Transitions for exceptional times in dynamical first-passage percolation

2021/08/30 by Michael Damron, Jack Hanson, Damron, Michael +5
Mathematics · #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2108.13248

Abstract

In first-passage percolation (FPP), we let (τ_v) be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of τ_v, there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. We study a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We prove that if ∑ F^-1(1/2+1/2^k) = ∞, then a.s. there are exceptional times at which the weight grows atypically, but if ∑ k^7/8 F^-1(1/2+1/2^k)

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