Speed of convergence in first passage percolation and geodesicity of the average distance

TL;DR

This paper provides an elementary proof linking Talagrand's concentration inequality to a limit shape theorem in first passage percolation, improving convergence bounds by analyzing the geodesic properties of the average distance.

Contribution

It introduces a novel approach that avoids the subadditive theorem, connecting convergence speed to the geodesicity of the average distance in percolation models.

Findings

Bound on the speed of convergence slightly better than previous results.

Average distance is shown to be close to geodesic, facilitating the limit shape theorem.

Elementary proof technique applicable to Cayley graphs of Z^d.

Abstract

We give an elementary proof that Talagrand's sub-Gaussian concentration inequality implies a limit shape theorem for first passage percolation on any Cayley graph of Z^d, with a bound on the speed of convergence that slightly improves Alexander's bounds. Our approach, which does not use the subadditive theorem, is based on proving that the average distance is close to being geodesic. Our key observation, of independent interest, is that the problem of estimating the rate of convergence for the average distance is equivalent (in a precise sense) to estimating its "level of geodesicity".