Rate of convergence in first-passage percolation under low moments

Abstract

We consider first-passage percolation on the d dimensional cubic lattice for d≥2; that is, we assign independently to each edge e a nonnegative random weight te with a common distribution and consider the induced random graph distance (the passage time), T(x,y). It is known that for each x∈Zd, μ(x)=limnT(0,nx)/n exists and that 0≤ET(0,x)−μ(x)≤C‖x‖11/2log‖x‖1 under the condition Eeαte<∞ for some α>0. By combining tools from concentration of measure with Alexander’s methods, we show how such bounds can be extended to te’s with distributions that have only low moments. For such edge-weights, we obtain an improved bound C(‖x‖1log‖x‖1)1/2 and bounds on the rate of convergence to the limit shape.