In the loop soup on the complete graph K_n, the number of clusters of size d converges to a mixed Poisson law with mixing variable exp(-dZ/κ), and clusters larger than n^{1-ε} appear almost surely.
Loop cluster on the discrete circle
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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 $n\rightarrow\infty$, 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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Loop clusters on complete graphs
In the loop soup on the complete graph K_n, the number of clusters of size d converges to a mixed Poisson law with mixing variable exp(-dZ/κ), and clusters larger than n^{1-ε} appear almost surely.