  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 $\mathbb{S}^{1}$.
