Transitions for exceptional times in dynamical first-passage percolation

TL;DR

This paper investigates the behavior of dynamical first-passage percolation on the triangular lattice, identifying conditions for the existence of exceptional times with atypical growth and analyzing the fractal dimensions of these times.

Contribution

It establishes new criteria for the occurrence of exceptional times in dynamical FPP and computes their Hausdorff and Minkowski dimensions, revealing richer dynamical phenomena.

Findings

Exceptional times exist under certain divergence conditions.

Hausdorff and Minkowski dimensions of exceptional sets can differ.

Dynamical behavior extends beyond subcritical FPP regimes.

Abstract

In first-passage percolation (FPP), we let $(\tau_v)$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. We study a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We prove that if $\sum F^{-1}(1/2+1/2^k) = \infty$, then a.s. there are exceptional times at which the weight grows atypically, but if $\sum k^{7/8} F^{-1}(1/2+1/2^k) <\infty$, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP.