Lower large deviations for the maximal flow through a domain of ℝ^d in first passage percolation

2009/07/31 by Raphaël Cerf, Cerf, Raphaël, Marie Théret +1
Mathematics · #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0907.5501

openalex publication_date 2009/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the standard first passage percolation model in the rescaled graph ℤ^d/n for d≥ 2, and a domain Ω of boundary Γ in ℝ^d. Let Γ¹ and Γ² be two disjoint open subsets of Γ, representing the parts of Γ through which some water can enter and escape from Ω. We investigate the asymptotic behaviour of the flow ϕ_n through a discrete version Ω_n of Ω between the corresponding discrete sets Γ¹_n and Γ²_n. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of ϕ_n/ n^d-1 below a certain constant are of surface order.

Related

• Law of large numbers for the maximal flow through a domain of ℝ^d in first passage percolation
• Large deviation principle for the cutsets and lower large deviation principle for the maximal flow in first passage percolation
• Large deviation principle for the streams and the maximal flow in first passage percolation
• Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation
• Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation