Large deviation principle for the cutsets and lower large deviation principle for the maximal flow in first passage percolation

2021/02/23 by Dembin, Barbara, Théret, Marie
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2102.11601

Abstract

We consider the standard first passage percolation model in the rescaled lattice \mathbb Z^d/n 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 sources and the sinks, i.e., where the water can enter in Ω and escape from Ω. A cutset is a set of edges that separates Γ¹ from Γ² in Ω, it has a capacity given by the sum of the capacities of its edges. Under some assumptions on Ω and the distribution of the capacities of the edges, we already know a law of large numbers for the sequence of minimal cutsets (\mathcal E_n^min)_{n≥ 1}: the sequence (\mathcal E_n^min)_{n≥ 1} converges almost surely to the set of solutions of a continuous deterministic problem of minimal cutset in an anisotropic network. We aim here to derive a large deviation principle for cutsets and deduce by contraction principle a lower large deviation principle for the maximal flow in Ω.

Related

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