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

We consider the first passage percolation model on the Z d lattice. In this model, we assign independently to each edge e a non-negative passage time t ( e ) with a common distribution F . Let a 0 , n be the passage time from the origin to ( n , 0 , … , 0 ) . Under the exponential tail assumption, Kesten (1993) [9] and Talagrand (1995) [12] investigated the concentration of a 0 , 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 E a 0 , n . In this paper, focusing on a moment condition, we reinvestigate the concentration and the convergence rate for a 0 , n using a special martingale structure.