  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 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) and Talagrand (1995) investigated the concentration of
$a_{0,n}$ from its mean using different methods. With this concentration and
the exponential tail assumption, Alexander gave an estimate for the convergence
rate for ${\bf E} a_{0,n}$. 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.
