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

We study planar first-passage percolation with independent weights whose common distribution is supported in $(0,\mathrm{\infty })$ 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 $$\underset{a\ge 0}{\max }\mathbb{P}(T(x,y)\in [a,a+1])\le \frac{C}{\sqrt{\log \mathrm{\parallel }x-y\mathrm{\parallel }}},$$ 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.