On the Speed of Convergence in First-Passage Percolation

Abstract

We consider the standard first-passage percolation problem on $\mathbb{Z}^d: \{t(e): e \text{an edge of} \mathbb{Z}^d\}$ is an i.i.d. family of random variables with common distribution $F, a_{0,n} := \inf\{\sum^k_1 t(e_1): (e_1, \cdots, e_k)$ a path on $\mathbb{Z}^d$ from 0 to $n \xi_1\}$, where $\xi_1$ is the first coordinate vector. We show that $\sigma^2(a_{0,n}) \leq C_1 n$ and that $P\{|a_{0,n} - Ea_{0,n}| \geq x\sqrt{n}\} \leq C_2 \exp(-C_3 x)$ for $x \leq C_4 n$ and for some constants $0 < C_i < \infty$. It is known that $\mu := \lim(1/n)Ea_{0,n}$ exists. We show also that $C_5 n^{-1} \leq Ea_{0,n} - n\mu \leq C_6 n^{5/6}(\log n)^{1/3}$. There are corresponding statements for the roughness of the boundary of the set $\tilde{B}(t) = \{\nu: \nu$ a vertex of $\mathbb{Z}^d$ that can be reached from the origin by a path $(e_1, \cdots, e_k)$ with $\sum t(e_i) \leq t\}$.