Fully-connected bond percolation on $\mathbb{Z}^d$

Dereudre, David

Type de document :

Article dans une revue scientifique: Article original

DOI :

10.1007/s00440-021-01088-8

URL permanente :

http://hdl.handle.net/20.500.12210/121651

Titre :

Fully-connected bond percolation on $\mathbb{Z}^d$

Auteur(s) :

Dereudre, David [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Titre de la revue :

Probability Theory and Related Fields

Pagination :

547–579

Éditeur :

Springer Verlag

Date de publication :

2022-06

ISSN :

0178-8051

Mot(s)-clé(s) en anglais :

FK-percolation
random cluster model
phase transition
FKG inequalities
DLR equations

Discipline(s) HAL :

Mathématiques [math]

Résumé en anglais : [en]

We consider the bond percolation model on the lattice $\mathbb{Z}^ d$ ($d \ge 2$) with the constraint to be fully connected. Each edge is open with probability $p \in (0, 1)$, closed with probability $1-p$ and then the ...
Lire la suite >We consider the bond percolation model on the lattice $\mathbb{Z}^ d$ ($d \ge 2$) with the constraint to be fully connected. Each edge is open with probability $p \in (0, 1)$, closed with probability $1-p$ and then the process is conditioned to have a unique open connected component (bounded or unbounded). The model is defined on $\mathbb{Z}^ d$ by passing to the limit for a sequence of finite volume models with general boundary conditions. Several questions and problems are investigated: existence, uniqueness, phase transition, DLR equations. Our main result involves the existence of a threshold $0 < p^* (d) < 1$ such that any infinite volume model is necessary the vacuum state in subcritical regime (no open edges) and is non trivial in the supercritical regime (existence of a stationary unbounded connected cluster). Bounds for $p^* (d)$ are given and show that it is drastically smaller than the standard bond percolation threshold in$\mathbb{Z}^ d$. For instance $0.128 < p^* (2) < 0.202$ (rigorous bounds) whereas the 2D bond percolation threshold is equal to $1/2$.Lire moins >

Langue :

Anglais

Comité de lecture :

Oui

Audience :

Internationale

Vulgarisation :

Non

Projet ANR :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

Date de dépôt :

2025-01-24T14:17:11Z

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Fully-connected bond percolation on $\mathbb{Z}^d$

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