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 on (0,∞) 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≥ 0 P ( T(x,y)∈ [a,a+1] ) ≤ C√log \|x-y\|, answering a question posed by of Ahlberg and de la Riva. This estimate recovers earlier results on the fluctuations of the passage time by Newman and Piza, Pemantle and Peres, and Chatterjee.