First Passage Percolation with nonidentical passage times

Abstract

In this paper we consider first passage percolation on the square lattice \(\mathbb{Z}^d\) with passage times that are independent and have bounded \(p^{th}\) moment for some \(p > 6(1+d),\) but not necessarily identically distributed. For integer \(n \geq 1,\) let \(T(0,n)\) be the minimum time needed to reach the point \((n,\mathbf{0})\) from the origin. We prove that \(\frac{1}{n}\left(T(0,n) - \mathbb{E}T(0,n)\right)\) converges to zero in \(L^2\) and use a subsequence argument to obtain almost sure convergence. As a corollary, for i.i.d. passage times, we also obtain the usual almost sure convergence of \(\frac{T(0,n)}{n}\) to a constant \(\mu.\)