Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Size of a minimal cutset in supercritical first passage percolation

Barbara Dembin and Marie Théret

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Abstract

We consider the standard model of i.i.d. first passage percolation on $\mathbb{Z}^{d}$ given a distribution $G$ on $[0,+\infty ]$ (including $+\infty $). 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 $\mathbb{Z}^{d}$. 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., $h(n)/n\rightarrow 0$ 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 $\psi_{n}$ the minimal cardinality of such a set of edges, we prove here that $\psi_{n}/n^{d-1}$ converges almost surely towards a constant.

Résumé

Considérons le modèle de percolation de premier passage i.i.d. dans $\mathbb{Z}^{d}$ associé à la distribution $G$ sur $[0,+\infty ]$ (en incluant $+\infty $). Supposons que $G(\{0\})>1-p_{c}(d)$, i.e., les arêtes ayant un temps de passage strictement positif sont dans un régime sous-critique de percolation dans $\mathbb{Z}^{d}$. Considérons un cylindre ayant pour base un hyperrectangle de dimension $d-1$ de côté de longueur $n$ et de hauteur $h(n)$ avec $h(n)$ négligeable devant $n$ (i.e., $h(n)/n\rightarrow 0$ quand $n$ tend vers l’infini). Nous nous intéressons à la quantité maximale de flux pouvant circuler de haut en bas dans le cylindre. Le flux maximal renormalisé par $n^{d-1}$ converge vers la constante de flux qui est nulle dans le cas où $G(\{0\})>1-p_{c}(d)$. L’étude du flux maximal est équivalente à l’étude des ensembles d’arêtes de capacité minimale séparant le haut du bas du cylindre. Notons $\psi_{n}$ le cardinal minimal de tels ensembles d’arêtes, nous prouvons ici que $\psi_{n}/n^{d-1}$ converge presque sûrement vers une constante.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1419-1439.

Dates
Received: 8 March 2018
Revised: 7 February 2019
Accepted: 28 May 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345643

Digital Object Identifier
doi:10.1214/19-AIHP1008

Mathematical Reviews number (MathSciNet)
MR4076789

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
First passage percolation Maximal flow Minimal cutset Size of a cutset

Citation

Dembin, Barbara; Théret, Marie. Size of a minimal cutset in supercritical first passage percolation. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1419--1439. doi:10.1214/19-AIHP1008. https://projecteuclid.org/euclid.aihp/1584345643


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