Preprints, Working Papers, ... Year : 2019

Size of a minimal cutset in supercritical first passage percolation

Marie Théret

Abstract

We consider the standard model of i.i.d. first passage percolation on Z^d given a distribution G on [0, +∞] (including +∞). We suppose that G({0}) > 1 − p_c(d), i.e., the edges of positive passage time are in the subcritical regime of percolation on Z^d. We consider a cylinder of basis an hyperrectangle of dimension d − 1 whose sides have length n and of height h(n) with h(n) negligible compared to n (i.e., h(n)/n → 0 when n goes to infinity). We study the maximal flow from the top to the bottom of this cylinder. We already know that the maximal flow renormalized by n^(d−1) converges towards the flow constant which is null in the case G({0}) > 1 − p_c (d). The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut the top from the bottom of the cylinder. If we denote by ψ_n the minimal cardinal of such a set of edges, we prove here that ψ_n /n^(d−1) converges almost surely towards a constant.
Fichier principal
Vignette du fichier
Size of a minimal cutset in supercritical first passage percolation.pdf (530) Télécharger le fichier
Origin Files produced by the author(s)
Loading...
HAL

Has version hal-03158726 Journal article Barbara Dembin, Marie Théret. Size of a minimal cutset in supercritical first passage percolation. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2020, 56 (2), pp.1419-1439. ⟨10.1214/19-AIHP1008⟩. ⟨hal-03158726⟩

Dates and versions

hal-01726599 , version 1 (09-03-2018)
hal-01726599 , version 2 (12-04-2019)

Identifiers

Cite

Barbara Dembin, Marie Théret. Size of a minimal cutset in supercritical first passage percolation. 2019. ⟨hal-01726599v2⟩
182 View
392 Download

Altmetric

Share

More