First Passage Percolation Has Sublinear Distance Variance

Let 0 < a < b < ∞, and for each edge e of Zdlet ωe=a or ωe=b, each with probability 1/2, independently. This induces a random metric distωon the vertices of Zd, called first passage percolation. We prove that for d > 1, the distance distω(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.