Subgaussian rates of convergence of means in directed first passage   percolation
  We consider directed first passage percolation on the integer lattice, with time constant $\mu$ and passage time $a_{0n}$ from the origin to $(n,0,...,0)$. It is shown that under certain conditions on the passage time distribution, $Ea_{0n} - n\mu = O(n^{1/2}(\log\log n)/\log n)$. 
Kenneth S. Alexander
math.PR