First passage percolation has sublinear distance variance

Let 0 < a < b < ∞, and for each edge e of ℤ ^d let ω _e = a or ω _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 dist _ω(0, ν) from the origin to a vertex ν, |ν| > 2, has variance bounded by C|ν|/log |ν|, where C = C(a,b,d) is a constant which may only depend on a, b and d. Some related variants are also discussed.