A convergence result on the lengths of Markovian loops

Abstract

Consider a sequence of Poisson point processes of non-trivial loops with certain intensity measures $(\mu^{(n)})_n$, where each $\mu^{(n)}$ is explicitly determined by transition probabilities $p^{(n)}$ of a random walk on a finite state space $V^{(n)}$ together with an additional killing parameter $c^{(n)}=e^{-a\cdot\sharp V^{(n)}+o(\sharp V^{(n)})}$. We are interested in asymptotic behavior of typical loops. Under general assumptions, we study the asymptotics of the length of a loop sampled from the normalized intensity measure $\bar{\mu}^{(n)}$ as $n\rightarrow\infty$. A typical loop is small for $a=0$ and extremely large for $a=\infty$. For $a=(0,\infty)$, we observe both small and extremely large loops. We obtain explicit formulas for the asymptotics of the mass of intensity measures, the asymptotics of the proportion of big loops, limit results on the number of vertices (with multiplicity) visited by a loop sampled from $\bar{\mu}^{(n)}$. We verify our general assumptions for random walk loop soups on discrete tori and truncated regular trees. Finally, we consider random walk loop soups on complete graphs. Here, our general assumptions are violated. In this case, we observe different asymptotic behavior of the length of a typical loop.