Open Access
2023 Models of random subtrees of a graph
Luis Fredes, Jean-François Marckert
Author Affiliations +
Probab. Surveys 20: 722-801 (2023). DOI: 10.1214/23-PS22
Abstract

Consider a connected graph G=(E,V) with N=|V| 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 nN (the spanning tree case correspond to n=N, 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 nN. We highlight the case of the uniform subtree of Z2 with n nodes, containing the origin (0,0) 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.

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Luis Fredes and Jean-François Marckert "Models of random subtrees of a graph," Probability Surveys 20(none), 722-801, (2023). https://doi.org/10.1214/23-PS22
Received: 1 October 2021; Published: 2023
Vol.20 • 2023
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