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 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.