Convergence in First Passage Percolation with nonidentical passage times

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.