Fluctuation lower bounds in planar random growth models

Bates, Erik; Chatterjee, Sourav

doi:10.1214/19-aihp1043

Cited by 2 publications

(1 citation statement)

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“…A logarithmic lower bound for the variance of the point-to-point minimal passage time T (x, y) in the standard model (FPP on the infinite discrete lattice Z 2 ) is well-known for a large class of distributions of edge weights; see [4,Sec. 3.3] and [5,8,11] for fluctuation bounds. A lower bound of polynomial order can even be shown under the curvature assumption we make below.…”

Section: Introductionmentioning

confidence: 99%

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

Damron¹,

Houdré²,

Özdemir³

2022

Preprint

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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.

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