Open Access
2015 Loop cluster on discrete circles
Yinshan Chang
Author Affiliations +
Electron. J. Probab. 20: 1-32 (2015). DOI: 10.1214/EJP.v20-3176


The loop clusters of a Poissonian ensemble of Markov loops on a finite or countable graph have been studied in \cite{Markovian-loop-clusters-on-graphs}. In the present article, we study the loop clusters associated with a rotation invariant nearest neighbor walk on the discrete circle $G^{(n)}$ with $n$ vertices. We prove a convergence result of the loop clusters on $G^{(n)}$, as $n\rightarrow\infty$, under suitable condition of the parameters. These parameters are chosen in such a way that the rotation invariant nearest neighbor walk on $G^{(n)}$, as $n\rightarrow\infty$, converges to a Brownian motion on circle $\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$ with certain drift and killing rate. In the final section, we show that several limit results are predicted by Brownian loop-soup on $\mathbb{S}^{1}$.


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Yinshan Chang. "Loop cluster on discrete circles." Electron. J. Probab. 20 1 - 32, 2015.


Accepted: 6 January 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1334.60206
MathSciNet: MR3311215
Digital Object Identifier: 10.1214/EJP.v20-3176

Primary: 60K15
Secondary: 60K35

Keywords: loop clusters , loop soup , Renewal process , subordinator

Vol.20 • 2015
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