Open Access
September 2017 Asymptotics for 2D critical first passage percolation
Michael Damron, Wai-Kit Lam, Xuan Wang
Ann. Probab. 45(5): 2941-2970 (September 2017). DOI: 10.1214/16-AOP1129

Abstract

We consider first passage percolation on Z2 with i.i.d. weights, whose distribution function satisfies F(0)=pc=1/2. This is sometimes known as the “critical case” because large clusters of zero-weight edges force passage times to grow at most logarithmically, giving zero time constant. Denote T(0,B(n)) as the passage time from the origin to the boundary of the box [n,n]×[n,n]. We characterize the limit behavior of T(0,B(n)) by conditions on the distribution function F. We also give exact conditions under which T(0,B(n)) will have uniformly bounded mean or variance. These results answer several questions of Kesten and Zhang from the 1990s and, in particular, disprove a conjecture of Zhang from 1999. In the case when both the mean and the variance go to infinity as n, we prove a CLT under a minimal moment assumption. The main tool involves a new relation between first passage percolation and invasion percolation: up to a constant factor, the passage time in critical first passage percolation has the same first-order behavior as the passage time of an optimal path constrained to lie in an embedded invasion cluster.

Citation

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Michael Damron. Wai-Kit Lam. Xuan Wang. "Asymptotics for critical first passage percolation." Ann. Probab. 45 (5) 2941 - 2970, September 2017. https://doi.org/10.1214/16-AOP1129

Information

Received: 1 August 2015; Revised: 1 June 2016; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 1378.60115
MathSciNet: MR3706736
Digital Object Identifier: 10.1214/16-AOP1129

Subjects:
Primary: 60K35
Secondary: 60F05 , 82B43

Keywords: central limit theorem , correlation length , Critical percolation , first passage percolation , Invasion percolation

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 5 • September 2017
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