Markovian loop clusters on graphs

2012/11/01 by Yves Le Jan, Jan, Yves Le, Sophie Lemaire +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Complex Network Analysis Techniques #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1211.0300

openalex publication_date 2012/11/01 · openalex created_date 2022/08/13 · openalex updated_date 2026/07/28

Abstract

We study the loop clusters induced by Poissonian ensembles of Markov loops on a finite or countable graph (Markov loops can be viewed as excursions of Markov chains with a random starting point, up to re-rooting). Poissonian ensembles are seen as a Poisson point process of loops indexed by 'time'. The evolution in time of the loop clusters defines a coalescent process on the vertices of the graph. After a description of some general properties of the coalescent process, we address several aspects of the loop clusters defined by a simple random walk killed at a constant rate on three different graphs: the integer number line ℤ, the integer lattice ℤ^d with d≥ 2 and the complete graph. These examples show the relations between Poissonian ensembles of Markov loops and other models: renewal process, percolation and random graphs.

Citations

• On random graphs. I.

Cited by

• A multiplicative coalescent with asynchronous multiple mergers
• 1+O(1) asymptotics for loop percolation in five and higher dimensions
• Convergence of the KMP model to the KPZ equation

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