Sublinear variance in first-passage percolation for general distributions

TL;DR

This paper proves that in first-passage percolation on Z^d, the variance of passage time grows slower than linearly with distance, under minimal assumptions on the distribution of edge weights.

Contribution

It extends previous results by establishing sublinear variance bounds for general distributions with minimal moment conditions.

Findings

Variance of passage time is sublinear in distance, bounded by Cx/(log x).

Results apply to various distribution types, including discrete and continuous.

Main theorem generalizes prior work to broader distribution classes.

Abstract

We prove that the variance of the passage time from the origin to a point x in first-passage percolation on Z^d is sublinear in the distance to x when d \geq 2, obeying the bound Cx/(log x), under minimal assumptions on the edge-weight distribution. The proof applies equally to absolutely continuous, discrete and singular continuous distributions and mixtures thereof, and requires only 2+log moments. The main result extends work of Benjamini-Kalai-Schramm and Benaim-Rossignol.