Small ball probabilities for the passage time in planar first-passage percolation

TL;DR

This paper establishes bounds on the probability that the passage time in planar first-passage percolation falls within a unit interval, revealing slow decay related to the logarithm of the distance.

Contribution

It proves a novel bound on small ball probabilities for passage times, answering a question by Ahlberg and de la Riva, and extends understanding of passage time fluctuations.

Findings

Bound on the maximum probability of passage time in an interval decreases as 1/√log distance.

Recovers and extends previous fluctuation results by Newman--Piza, Pemantle--Peres, and Chatterjee.

Provides a new quantitative estimate for passage time distribution in planar first-passage percolation.

Abstract

We study planar first-passage percolation with independent weights whose common distribution is supported in $(0,\infty)$ and is absolutely continuous with respect to Lebesgue measure. We prove that the passage time from $x$ to $y$ denoted by $T(x,y)$ satisfies $$\max _{a\ge 0} \mathbb P \big( T(x,y)\in [a,a+1] \big) \le \frac{C}{\sqrt{\log \|x-y\|}},$$ answering a question posed by Ahlberg and de la Riva. This estimate recovers earlier results on the fluctuations of the passage time by Newman--Piza, Pemantle--Peres, and Chatterjee.