Models of random subtrees of a graph

TL;DR

This paper develops new Markov chain-based methods for uniformly sampling subtrees of a connected graph, explores specific cases like the lattice $ ext{Z}^2$, and surveys various models of random subtrees with new insights.

Contribution

It introduces asymptotically exact simulation techniques for subtrees of any size in general graphs and surveys existing models with new conjectures and models.

Findings

New simulation methods for subtrees in general graphs

Analysis of uniform subtrees in $ ext{Z}^2$ lattice

Survey of random subtree models with new conjectures

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 $n\leq N$ (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 $n\leq N$. We highlight the case of the uniform subtree of $\mathbb{Z}^2$ 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.