Random two-component spanning forests

TL;DR

This paper investigates properties of random two-component spanning forests in finite and infinite graphs, providing formulas for moments, inclusion probabilities, and edge separation probabilities, with limits on infinite periodic graphs.

Contribution

It introduces explicit formulas for moments and probabilities related to 2SFs and analyzes their limits on infinite periodic graphs, advancing understanding of their structure.

Findings

Formulas for first and second moments of component sizes

Vertex-inclusion probabilities for one or two vertices

Edge separation probabilities in 2SFs

Abstract

We study random two-component spanning forests ($2$SFs) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an edge separates the components. We compute the limit of these quantities when the graph tends to an infinite periodic graph in ${\mathbb R}^d$.