Fluctuation bounds for first-passage percolation on the square, tube, and torus

Abstract

In first-passage percolation, one assigns i.i.d. nonnegative weights (te) to the edges of Z d and studies the induced distance (passage time) T (x, y) between vertices x and y. It is known that for d = 2, the fluctuations of T (x, y) are at least order log |x − y| under mild assumptions on te. We study the question of fluctuation lower bounds for Tn, the minimal passage time between two opposite sides of an n by n square. The main result is that, under a curvature assumption, this quantity has fluctuations at least of order n 1/8−ǫ for any ǫ > 0 when the te are exponentially distributed. As previous arguments to bound the fluctuations of T (x, y) only give a constant lower bound for those of Tn (even assuming curvature), a different argument, representing Tn as a minimum of cylinder passage times, and deriving more detailed information about the distribution of cylinder times using the Markov property, is developed. As a corollary, we obtain the first polynomial lower bounds on higher central moments of the discrete torus passage time, under the same curvature assumption.