Rate of convergence in first-passage percolation under low moments

2014/06/12 by Michael Damron, Damron, Michael, Naoki Kubota +1
Mathematics · #60F99 #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60F99 #msc:60K35

paper · pdf · doi:10.48550/arxiv.1406.3105

This is the corrected version of the paper. 13 pages, title changed

openalex publication_date 2014/06/12 · arxiv created 2016/04/20 · arxiv updated 2016/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 t_e with a common distribution and consider the induced random graph distance (the passage time), T(x,y). It is known that for each x ∈ ℤ^d, μ(x) = lim_n T(0,nx)/n exists and that 0 ≤ 𝔼T(0,x) - μ(x) ≤ C‖x‖₁^1/2log ‖x‖₁ under the condition 𝔼e^αt_e<∞ for some α>0. By combining tools from concentration of measure with Alexander's methods, we show how such bounds can be extended to t_e's with distributions that have only low moments. For such edge-weights, we obtain an improved bound C (‖x‖₁ log ‖x‖₁)^1/2 and bounds on the rate of convergence to the limit shape.

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