Exceptional behavior in critical first-passage percolation and random sums

TL;DR

This paper investigates the critical case of first-passage percolation on a lattice, establishing the existence of an incipient infinite cluster measure and analyzing the limiting behavior of conditioned sums of random variables.

Contribution

It introduces an IIC measure for critical FPP and characterizes the limits of conditioned sums of independent nonnegative random variables.

Findings

Existence of an IIC measure in critical FPP when passage times are unbounded.

Conditions for trivial and nontrivial limits of conditioned sums of random variables.

Analysis of the behavior of passage times near criticality.

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.