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• W2964138760 abstract "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 $ntoinfty$, 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." @default.
• W2964138760 created "2019-07-30" @default.
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• W2964138760 date "2017-09-01" @default.
• W2964138760 modified "2023-09-26" @default.
• W2964138760 title "Asymptotics for $2D$ critical first passage percolation" @default.
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• W2964138760 doi "https://doi.org/10.1214/16-aop1129" @default.
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