Convergence in First Passage Percolation with nonidentical passage times (1704.00459v1)

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.