally among its neighbors, subject to the condition that no eroded site attains mass > 1. If an eroded site x attains mass 1, all of its neighbors become eroded. Claim: When this process stops, the set C of uneroded sites satisfies A = B C.

30 Half of a Domain To find a domain D such that A = D D, use the same erosion algorithm, starting with mass 1 2 at every site in A. Half of a square: Why it works: If m(x) is the final amount of mass at x, then m = A + u where u is the odometer of the erosion process. Also m = 1 A 1 D, where D is the set of uneroded sites. Hence 1 A = 2 1 D + 2 u.

31 Internal Erosion, or Negative Sources Given a finite set A Z d containing the origin. Start a simple random walk at the origin. Stop the walk when it reaches a site x A adjacent to the complement of A. Let e(a) = A {x}. We say that x is eroded from A. Iterate this operation until the origin is eroded. The resulting random set is called the internal erosion of A.

32 Internal erosion of a disk of radius 250 in Z 2.

33 Internal erosion of a box of side length 500 in Z 2.

34 Questions How many sites are eroded? If A is (say) a square of side length n in Z 2, we would guess that E#eroded sites = Θ(n α ) for some 1 < α < 2. Analogy with diffusion-limited aggregation. What is the probability that a given site x is eroded? For some sites, is this probability o(n α 2 )?

35 Probability of a given site being eroded from a box in Z 2.

36 Internal Erosion in One Dimension Interval A = [ m,n] Z with m 0 n. Transition probabilities are given by gambler s ruin: P([ m,n],[ m,n 1]) = P([ m,n],[1 m,n]) = m m + n ; n m + n. How large is the remaining interval when the origin gets eroded? Urn model: choose a ball at random, then remove a ball from the other urn. OK Corral Process: Gunfight with m fighters on one side and n on the other. Williams-McIlroy 98, Kingman 99, Kingman-Volkov 03.

37 The Number of Surviving Gunners Let m = n (an equal gunfight). Theorem (Kingman-Volkov 03) Starting from the interval [ n,n], let R(n) be the number of sites remaining when the origin is eroded. Then as n R(n) = n3/4 where Z is a standard Gaussian. ( ) 8 1/4 Z (1) 3

38 Two Mystery Shapes Cardioid??