First Passage Percolation Has Sublinear Distance Variance

Abstract

Let $0<a<b<\infty$, and for each edge $e$ of $Z^d$ let ${\omega }_e=a$ or ${\omega }_e=b$, each with probability 1/2, independently. This induces a random metric $\dist_ω$ on the vertices of $Z^d$, called first passage percolation. We prove that for $d>1$ the distance $dis{t}_{\omega }(0,v)$ from the origin to a vertex $v$, $|v|>2$, has variance bounded by $C|v|/\log |v|$, where $C=C(a,b,d)$ is a constant which may only depend on $a$, $b$ and $d$. Some related variants are also discussed

Replaced theorem 2 (which was incorrect) by a new theorem