ASYMPTOTICS FOR 2D CRITICAL FIRST PASSAGE PERCOLATION

We consider first passage percolation on 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 (0, partial derivative B(n)) as the passage time from the origin to the boundary of the box [-n, n] x [-n, n]. We characterize the limit behavior of T (0, partial derivative B (n)) by conditions on the distribution function F. We also give exact conditions under which T (0, partial derivative 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 -> infinity, 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.