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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


We consider first passage percolation on $\mathbb{Z}^{2}$ with i.i.d. weights, whose distribution function satisfies $F(0)=p_{c}=1/2$. 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 $T(\mathbf{0},\partial B(n))$ as the passage time from the origin to the boundary of the box $[-n,n]\times[-n,n]$. We characterize the limit behavior of $T(\mathbf{0},\partial B(n))$ by conditions on the distribution function $F$. We also give exact conditions under which $T(\mathbf{0},\partial B(n))$ 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 $n\to\infty$, 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.


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Michael Damron. Wai-Kit Lam. Xuan Wang. "Asymptotics for $2D$ critical first passage percolation." Ann. Probab. 45 (5) 2941 - 2970, September 2017.


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

Primary: 60K35
Secondary: 60F05, 82B43

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 5 • September 2017
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