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November 2022 Constrained-degree percolation in random environment
Rémy Sanchis, Diogo C. dos Santos, Roger W. C. Silva
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Ann. Inst. H. Poincaré Probab. Statist. 58(4): 1887-1899 (November 2022). DOI: 10.1214/21-AIHP1231


We consider the Constrained-degree percolation model in random environment on the square lattice. In this model, each vertex v has an independent random constraint κv which takes the value j{0,1,2,3} with probability ρj. Each edge e attempts to open at a random uniform time Ue in [0,1], independently of all other edges. It succeeds if at time Ue both its end-vertices have degrees strictly smaller than their respectively attached constraints. We show that this model undergoes a non-trivial phase transition when ρ3 is sufficiently large. The proof consists of a decoupling inequality, the continuity of the probability for local events, and a coarse-graining argument.

Nous considérons le modèle de percolation à degré contraint dans un environnement aléatoire sur le réseau carré. Dans ce modèle, chaque sommet v a une contrainte aléatoire indépendante κv qui prend la valeur j{0,1,2,3} avec probabilité ρj. Chaque arête e tente de s’ouvrir à un temps uniforme aléatoire Ue dans [0,1], indépendamment de toutes les autres arêtes. Il réussit si à l’instant Ue ses deux sommets d’extrémité ont des degrés strictement inférieurs à leurs contraintes respectives attachées. Nous montrons que ce modèle subit une transition de phase non triviale lorsque ρ3 est suffisamment grand. La preuve consiste en une inégalité de découplage, la continuité de la probabilité des événements locaux et un argument de renormalisation.

Funding Statement

The research of Rémy Sanchis was partially supported by FAPEMIG (PPM 0600-16) and CNPq.
Diogo dos Santos was partially supported by PNPD/CAPES.
Roger Silva was partially supported by FAPEMIG (Universal APQ-00774-21).
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brasil (CAPES), Finance Code 001.


The authors are grateful to two anonymous referees for their valuable comments and suggestions.


Download Citation

Rémy Sanchis. Diogo C. dos Santos. Roger W. C. Silva. "Constrained-degree percolation in random environment." Ann. Inst. H. Poincaré Probab. Statist. 58 (4) 1887 - 1899, November 2022.


Received: 4 November 2020; Revised: 30 October 2021; Accepted: 9 November 2021; Published: November 2022
First available in Project Euclid: 6 October 2022

Digital Object Identifier: 10.1214/21-AIHP1231

Primary: 60K35
Secondary: 82B43

Keywords: Constrained-degree percolation , continuity , Exponential decay of correlations , renormalization

Rights: Copyright © 2022 Association des Publications de l’Institut Henri Poincaré


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Vol.58 • No. 4 • November 2022
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