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

2009/07/31 by Cerf, Raphaël, Théret, Marie
#60K35 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0907.5504

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, ϕ_n converges almost surely towards a constant ϕ_Ω, which is the solution of a continuous non-random min-cut problem. Moreover, we give a necessary and sufficient condition on the law of the capacity of the edges to ensure that ϕ_Ω >0.

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