Rate of convergence in first-passage percolation under low moments

Michael Damron and Naoki Kubota

Stochastic Processes and their Applications, 2016, vol. 126, issue 10, 3065-3076

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.

Keywords: First-passage percolation; Concentration inequalities; Low moments; Rate of convergence (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:126:y:2016:i:10:p:3065-3076

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DOI: 10.1016/j.spa.2016.04.001

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