Obstacle Problems and Lattice Growth Models

Transcription

1 (MIT) June 4, 2009 Joint work with Yuval Peres

2 Talk Outline Three growth models Internal DLA Divisible Sandpile Rotor-router model Discrete potential theory and the obstacle problem. Scaling limit and quadrature domains. The abelian sandpile as a growth model Conjectures about pattern formation: Scale invariance Dimensional reduction

3 Internal DLA with Multiple Sources Finite set of points x 1,...,x k Z d. Start with m particles at each site x i.

4 Internal DLA with Multiple Sources 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.

5 Internal DLA with Multiple Sources 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).

6 100 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.

7 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.

8 Questions Fix sources x 1,...,x k R d. Run internal DLA on 1 n Zd with n d particles per source.

9 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?

10 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?

11 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.

12 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.

13 Overlapping Internal DLA Clusters Idea: First let the particles at each source x i perform internal DLA ignoring the particles from the other sources.

14 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.

15 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?

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

17 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.

18 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 }.

19 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.

20 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.

21 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.

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

23 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.

24 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.

25 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.

26 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.

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

28 Diaconis-Fulton sum Divisible sandpile sum

29 Odometer Function u(x) = total mass emitted from x.

30 Odometer Function u(x) = total mass emitted from x. Discrete Laplacian: u(x) = 1 2d u(y) u(x) y x

31 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

32 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.

33 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).

34 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.

35 Free Boundary Problem Unknown function u, unknown domain D = {u > 0}. u 0 u 1 1 A 1 B

36 Free Boundary Problem Unknown function u, unknown domain D = {u > 0}. u 0 u 1 1 A 1 B u( u A + 1 B ) = 0.

37 Least Superharmonic Majorant Given A,B Z d, let γ(x) = x 2 g(x,y) g(x,y), y A y B

38 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}.

39 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}.

40 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}. Then the odometer function is u = s γ.

41 The Smash Sum of Two Domains in R d A,B R d bounded open sets such that A, B have zero d-dimensional Lebesgue measure.

42 The Smash Sum of Two Domains in R d A,B R d bounded open sets such that A, B have zero d-dimensional Lebesgue measure. The smash sum of A and B is the domain A B = A B {s > γ}

43 The Smash Sum of Two Domains in R d A,B R d bounded open sets such that A, B have zero d-dimensional Lebesgue measure. The smash sum of A and B is the domain A B = A B {s > γ} where Z Z γ(x) = x 2 g(x,y)dy g(x,y)dy A B

44 The Smash Sum of Two Domains in R d A,B R d bounded open sets such that A, B have zero d-dimensional Lebesgue measure. The smash sum of A and B is the domain A B = 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.

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

46 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 )

47 The smash sum A B = A B {s > γ} of two overlapping disks A,B R 2.

48 Properties of the Smash Sum Associativity: (A B) C = A (B C). Analogous to the abelian property of the divisible sandpile. Volume conservation: vol(a B) = vol(a) + vol(b). Analogous to mass conservation for the divisible sandpile.

49 Properties of the Smash Sum Associativity: (A B) C = A (B C). Analogous to the abelian property of the divisible sandpile. Volume conservation: vol(a B) = vol(a) + vol(b). Analogous to mass conservation for the divisible sandpile. Quadrature identity: If h is an integrable superharmonic function on A B, then Z Z Z h(x)dx h(x)dx + h(x)dx. A B A B One can also take this as the defining property of the smash sum (Gustafsson 88).

50 Two Physical Interpretations of the Smash Sum Hele-Shaw stamping problem: Blob of incompressible fluid in the narrow gap between two plates. Initial shape of the blob is A B. Stamp the plates together on A B. Fluid will expand to fill A B.

51 Two Physical Interpretations of the Smash Sum Hele-Shaw stamping problem: Blob of incompressible fluid in the narrow gap between two plates. Initial shape of the blob is A B. Stamp the plates together on A B. Fluid will expand to fill A B. Electrostatic interpretation (S. Sheffield): Positively charged solid in A B (charge density +1). Neutral solid in A B A B. Negatively charged fluid (charge density 1) outside A B. Total energy is minimized when the fluid occupies A B A B.

52 Scaling Limit of the Discrete Models Let A,B R d be bounded open sets such that A, B have measure zero.

53 Scaling Limit of the Discrete Models Let A,B R d be bounded open sets such that A, B have measure zero. Lattice spacing δ n 0.

54 Scaling Limit of the Discrete Models 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 D n,r n,i n D as n,

55 Scaling Limit of the Discrete Models 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.

56 Scaling Limit of the Discrete Models 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}.

57 Scaling Limit of the Discrete Models 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.

58 Internal DLA Divisible Sandpile Rotor-Router Model

59 Steps of the Proof convergence of densities convergence of obstacles

60 Steps of the Proof convergence of densities convergence of obstacles convergence of odometer functions

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

62 Multiple Point Sources Fix centers x 1,...,x k R d and λ 1,...,λ k > 0.

63 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, For any ε > 0, with probability one for all sufficiently large n, D :: ε D n,r n,i n D ε::

64 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, For any ε > 0, with probability one D :: ε D n,r n,i n D ε:: for all sufficiently large 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.

65 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, For any ε > 0, with probability one D :: ε D n,r n,i n D ε:: for all sufficiently large 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 balls B(x i,r i ), where λ i = ω d ri d.

66 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, For any ε > 0, with probability one D :: ε D n,r n,i n D ε:: for all sufficiently large 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 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.

67 Quadrature Domains Given x 1,...x k R d and λ 1,...,λ k > 0. A domain D R d satisfying Z D h(x)dx k i=1 λ i h(x i ). for all integrable superharmonic functions h on D is called a quadrature domain. (Aharonov-Shapiro 76, Gustafsson, Sakai, Putinar,...)

68 Quadrature Domains Given x 1,...x k R d and λ 1,...,λ k > 0. A domain D R d satisfying Z D h(x)dx k i=1 λ i h(x i ). for all integrable superharmonic functions h on D is called a quadrature domain. (Aharonov-Shapiro 76, Gustafsson, Sakai, Putinar,...) 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.

69 Quadrature Domains Given x 1,...x k R d and λ 1,...,λ k > 0. A domain D R d satisfying Z D h(x)dx k i=1 λ i h(x i ). for all integrable superharmonic functions h on D is called a quadrature domain. (Aharonov-Shapiro 76, Gustafsson, Sakai, Putinar,...) 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. In dimension two, the boundary of B 1... B k lies on an algebraic curve of degree 2k.

70 ZZ 1 πr 2 h(x,y)dx dy h( 1,0) + h(1,0) D

71 The Abelian Sandpile as a Growth Model Start with n chips at the origin in Z d. If a site has at least 2d chips, it topples by sending one chip to each of the 2d neighboring sites.

72 The Abelian Sandpile as a Growth Model Start with n chips at the origin in Z d. If a site has at least 2d chips, it topples by sending one chip to each of the 2d neighboring sites. Abelian property: The final chip configuration does not depend on the order of the firings.

73 The Abelian Sandpile as a Growth Model Start with n chips at the origin in Z d. If a site has at least 2d chips, it topples by sending one chip to each of the 2d neighboring sites. Abelian property: The final chip configuration does not depend on the order of the firings. Bak-Tang-Wiesenfeld 87, Dhar 90,...

74 Sandpile of 1,000,000 chips in Z 2

75 Growth on a General Background Let each site x Z d start with σ(x) chips. (σ(x) 2d 1) We call σ the background configuration. Place n additional chips at the origin. Let S n,σ be the set of sites that topple.

76 Constant Background σ h h = 2 h = 1 h = 0

77 The Square Sandpile: d = h = 2

78 Closeup of the Lower Left Corner

79 Odometer Function u(x) = number of times x topples.

80 Odometer Function u(x) = number of times x topples. Discrete Laplacian: u(x) = u(y) 2d u(x) y x

81 Odometer Function u(x) = number of times x topples. Discrete Laplacian: u(x) = u(y) 2d u(x) y x = chips received chips emitted

82 Odometer Function u(x) = number of times x topples. Discrete Laplacian: u(x) = u(y) 2d u(x) y x = chips received chips emitted = τ (x) τ(x) where τ is the initial unstable chip configuration and τ is the final stable configuration.

83 Stabilizing Functions Given a chip configuration τ on Z d and a function u 1 : Z d Z, call u 1 stabilizing for τ if τ + u 1 2d 1.

84 Stabilizing Functions Given a chip configuration τ on Z d and a function u 1 : Z d Z, call u 1 stabilizing for τ if τ + u 1 2d 1. If u 1 and u 2 are stabilizing for τ, then τ + min(u 1,u 2 ) τ + max( u 1, u 2 ) = max(τ + u 1,τ + u 2 ) 2d 1 so min(u 1,u 2 ) is also stabilizing for τ.

85 Least Action Principle Let τ be a chip configuration on Z d that stabilizes after finitely many topplings, and let u be its odometer function. Least Action Principle: If u 1 : Z d Z 0 is stabilizing for τ, then u u 1.

86 Least Action Principle Let τ be a chip configuration on Z d that stabilizes after finitely many topplings, and let u be its odometer function. Least Action Principle: If u 1 : Z d Z 0 is stabilizing for τ, then u u 1. So the odometer is minimal among all nonnegative stabilizing functions: u(x) = min{u 1 (x) u 1 0 is stabilizing for τ}. Interpretation: Sandpiles are lazy.

87 Proof of LAP Odometer function u, stabilizing function u 1. Want u u 1. Perform legal topplings in any order, without allowing any site x to topple more than u 1 (x) times, until no such toppling is possible.

88 Proof of LAP Odometer function u, stabilizing function u 1. Want u u 1. Perform legal topplings in any order, without allowing any site x to topple more than u 1 (x) times, until no such toppling is possible. Get a function u u 1 and chip configuration τ = τ + u. If τ is stable, then u = u by the abelian property.

89 Proof of LAP Odometer function u, stabilizing function u 1. Want u u 1. Perform legal topplings in any order, without allowing any site x to topple more than u 1 (x) times, until no such toppling is possible. Get a function u u 1 and chip configuration τ = τ + u. If τ is stable, then u = u by the abelian property. Otherwise, τ has some unstable site y, and u (y) = u 1 (y).

90 Proof of LAP Odometer function u, stabilizing function u 1. Want u u 1. Perform legal topplings in any order, without allowing any site x to topple more than u 1 (x) times, until no such toppling is possible. Get a function u u 1 and chip configuration τ = τ + u. If τ is stable, then u = u by the abelian property. Otherwise, τ has some unstable site y, and u (y) = u 1 (y). Further topplings according to u 1 u can only increase the number of chips at y. But y is stable in τ + u 1.

91 Obstacle Problem with an Integrality Condition Lemma. The abelian sandpile odometer function is given by u = s γ where { } f : Z d R is superharmonic s(x) = min f (x). and f γ is Z 0 -valued

92 Obstacle Problem with an Integrality Condition Lemma. The abelian sandpile odometer function is given by u = s γ where { } f : Z d R is superharmonic s(x) = min f (x). and f γ is Z 0 -valued The obstacle γ is given by γ(x) = (2d 1) x 2 + n g(o,x) 2d where g is the Green s function for simple random walk in Z d g(o,x) = E o #{k X k = x}.

93

94 Bootstrapping From Small Values of h Theorem (L.-Peres) Let S n,h be the set of sites visited by the abelian sandpile in Z d, starting from n chips at the origin and constant background h d 1.

95 Bootstrapping From Small Values of h Theorem (L.-Peres) Let S n,h be the set of sites visited by the abelian sandpile in Z d, starting from n chips at the origin and constant background h d 1. Then ( Ball of volume n o(n) 2d 1 h ) S n,h ( Ball of volume n + o(n) ). d h

96 Bootstrapping From Small Values of h Theorem (L.-Peres) Let S n,h be the set of sites visited by the abelian sandpile in Z d, starting from n chips at the origin and constant background h d 1. Then ( Ball of volume n o(n) 2d 1 h ) S n,h ( Ball of volume n + o(n) ). d h Improves earlier bounds of Le Borgne and Rossin, Fey and Redig.

97 Bounds for the Abelian Sandpile Shape (Disk of area n/3) S n (Disk of area n/2)

98 Growth Rate of The Square Sandpile Theorem (Fey-L.-Peres) Let S n,2 be the set of sites in Z 2 that topple if n + 2 chips start at the origin and 2 chips start at every other site in Z 2.

99 Growth Rate of The Square Sandpile Theorem (Fey-L.-Peres) Let S n,2 be the set of sites in Z 2 that topple if n + 2 chips start at the origin and 2 chips start at every other site in Z 2. Then for any ε > 0, we have S n,2 Q r for all sufficiently large n, where ( 2 n r = π + ε) and Q r = {x Z 2 : x 1, x 2 r}.

100 Growth Rate of The Square Sandpile Theorem (Fey-L.-Peres) Let S n,2 be the set of sites in Z 2 that topple if n + 2 chips start at the origin and 2 chips start at every other site in Z 2. Then for any ε > 0, we have S n,2 Q r for all sufficiently large n, where ( 2 n r = π + ε) and Q r = {x Z 2 : x 1, x 2 r}. Similar bound with r = Θ(n 1/d ) in d dimensions, for any constant background h 2d 2.

101 A Mystery: Scale Invariance Big sandpiles look like scaled up small sandpiles! Let σ n (x) be the final number of chips at x in the sandpile of n particles on Z d.

102 A Mystery: Scale Invariance Big sandpiles look like scaled up small sandpiles! Let σ n (x) be the final number of chips at x in the sandpile of n particles on Z d. Squint your eyes: for x R d let f n (x) = 1 a 2 n y Z d y nx a n σ n (y). where a n is a sequence of integers such that a n and a n n 0.

103 Scale Invariance Conjecture Conjecture: There is a sequence a n and a function f : R d R 0 which is locally constant almost everywhere, such that f n f at all continuity points of f.

104 Two Sandpiles of Different Sizes n = 100,000 n = 200,000 (scaled down by 2)

105 Locally constant steps of f correspond to periodic patterns:

106 A Mystery: Dimensional Reduction Our argument used simple properties of one-dimensional sandpiles to bound the diameter of higher-dimensional sandpiles. Deepak Dhar pointed out that there seems to be a deeper relationship between sandpiles in d and d 1 dimensions...

107 Dimensional Reduction Conjecture σ n,d : sandpile of n chips on background h = 2d 2 in Z d. Conjecture: For any n there exists m such that σ n,d (x 1,...,x d 1,0) = 2 + σ m,d 1 (x 1,...,x d 1 ) for almost all x sufficiently far from the origin.

108 A Two-Dimensional Slice of A Three-Dimensional Sandpile d = 3 (slice through origin) d = 2 h = 4 h = 2 n = 5,000,000 m = 46,490

109 Thank You! arxiv: arxiv: