Exceptional behavior in critical first-passage percolation and random sums

Abstract: We study first-passage percolation (FPP) on the square lattice. The model is defined using i.i.d. nonnegative random edge-weights $(t_e)$ associated to the nearest neighbor edges of $\mathbb{Z}^2$. The passage time between vertices $x$ and $y$, $T(x,y)$, is the minimal total weight of any lattice path from $x$ to $y$. The growth rate of $T(x,y)$ depends on the value of $F(0) = \mathbb{P}(t_e=0)$: if $F(0) < 1/2$ then $T(x,y)$ grows linearly in $|x-y|$, but if $F(0) > 1/2$ then it is stochastically bounded. In the critical case, where $F(0) = 1/2$, $T(x,y)$ can be bounded or unbounded depending on the behavior of the distribution function $F$ of $t_e$ near 0. In this paper, we consider the critical case in which $T(x,y)$ is unbounded and prove the existence of an incipient infinite cluster (IIC) type measure, constructed by conditioning the environment on the event that the passage time from $0$ to a far distance remains bounded. This IIC measure is a natural candidate for the distribution of the weights at a typical exceptional time in dynamical FPP. A major part of the analysis involves characterizing the limiting behavior of independent nonnegative random variables conditioned to have small sum. We give conditions on random variables that ensure that such limits are trivial, and several examples that exhibit nontrivial limits.