Bernoulli hyper-edge percolation on Zd

We consider Bernoulli hyper-edge percolation on \(\mathbb{Z}^d\). This model is a generalization of Bernoulli bond percolation. An edge connects exactly two vertices and a hyper-edge connects more than two vertices. As in the classical Bernoulli bond percolation, we open hyper-edges independently in...

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Bibliographic Details
Published in:arXiv.org 2022-02
Main Author: Chang, Yinshan
Format: Article
Language:English
Subjects:
Apexes
Graph theory
Percolation
Phase transitions
Online Access:Get full text
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Summary:We consider Bernoulli hyper-edge percolation on \(\mathbb{Z}^d\). This model is a generalization of Bernoulli bond percolation. An edge connects exactly two vertices and a hyper-edge connects more than two vertices. As in the classical Bernoulli bond percolation, we open hyper-edges independently in a homogeneous manner with certain probabilities parameterized by a parameter \(u\in[0,1]\). We discuss conditions for non-trivial phase transitions when \(u\) varies. We discuss the conditions for the uniqueness of the infinite cluster. Also, we provide conditions under which the Grimmett-Marstrand type theorem holds in the supercritical regime.
ISSN:2331-8422