Loop cluster on the discrete circle

Abstract

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 $n\to \infty$, 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 $n\to \infty$, 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 ${\mathbb{S}}^1$.

References in corpus (2)

• Markov loops in discrete spaces
• Markovian loop clusters on graphs

Cited by in corpus (2)

• Poisson ensembles of loops of one-dimensional diffusions
• Markov loops, complex free field and Eulerian circuits