Large deviation principle for the streams and the maximal flow in first passage percolation

2020/10/09 by Barbara Dembin, Dembin, Barbara, Marie Théret +1
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2010.05526

openalex publication_date 2020/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the standard first passage percolation model in the rescaled lattice ℤ^d for d≥ 2 and a bounded domain Ω in \mathbb R ^d. We denote by Γ¹ and Γ² two disjoint subsets of ∂ Ω representing respectively the source and the sink, i.e., where the water can enter in Ω and escape from Ω. A maximal stream is a vector measure \overrightarrowμ_n^max that describes how the maximal amount of fluid can enter through Γ¹ and spreads in Ω. Under some assumptions on Ω and G, we already know a law of large number for \overrightarrowμ_n^max. The sequence (\overrightarrowμ_n^max)_{n≥ 1} converges almost surely to the set of solutions of a continuous deterministic problem of maximal stream in an anisotropic network. We aim here to derive a large deviation principle for streams and deduce by contraction principle the existence of a rate function for the upper large deviations of the maximal flow in Ω.

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