Convergence in First Passage Percolation with nonidentical passage times

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 $n\ge 1,$ let $T_n$ be the minimum passage time between the origin and the point $(n,0,\dots ,0)\mathrm{.}$ We prove that $\frac{1}{n}(T_n-\mathbb{E}{T}_n)$ converges to zero almost surely and in $L^2$ as $n\to \mathrm{\infty }\mathrm{.}$ 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 ${\widehat{T}}_n^{(n)}$ that is asymptotically equivalent to~$T_n\mathrm{.}$ We then use a finite box modification of the martingale method of Kesten~(1993) to estimate the variance of ${\widehat{T}}_n^{(n)}\mathrm{.}$ 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 ${\widehat{T}}_n^{(n)}\mathrm{.}$ 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.