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 originto 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, band d. Some related variants are also discussed.
"First passage percolation has sublinear distance variance." Ann. Probab. 31 (4) 1970 - 1978, October 2003. https://doi.org/10.1214/aop/1068646373