Open Access
2015 A convergence result on the lengths of Markovian loops
Yinshan CHANG
Author Affiliations +
Electron. Commun. Probab. 20: 1-11 (2015). DOI: 10.1214/ECP.v20-4263


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 $a=}$. 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.


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Yinshan CHANG. "A convergence result on the lengths of Markovian loops." Electron. Commun. Probab. 20 1 - 11, 2015.


Accepted: 14 October 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1329.60249
MathSciNet: MR3417445
Digital Object Identifier: 10.1214/ECP.v20-4263

Primary: 60J10

Keywords: Markovian loop , Random walk

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