Logarithmic Fluctuations From Circularity

Transcription

1 (MIT) Southeast Probability Conference May 16, 2011

2 Talk Outline Part 1: Logarithmic fluctuations Part 2: Limiting shapes Part 3: Integrality wreaks havoc Part 1: Joint work with David Jerison and Scott Sheffield. Parts 2 & 3: Joint work with Anne Fey and Yuval Peres.

3 Part 1: Logarithmic Fluctuations

4 From random walk to growth model Internal DLA Start with n particles at the origin in the square grid Z 2. Each particle in turn performs a simple random walk until it finds an unoccupied site, stays there. A(n): the resulting random set of n sites in Z 2. Growth rule: Let A(1) = {o}, and A(n + 1) = A(n) {X n (τ n )} where X 1,X 2,... are independent random walks, and τ n = min{t X n (t) A(n)}.

5 Internal DLA cluster in Z 2. Closeup of the boundary.

6 Questions Limiting shape Fluctuations

7 Meakin & Deutch, J. Chem. Phys It is also of some fundamental significance to know just how smooth a surface formed by diffusion limited processes may be. Initially, we plotted ln(ξ) vs ln(l) but the resulting plots were quite noticably curved. Figure 2 shows the dependence of ln(ξ) on ln[ln(l)].

8 History of the Problem Diaconis-Fulton 1991: Addition operation on subsets of Z d. Lawler-Bramson-Griffeath 1992: w.p.1, B (1 ε)r A(πr 2 ) B (1+ε)r eventually. Lawler 1995: w.p.1, B r r 1/3 log 2 r A(πr 2 ) B r+r 1/3 log 4 r eventually. A more interesting question... is whether the errors are o(n α ) for some α < 1/3.

9 Logarithmic Fluctuations Theorem Jerison - L. - Sheffield 2010: with probability 1, B r Clog r A(πr 2 ) B r+clog r eventually. Asselah - Gaudillière 2010 independently obtained B r Clog r A(πr 2 ) B r+clog 2 r eventually.

10 Logarithmic Fluctuations in Higher Dimensions In dimension d 3, let ω d be the volume of the unit ball in R d. Then with probability 1, B r C log r A(ω d r d ) B r+c log r eventually for a constant C depending only on d. (Jerison - L. - Sheffield 2010; Asselah - Gaudillière 2010)

11 Elements of the proof Thin tentacles are unlikely. Martingales to detect fluctuations from circularity. Self-improvement

12 Thin tentacles are unlikely B(z, m) A(n) z A thin tentacle. Lemma. If 0 / B(z,m), then { { P z A(n), #(A(n) B(z,m)) bm d} Ce cm2 /log m, d = 2 Ce cm2, d 3 for constants b,c,c > 0 depending only on the dimension d.

13 Early and late points in A(n), for n = πr 2 B r B r+m B r l A(n) m-early point l-late point

14 Early and late points Definition 1. z is an m-early point if: z A(n), n < π( z m) 2 Definition 2. z is an l-late point if: z / A(n), n > π( z + l) 2 E m [n] = event that some point in A(n) is m-early L l [n] = event that some point in B n/π l is l-late

15 Structure of the argument: Self-improvement LEMMA 1. No l-late points implies no m-early points: If m Cl, then P(E m [n] L l [n] c ) < n 10. LEMMA 2. No m-early points implies no l-late points: If l C(logn)m, then P(L l [n] E m [n] c ) < n 10. Iterate, l C(logn)Cl, which is decreasing until l = C 2 logn.

16 Iterating Lemmas 1 and 2 Fix n and let l,m be the maximal lateness and earliness occurring by time n. Iterate starting from m 0 = n: (l,m) unlikely to belong to a vertical rectangle by Lemma 1. (l,m) unlikely to belong to a horizontal rectangle by Lemma 2.

17 Early and late point detector To detect early points near ζ Z 2, we use the martingale M ζ (n) = z Ã(n) (H ζ (z) H ζ (0)) where H ζ is a discrete harmonic function approximating Re ( ) ζ/ ζ. ζ z B ζ ζ The fine print: Discrete harmonicity fails at three points z = ζ,ζ + 1,ζ i. Modified growth process Ã(n) stops at B ζ (0).

18 Time change of Brownian motion To get a continuous time martingale, we use Brownian motions on the grid Z R R Z instead of random walks. Then there is a standard Brownian motion B ζ such that where s ζ (t) = lim M ζ (t) = B ζ (s ζ (t)) N i=1 is the quadratic variation of M ζ. (M(t i ) M(t i 1 )) 2

19 LEMMA 1. No l-late implies no m = Cl-early Event Q[z, k]: z A(k) \ A(k 1). z is m-early (z A(πr 2 ) for r = z m). E m [k 1] c : No previous point is m-early. L l [n] c : No point is l-late. We will use M ζ for ζ = (1 + 4m/r)z to show for 0 < k n, P(Q[z,k]) < n 20.

20 Main idea: Early but no late would be a large deviation! Recall there is a Brownian motion B ζ such that On the event Q[z,k] and M ζ (n) = B ζ (s ζ (n)). P (M ζ (k) > c 0 m) > 1 n 20 (1) P (s ζ (k) < 100logn) > 1 n 20. (2) On the other hand, (s = 100log n) ( ) P sup B ζ (s ) s s [0,s] e s/2 = n 50.

21 Proof of (1) On the event Q[z,k] P(M ζ (k) > c 0 m) > 1 n 20. Since z A(k) and thin tentacles are unlikely, we have with high probability, #(A(k) B(z,m)) bm 2. For each of these bm 2 points, the value of H ζ is order 1/m, so their total contribution to M ζ (k) is order m. No l-late points means that points elsewhere cannot compensate.

22 Proof of (2): Controlling the Quadratic Variation On the event Q[z,k] P(s ζ (k) < 100log n) > 1 n 20. Lemma: There are independent standard Brownian motions B 1,B 2,... such that s ζ (i + 1) s ζ (i) τ i where τ i is the first exit time of B i from the interval (a i,b i ). a i = min H ζ (z) H ζ (0) z Ã(i) b i = max H ζ (z) H ζ (0). z Ã(i)

23 Proof of (2): Controlling the Quadratic Variation On the event Q[z,k] By independence of the τ i, P(s ζ (k) < 100log n) > 1 n 20. Ee s ζ(k) Ee (τ 1+ +τ k ) = (Ee τ 1 ) (Ee τ k ). By large deviations for Brownian exit times, Ee τ( a,b) ab. Easy to estimate a i, and use the fact that no previous point is m-early to bound b i. Conclude that ] E [e sζ(k) 1 Q n 50.

24 What changes in higher dimensions? In dimension d 3 the quadratic variation s ζ (n) is constant order instead of logn. So the fluctuations are instead dominated by thin tentacles, which can grow to length logn. Still open: prove matching lower bounds on the fluctuations of order logn in dimension 2 and logn in dimensions d 3.

25 Part 2: Limiting Shapes

26 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).

27 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? Recall from part 1: If k = 1, then the limiting shape is a ball in R d. (Lawler-Bramson-Griffeath 92)

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

29 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.

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

31 Divisible Sandpile Given A,B Z d, start with 2 units of mass on each site in A B; and 1 unit of mass 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.

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

33 Diaconis-Fulton sum Divisible sandpile sum

34 The Odometer Function u(x) = total mass emitted from x. (gross, not net) 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 u( u A + 1 B ) = 0.

36 The Obstacle Problem Given A,B Z d, we define the obstacle: γ(x) = x 2 g(x,y) g(x,y), y A y B where g is the Green function for simple random walk g(x,y) = E x #{k X k = y}. (In Z 2, we use the negative of the potential kernel instead.) Let s(x) = inf{φ(x) φ is superharmonic on Z d and φ γ}. Then the odometer function = s γ.

37 Obstacle for two overlapping disks A and B: 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 )

38 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. We define their smash sum A B to be the domain A B = A B {s > γ} where 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}.

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

40 Properties of the Smash Sum A B A B. Associativity: (A B) C = A (B C). Volume conservation: vol(a B) = vol(a) + vol(b). Quadrature identity: If h is an integrable superharmonic function on A B, then h(x)dx h(x)dx + h(x)dx. A B A B

41 Scaling Limit of the Discrete Models Let A,B R d be bounded open sets such that A, B have measure zero. Theorem (L.-Peres) With probability one where D n,r n,i n A B as n, D n, R n, I n are the smash sums of A 1 n Zd and B 1 n Zd, computed using divisible sandpile, rotor-router, and Diaconis-Fulton dynamics, respectively. Convergence is in the sense of ε-neighborhoods: for all ε > 0 (A B) :: ε D n,r n,i n (A B) ε:: for all sufficiently large n.

42 Internal DLA Divisible Sandpile Rotor-Router Model

43 Part 3: Integrality wreaks havoc

44 The Abelian Sandpile as a Growth Model Start with a pile of n chips at the origin in Z d. Each site x = (x 1,...,x d ) Z d has 2d neighbors x ± e i, i = 1,...,d. Any site with at least 2d chips is unstable, and topples by sending one chip to each neighbor. This may create further unstable sites, which also topple. Continue until there are no more unstable sites.

45 Toppling to Stabilize A Sandpile Example: n=16 chips in Z 2. Sites with 4 or more chips are unstable Stable.

46 Abelian Property The final stable configuration does not depend on the order of topplings. Neither does the number of times a given vertex topples.

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

48 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.

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

50 What about background h = 3?

51 Never stops toppling!

52 The 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.

53 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.

54 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.

55 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}.

56 Abelian sandpile (Integrality constraint) Divisible sandpile (No integrality constraint)

57 Sandpile growth rates Let S n,d,h be the set of sites in Z d that topple, if n + h chips start at the origin and h chips start at every other site in Z d. Theorem (Fey-L.-Peres) If h 2d 2, then B cn 1/d S n,d,h B Cn 1/d. Extends earlier work of Fey-Redig and Le Borgne-Rossin.

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

59 A Few Extra Chips Produce An Explosion Let (β(x)) x Z d be independent Bernoulli random variables { 1 with probability ε β(x) = 0 with probability 1 ε. Theorem (Fey-L.-Peres) For any ε > 0, with probability 1, the background 2d 2 + β on Z d is explosive. i.e., for large enough n, adding n chips at the origin causes every site in Z d to topple infinitely many times. Same is true if the extra chips start on an arbitrarily sparse lattice L Z d, provided L meets every coordinate plane {x i = k}.

60 How to Prove An Explosion Claim: If every site in Z d topples at least once, then every site topples infinitely often. Otherwise, let x be the first site to finish toppling. Each neighbor of x topples at least one more time, so x receives at least 2d additional chips. So x must topple again.

61 Straley s Argument for Bootstrap Percolation Let E k be the event that each face of the cube Q k starts with at least one extra chip. Then P(E c k ) 2d(1 ε)k. By Borel-Cantelli, with probability 1 almost all E k occur.

62 An Explosion In Progress Sites colored black are unstable. All sites in Z 2 will topple infinitely often!

63 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.

64 Scale Invariance Conjecture Conjecture: There is a sequence a n and a function f : R d R 0 which is locally constant on an open dense set, such that f n f at all continuity points of f. Now partly proved! Pegden and Smart (arxiv: ) show existence of a weak- limit for f n!

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

66 Locally constant steps of f correspond to periodic patterns:

67 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...

68 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.

69 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

70 Thank You! References: D. Jerison, L. Levine and S. Sheffield, Logarithmic fluctuations for internal DLA. arxiv: L. Levine and Y. Peres, Scaling limits for internal aggregation models with multiple sources, J. d Analyse Math., A. Fey, L. Levine and Y. Peres, Growth rates and explosions in sandpiles, J. Stat. Phys., 2010.