Limit theorems for critical first-passage percolation on the triangular lattice

Abstract

Consider (independent) first-passage percolation on the sites of the triangular lattice T embedded in C. Denote the passage time of the site v in T by t(v), and assume that P (t(v) = 0) = P (t(v) = 1) = 1/2. Denote by b0,n the passage time from 0 to the halfplane {v ∈ T : Re(v) ≥ n}, and by T (0, nu) the passage time from 0 to the nearest site to nu, where |u| = 1. We prove that as n → ∞, b0,n/ log n → 1/(2 √ 3π) a.s., E[b0,n]/ log n → 1/(2 √ 3π) and Var[b0,n]/ log n → 2/(3This answers a question of Kesten and Zhang (1997) and improves our previous work (2014). From this result, we derive an explicit form of the central limit theorem for b0,n and T (0, nu). A key ingredient for the proof is the moment generating function of the conformal radii for conformal loop ensemble CLE6, given by Schramm, Sheffield and Wilson (2009).

Asymptotics for $2D$ critical first passage percolation

“…In this paper, we will study the critical case, namely F (0) = p c = 1/2, where p c is the critical threshold for site percolation on T. In this case, it is shown by Damron-Lam-Wang in [3] that under suitable moment assumptions on t x , one has ET (0, ∂B(n)) log n k=2 F −1 (p c + 2 −k ) and Var(T (0, ∂B(n))) log n k=2 F −1 (p c + 2 −k ) 2 , (1.1) and further if Var(T (0, ∂B(n))) → ∞ as n → ∞, then one also has a Gaussian central limit theorem for the variables (T (0, ∂B(n))) 1 . Sharper asymptotics were proved by C.-L. Yao in [12,13] (with further development in [5,11]) in the special case where t x is Bernoulli (that is, t x = 0 with probability 1/2 and t x = 1 with probability 1/2):…”

Asymptotics for 2D critical and near-critical first-passage percolation

“…Proof. The proof of (48) is essentially the same as that of (29) in [27]. For completeness we give it here.…”

“…We will use the martingale method introduced in [14]. This approach has been used in [8,27] also. We start with some notation.…”

“…Theorem 1 ( [27]). Under the critical Bernoulli FPP measure P 1/2 on T, lim n→∞ c n log n = 1 2 √ 3π a.s.,…”

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

“…As shown in [61], the continuum nonsimple loop process constructed in [57,59] is a Conformal Loop Ensemble (CLE) [256,257]. The construction of the continuum scaling limit of planar critical percolation [57,59] has had several interesting applications, including [27,280,283,281,282].…”