Speed of convergence in first passage percolation and geodesicity of the average distance

2014/10/07 by Romain Tessera, Tessera, Romain · 2 citations
Mathematics · #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60K35

paper · pdf · doi:10.48550/arxiv.1410.1701

The proof of the main theorem contained a few mistakes that have been corrected. The presentation has also been improved

openalex publication_date 2014/10/07 · arxiv created 2015/05/11 · arxiv updated 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an elementary proof that Talagrand's sub-Gaussian concentration inequality implies a limit shape theorem for first passage percolation on any Cayley graph of Z^d, with a bound on the speed of convergence that slightly improves Alexander's bounds. Our approach, which does not use the subadditive theorem, is based on proving that the average distance is close to being geodesic. Our key observation, of independent interest, is that the problem of estimating the rate of convergence for the average distance is equivalent (in a precise sense) to estimating its "level of geodesicity".

Cited by

• Lower bounds for fluctuations in first-passage percolation for general distributions
• Evolution and Limiting Configuration of a Long-Range Schelling-Type Spin System

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