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

Abstract

We consider the standard first passage percolation model in the rescaled lattice Z d /n for d ≥ 2 and a bounded domain Ω in R d . We denote by Γ 1 and Γ 2 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 − → µ max n that describes how the maximal amount of fluid can enter through Γ 1 and spreads in Ω. Under some assumptions on Ω and G, we already know a law of large number for − → µ max n . The sequence ( − → µ max n ) 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.

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

“…Let (Ω, Γ 1 , Γ 2 ) that satisfies hypothesis 2. The sequence 3) is used to prove upper large deviation principle for the maximal flow in the companion paper [5]. It also gives the precise role of λ min in our study.…”

“…We won't define rigorously what a stream is and its link with maximal flow. We refer for instance to the companion paper [5] where we study large deviation principle for admissible streams to obtain an upper large deviation principle for the maximal flow.…”

“…We refer to [5] for a precise definition of J u . Theorems 1.4 and 1.6 give the full picture of large deviations of φ n (Γ 1 , Γ 2 , Ω).…”

“…This is the reason why to study lower large deviations, it is natural to study cutsets that are (d − 1)-dimensional objects, whereas to study the upper large deviations, we study streams (functions on the edges that describe how the water flows in the lattice) that are d-dimensional objects. Actually, theorem 1.4 is used in [5] to prove theorem 1.6. Theorem 1.4 justifies the fact that the lower large deviations are not of the same order as the order of the upper large deviations.…”

“…The remaining part of the proof required working with maximal streams (which is the dual object associated with minimal cutset). We refer to the companion paper [5] for a more precise study of maximal streams.…”