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 maxa≥0P(T(x,y)∈[a,a+1])≤C log‖x−y‖,(0.1) answering a question posed by Ahlberg and de la Riva [1]. This estimate recovers earlier results on the fluctuations of the passage time by Newman and Piza [12], Pemantle and Peres [13] and Chatterjee [5].