First passage percolation has sublinear distance variance
First passage percolation has sublinear distance variance
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 $\dist_\omega$ on the vertices of $\Z^d$, called first passage percolation. We prove that for $d>1$, the distance $\dist_\omega(0,v)$ from the origin …