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

Barbara Dembin (1) , Marie Théret (2, 3)

1 D-MATH - Department of Mathematics [ETH Zurich]
2 MODAL'X - Modélisation aléatoire de Paris X
3 FP2M - Fédération Parisienne de Modélisation Mathématique

We consider the standard first passage percolation model in the rescaled lattice $\mathbb Z^d/n$ for $d\geq 2$ and a bounded domain $\Omega$ in $\mathbb R^d$. We denote by $\Gamma^1$ and $\Gamma^2$ two disjoint subsets of $\partial \Omega$ representing respectively the sources and the sinks, \textit{i.e.}, where the water can enter in $\Omega$ and escape from $\Omega$. A cutset is a set of edges that separates $\Gamma ^1$ from $\Gamma^2$ in $\Omega$, it has a capacity given by the sum of the capacities of its edges. Under some assumptions on $\Omega$ and the distribution of the capacities of the edges, we already know a law of large numbers for the sequence of minimal cutsets $(\mathcal E_n^{min})_{n\geq 1}$: the sequence $(\mathcal E_n^{min})_{n\geq 1}$ 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 $\Omega$.

Mots clés: First passage percolation; minimal cutset; large deviation principle

Domaines: Probabilités [math.PR]

Barbara Dembin, Marie Théret. Large deviation principle for the cutsets and lower large deviation principle for the maximal flow in first passage percolation. 2021. ⟨hal-03149069⟩