First passage percolation and escape strategies

TL;DR

This paper investigates conditions under which semi-infinite paths in first passage percolation on integer lattices can consistently outperform direct routes, providing criteria based on passage time distributions and analyzing geodesic edge usage.

Contribution

It establishes necessary and sufficient conditions on passage time distributions for the existence of escape strategies in first passage percolation.

Findings

Characterization of when escape paths exist based on distribution F

Results on the number of large passage time edges in geodesics

Conditions for the existence of semi-infinite paths with specific properties

Abstract

Consider first passage percolation on $\mathbb{Z}^d$ with passage times given by i.i.d. random variables with common distribution $F$. Let $t_\pi(u,v)$ be the time from $u$ to $v$ for a path $\pi$ and $t(u,v)$ the minimal time among all paths from $u$ to $v$. We ask whether or not there exist points $x,y \in \mathbb{Z}^d$ and a semi-infinite path $\pi=(y_0=y,y_1,\dots)$ such that $t_\pi(y, y_{n+1})<t(x,y_n)$ for all $n$. Necessary and sufficient conditions on $F$ are given for this to occur. When the support of $F$ is unbounded, we also obtain results on the number of edges with large passage time used by geodesics.