Rate of convergence in first-passage percolation under low moments

We consider first-passage percolation on the $d$ dimensional cubic lattice for $d \geq 2$; 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 $x \in \mathbb{Z}^d$, $\mu(x) = \lim_n T(0,nx)/n$ exists and that $0 \leq \mathbb{E}T(0,x) - \mu(x) \leq C\|x\|_1^{1/2}\log \|x\|_1$ under the condition $\mathbb{E}e^{\alpha t_e}<\infty$ for some $\alpha>0$. 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 $C (\|x\|_1 \log \|x\|_1)^{1/2}$ and bounds on the rate of convergence to the limit shape.