On the Speed of Convergence in First-Passage Percolation

Abstract

We consider the standard first-passage percolation problem on Zd: {t(e): e an edge of Zd} 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 Zd from 0 to n ξ1}, where ξ1 is the first coordinate vector. We show that σ2(a0,n) ≤ C1 n and that $P\{|a_{0,n} - Ea_{0,n}| \geq x\sqrt{n}\} \leq C_2 \exp(-C_3 x)$ for x ≤ C4 n and for some constants $0 < C_i