First passage percolation on nilpotent Cayley graphs and beyond

2014/10/13 by Itaï Benjamini, Romain Tessera, Benjamini, Itai +1 · 1 citation
Mathematics · #20E22 #20F69 #22D10 #46B85 #FOS: Mathematics #Geometry and complex manifolds #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1410.3292

openalex publication_date 2014/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a previous work by the second author, the proof relies on Talagrand's concentration inequality, and on Pansu's theorem. Adapting a well-known argument for Z^d, we prove a sublinear estimate on the variance for virtually nilpotent groups which are not virtually isomorphic to Z. We further discuss the asymptotic cones of first-passage percolation on general infinite connected graphs: we prove that the asymptotic cones are a.e. deterministic if and only the volume growth is subexponential.

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