Rate of convergence in first-passage percolation under low moments

Abstract

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 $x\in {\mathbb{Z}}^d$, $\mu (x)={\lim }_{n}T(0,nx)/n$ exists and that $0\le \mathbb{E}T(0,x)-\mu (x)\le C\Vert x{\Vert }_1^{1/2}\log \Vert x{\Vert }_1$ 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 $C(\Vert x{\Vert }_1\log \Vert x{\Vert }_1{)}^{1/2}$ and bounds on the rate of convergence to the limit shape.

This is the corrected version of the paper. 13 pages, title changed

References in corpus (2)

• Sublinear variance in first-passage percolation for general distributions
• Subgaussian concentration and rates of convergence in directed polymers

Cited by in corpus (1)

• Subdiffusive concentration in first-passage percolation