For two-valued i.i.d. edge weights on Z^2, the event that the constrained left-right crossing time exceeds its median is noise sensitive up to width about n^{1/2}, improving the prior n^{1/22}, and conditional curvature assumptions extend this to all widths and yield a new variance lower bound.
Small ball probabilities for the passage time in planar first-passage percolation
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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.
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Noise sensitivity and variance lower bound for minimal left-right crossing of a square in first-passage percolation
For two-valued i.i.d. edge weights on Z^2, the event that the constrained left-right crossing time exceeds its median is noise sensitive up to width about n^{1/2}, improving the prior n^{1/22}, and conditional curvature assumptions extend this to all widths and yield a new variance lower bound.