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We consider the first passage percolation model on the ${\bf Z}^d$ lattice.In this model, we assign independently to each edge $e$ a non-negative passagetime $t(e)$ with a common distribution $F$. Let $a_{0,n}$ be the passage timefrom the origin to $(n,0,..., 0)$. Under the exponential tail assumption,  Kesten (1993) and Talagrand (1995) investigated the concentration of$a_{0,n}$ from its mean using different methods. With this concentration andthe exponential tail assumption, Alexander gave an estimate for the convergencerate for ${\bf E} a_{0,n}$. In this paper, focusing on a moment condition, wereinvestigate the concentration and the convergence rate for $a_{0,n}$ using aspecial martingale structure.

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