Convergence in First Passage Percolation with nonidentical passage times

TL;DR

This paper proves that in first passage percolation on ^d with independent, nonidentically distributed edge times, the normalized passage time converges almost surely and in L^2, extending previous results to more general settings.

Contribution

It introduces a new approach to establish almost sure convergence of normalized passage times without relying on the subadditive ergodic theorem for nonidentical distributions.

Findings

^d first passage times normalized by n converge almost surely.

The method applies to nonidentically distributed edge times with bounded second moments.

Almost sure convergence holds even when passage times are not identically distributed.

Abstract

In this paper we consider first passage percolation on the square lattice \(\mathbb{Z}^d\) with edge passage times that are independent and have uniformly bounded second moment, but not necessarily identically distributed. For integer \(n \geq 1,\) let \(T_n\) be the minimum passage time between the origin and the point \((n,0,\ldots,0).\) We prove that \(\frac{1}{n}(T_n-\mathbb{E}T_n)\) converges to zero almost surely and in \(L^2\) as \(n~\rightarrow~\infty.\) The convergence is nontrivial in the sense that \(\frac{T_n}{n}\) is asymptotically bounded away from zero and infinity almost surely. We first define a truncated version \(\hat{T}^{(n)}_n\) that is asymptotically equivalent to~\(T_n.\) We then use a finite box modification of the martingale method of Kesten~(1993) to estimate the variance of \(\hat{T}^{(n)}_n.\) Finally, we use a subsequence argument to obtain almost sure convergence for \(\frac{1}{n}(\hat{T}^{(n)}_n - \mathbb{E}\hat{T}^{(n)}_n).\) The corresponding result for \(T_n\) is then obtained using asymptotic equivalence of \(T_n\) and \(\hat{T}^{(n)}_n.\) For identically distributed passage times, our method alternately obtains almost sure convergence of~\(\frac{T_n}{n}\) to a positive constant~\(\mu_F,\) without invoking the subadditive ergodic theorem.