Abstract
Consider a connected graph with vertices. The main purpose of this paper is to explore the question of uniform sampling of a subtree of G with n nodes, for some (the spanning tree case correspond to , and is already deeply studied in the literature). We provide new asymptotically exact simulation methods using Markov chains for general connected graphs G, and any . We highlight the case of the uniform subtree of with n nodes, containing the origin for which Schramm asked several questions. We produce pictures, statistics, and some conjectures.
A second aim of the paper is devoted to surveying other models of random subtrees of a graph, among them, DLA models, the first passage percolation, the uniform spanning tree and the minimum spanning tree. We also provide new models, some statistics, and some conjectures.
Acknowledgments
We would like to thank the Reviewer for taking the necessary time and effort to review the manuscript. We sincerely appreciate all his/her valuable comments and suggestions, which helped us to improve the quality of the manuscript.
Citation
Luis Fredes. Jean-François Marckert. "Models of random subtrees of a graph." Probab. Surveys 20 722 - 801, 2023. https://doi.org/10.1214/23-PS22
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