Fluctuations in first-passage percolation

TL;DR

This paper surveys techniques for bounding the variance of passage times in first-passage percolation, highlighting a combination of concentration measure tools and an improved method based on Benjamini-Kalai-Schramm ideas.

Contribution

It introduces a novel approach that improves variance bounds from linear to logarithmic for general edge-weight distributions in first-passage percolation.

Findings

Logarithmic variance bounds achieved

Combination of concentration inequalities and BKS ideas

Applicable to general edge-weight distributions

Abstract

We present a survey of techniques to obtain upper bounds for the variance of the passage time in first-passage percolation. The methods discussed are a combination of tools from the theory of concentration of measure, some of which we briefly review. These are combined with variations on an idea of Benjamini-Kalai-Schramm to obtain a logarithmic improvement over the linear bound implied by the Efron-Stein/Poincare inequality, for general edge-weight distributions.