Limit shapes outside the percolation cone

Transcription

1 jointly with Tuca Auffinger and Mike Hochman Princeton University July 14, 2011

2 Definitions We consider first-passage percolation on the square lattice Z 2. x γ - We place i.i.d. non-negative passage times (τ e ) on the edges of the square lattice. - The passage time of a path γ is τ(γ) = τ e. e γ y

3 Definitions We consider first-passage percolation on the square lattice Z 2. x - The passage time between two vertices x and y is γ τ(x, y) = min γ:x y τ(γ). y - If τ e 0 then τ is a metric.

4 Definitions We consider first-passage percolation on the square lattice Z 2. B(t) - The ball B(t) = {x : τ(0, x) t} grows as t increases. - Richardson and Cox-Durrett proved a (very strong) law of large numbers for B(t).

5 Shape theorem Theorem (Cox-Durrett) Let µ have finite mean and be such that µ({0}) < p c, the critical probability for bond percolation on Z 2. There exists a deterministic compact, convex set B µ with non-empty interior such that for each ε > 0, ( P (1 ε)b µ 1 ) t B(t) (1 + ε)b µ for all large t = 1.

6 FPP questions QUESTIONS: (1) Which convex, compact sets B are realizable as B µ s? - Solved by Häggstrom-Meester in the case of stationary FPP. - Some insight in i.i.d. case by Durrett-Liggett, Marchand.

7 FPP questions QUESTIONS: (1) Which convex, compact sets B are realizable as B µ s? - Solved by Häggstrom-Meester in the case of stationary FPP. - Some insight in i.i.d. case by Durrett-Liggett, Marchand. (2) What is the relationship between µ and B µ? How exactly does B µ depend on µ? - Unlike in other lattice models, the lattice matters!

8 FPP questions QUESTIONS: (1) Which convex, compact sets B are realizable as B µ s? - Solved by Häggstrom-Meester in the case of stationary FPP. - Some insight in i.i.d. case by Durrett-Liggett, Marchand. (2) What is the relationship between µ and B µ? How exactly does B µ depend on µ? - Unlike in other lattice models, the lattice matters! (3) What are the fluctuations of B(t) about tb µ? We will focus on questions 1 and 3, though our technique sheds some light on question 2.

9 Possible limit shapes To study question 1, let sides(c) be the number of extreme points of a compact convex set C. - One might think that B µ is a disk (forgets the lattice). This is not true. - It is believed that at least sides(b µ ) = for all but the most degenerate cases (e.g., µ = δ a for a > 0).

10 Possible limit shapes - Previous results: - By symmetry, sides(b µ ) 4. - Marchand (2002) proved that for a class of µ s, sides(b µ ) 8. - With M. Hochman, we showed there are µ s such that sides(b µ ) =. (Proof in Examples of non-polygonal limit shapes for i.i.d. first-passage percolation and infinite coexistence in spatial growth models on arxiv.)

11 Flat edges We consider a subclass M p of measures µ such that µ({1}) = p > 0 and supp(µ) [1, ). p 1 - In 1981, Durrett and Liggett showed that for µ M p (for p large), B µ has some flat edges. - In 2002, Marchand found the exact location of these edges.

12 Flat edges Imagine that p is large: there are many 1-edges (e s with τ e = 1). - For n large, consider the line segment P x+y=n L n = {(x, y) Z 2 : x+y = n} [0, ) 2 in the first quadrant. - If z L n then we want to estimate the passage time τ(0, z).

13 Flat edges Imagine that p is large: there are many 1-edges (e s with τ e = 1). - Every path P from 0 to z has at least n edges, so P x+y=n τ(p) [inf supp(µ)] P = P n. Therefore τ(0, z) n.

14 Oriented percolation To examine the other inequality, we consider oriented percolation. For p [0, 1] called each edge e open with probability p and closed with probability 1 p, independently. - Two vertices x and y are connected (x y) if there is an oriented path from x to y. - An oriented path only moves up or right.

15 Oriented percolation To examine the other inequality, we consider oriented percolation. For p [0, 1] called each edge e open with probability p and closed with probability 1 p, independently. - For p = 0, the origin cannot be part of an infinite open path; when p = 1 it always is. - There is a critical probability: p c = sup{p : P p (0 ) = 0}.

16 Supercritical phase When p > p c, not only does {0 } have positive probability, on that event we have many more connections. - If P = 0, x 1, x 2,... is a path of vertices, P has direction θ if arg x n θ as n.

17 Supercritical phase When p > p c, not only does {0 } have positive probability, on that event we have many more connections. - For p > p c, there exists an interval I p = (π/4 a(p), π/4 + a(p)) such that for all θ I p, there is an infinite oriented path from 0 with direction θ. - This is true with conditional probability 1. - This forms the percolation cone.

18 Back to FPP If µ M p and p > p c then with positive probability, 1-edges percolate. - For any θ in the percolation cone, take an oriented path P with only 1-edges. - It intersects L n after only using n edges. - If z is the intersection point, then τ(0, z) τ(p) = n.

19 Back to FPP If µ M p and p > p c then with positive probability, 1-edges percolate. - Therefore τ(0, z) = n. - Using these ideas, Durrett and Liggett showed B µ has a flat edge in the percolation cone. - Marchand made this more explicit:

20 Marchand s theorem Let p c be the critical value for oriented bond percolation. Let B 1 be the closed l 1 unit ball and Q 1 the first quadrant. Let µ M p. w p(1) Theorem (Marchand) w p(2) 1. B µ B 1 2. If p = p c, then B µ B 1 Q 1 = {(1/2, 1/2)}. 3. If p > p c, then B µ B 1 Q 1 = [w p (1), w p (2)]

21 Differentiability What happens at the corner of the flat edge? - For 0 θ < θ p the point v θ B µ (in direction θ) is strictly inside B µ. - Two possibilities can occur at w p (2): wp(2) wp(2) θp θp Both derivatives are different. They are the same.

22 Main Theorem Theorem (Auffinger, D.) Let µ M p for p p c with finite mean. The boundary B µ is differentiable at w p (2). - Proof is given in. on arxiv.

23 Consequences (1) From Marchand s theorem, all B µ s for µ M p are non-polygonal. (2) Solves questions of Häggstrom-Pemantle about growth models: - Infinite coexistence for Richardson-type growth models. - Graph of infection has infinitely many ends. (3) Verifies predictions of Newman-Piza and Zhang: - New logarithmic lower bounds for fluctuations of passage time outside the percolation cone. - Completes the picture of convergence/divergence of shape fluctuations in 2d.