First Passage Percolation with nonidentical passage times (1409.2602v1)

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.)