Small ball probabilities for the passage time 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 $$\underset{a\ge 0}{\max }\mathbb{P}(T(x,y)\in [a,a+1])\le \frac{C}{\sqrt{\log \Vert x-y\Vert }},$$ 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.