Asymptotics for 2D2D critical first passage percolation

• Michael Damron ^[1] ; Wai-Kit Lam ^[2] ; Xuan Wang ^[3]
  1. [1] Georgia Institute of Technology

     Georgia Institute of Technology

     Estados Unidos

     
  2. [2] Indiana University (USA)
  3. [3] Databricks Inc. (USA)

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 5, 2017, págs. 2941-2970
• Idioma: inglés
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  • We consider first passage percolation on Z2Z2 with i.i.d. weights, whose distribution function satisfies F(0)=pc=1/2F(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))T(0,∂B(n)) as the passage time from the origin to the boundary of the box [−n,n]×[−n,n][−n,n]×[−n,n]. We characterize the limit behavior of T(0,∂B(n))T(0,∂B(n)) by conditions on the distribution function FF. We also give exact conditions under which T(0,∂B(n))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→∞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.