Supercritical loop percolation on \(\mathbb{Z}^d\) for \(d\geq 3\)

In this paper, we are interested in the loop cluster model on \(\mathbb{Z}^d\) for \(d\geq 3\). It is a long range model with two parameters \(\alpha\) and \(\kappa\), where the non-negative parameter \(\alpha\) measures the amount of loops, and \(\kappa\) plays the role of killing on vertices penal...

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Bibliographic Details
Published in:arXiv.org 2015-04
Main Author: Chang, Yinshan
Format: Article
Language:English
Subjects:
Apexes
Clusters
Mathematical models
Parameters
Percolation
Random walk
Random walk theory
Online Access:Get full text
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Summary:In this paper, we are interested in the loop cluster model on \(\mathbb{Z}^d\) for \(d\geq 3\). It is a long range model with two parameters \(\alpha\) and \(\kappa\), where the non-negative parameter \(\alpha\) measures the amount of loops, and \(\kappa\) plays the role of killing on vertices penalizing (\(\kappa\geq 0\)) or favoring (\(\kappa\alpha_c\) large balls in the infinite cluster are finally very regular in the sense of \cite{Sapozhnikov2014}, which implies that large balls are finally very good in the sense of \cite{BarlowMR2094438}. By \cite{BarlowMR2094438} and \cite{BarlowHamblyMR2471657}, we have Harnack's inequality and Gaussian type estimate for simple random walks on the infinite cluster for all \(\alpha>\alpha_c\).
ISSN:2331-8422