A central limit theorem for “critical” first-passage percolation in two dimensions

Abstract

Consider (independent) ÿrst-passage percolation on the edges of Z 2 . Denote the passage time of the edge e in Z 2 by t(e), and assume that P{t(e) = 0} = 1=2, P{0 ¡t(e)¡C 0 } = 0 for some constant C 0 ¿ 0 and that E[t (e)] ¡ ∞ for some ¿4. Denote by b 0; n the passage time from 0 to the halfplane {(x; y): x = n}, and by T (0; nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0 ¡C 1 , C 2 ¡∞ and n such that C 1 (log n) 1=2 5 n 5 C 2 (log n) 1=2 and such that −1 n [b 0; n − Eb 0; n ] and ( √ 2 n ) −1 [T (0; nu) − ET (0; nu)] converge in distribution to a standard normal variable (as n → ∞, u ÿxed). A similar result holds for the site version of ÿrst-passage percolation on Z 2 , when the common distribution of the passage times {t(v)} of the vertices satisÿes P{t(v) = 0} = 1 − P{t(v) = C 0 } = p c (Z 2 ; site) := critical probability of site percolation on Z 2 , and E[t (u)] ¡ ∞ for some ¿4. Mathematics Subject Classiÿcation (1991): 60K35, 60F05, 82B43

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

“…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 →This 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).…”

Critical first-passage percolation starting on the boundary

“…Let us mention that it is possible to use the SLE techniques in Section 4.3 of [6] to give a third proof. In order to prove the limit result for Var(c + n ), we use a martingale method from [12]. The organization of the paper is as follows.…”

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

“…Proof of (55). The proof is essentially the same as the proof of (2.24) in [14]. Recall that for j ≥ 0, ∆ j (ω) := E p [T (0, ∂B(L(p))) | F j ] − E p [T (0, ∂B(L(p))) | F j−1 ].…”

“…We now wish to bound the second term on the right-hand side of (32). We will use the martingale method introduced in [14]. This approach has been used in [8,27] also.…”

“…We need the following lemma, which was essentially proved in [14]. Note that (42) is standard and follows from RSW and FKG; (43) is the same as Lemma 2 of [14]. Lemma 7 ([14]).…”