On the concentration and the convergence rate with a moment condition in first passage percolation

We consider the first passage percolation model on the Zd lattice. In this model, we assign independently to each edge ee a non-negative passage time t(e)t(e) with a common distribution FF. Let a0,na0,n be the passage time from the origin to (n,0,…,0)(n,0,…,0). Under the exponential tail assumption, Kesten (1993) [9] and Talagrand (1995) [12] investigated the concentration of a0,na0,n from its mean using different methods. With this concentration and the exponential tail assumption, Alexander (1993) [1] gave an estimate for the convergence rate for Ea0,n. In this paper, focusing on a moment condition, we reinvestigate the concentration and the convergence rate for a0,na0,n using a special martingale structure.