Rate of convergence in first-passage percolation under low moments

We consider first-passage percolation on the $d$ dimensional cubic lattice for $d\ge 2$; that is, we assign independently to each edge $e$ a nonnegative random weight $t_e$ with a common distribution and consider the induced random graph distance (the passage time), $T(x,y)$. It is known that for each $x \in $\mathbb{Z}$^d$,$\mu(x) = \lim_n T(0,nx)/n$existsandthat$0 \leq $\mathbb{E}$T(0,x) - \mu(x) \leq C|x|_1^{1/2}\log |x|_1$underthecondition$$\mathbb{E}$e^{\alpha t_e}<\infty$forsome$\alpha>0$. By combining tools from concentration of measure with Alexander’s methods, we show how such bounds can be extended to $t_e$’s with distributions that have only low moments. For such edge-weights, we obtain an improved bound $C(\mathrm{\parallel }x{\mathrm{\parallel }}_1\log \mathrm{\parallel }x{\mathrm{\parallel }}_1{)}^{1\mathrm{/}2}$ and bounds on the rate of convergence to the limit shape.