Models of random spanning trees

TL;DR

This paper develops new tools for analyzing the properties of random minimum spanning trees (MST) generated by randomized algorithms, extending to various distributions of edge weights.

Contribution

It introduces methods for the quantitative study of random MSTs, including cases with independent weights from arbitrary distributions.

Findings

Provides a framework for analyzing random MSTs with i.i.d. weights.

Extends analysis to product measures with arbitrary independent distributions.

Enhances understanding of the mathematical properties of random MSTs.

Abstract

There are numerous randomized algorithms to generate spanning trees in a given ambient graph; several target the uniform distribution on trees (UST), while in practice the fastest and most frequently used draw random weights on the edges and then employ a greedy algorithm to choose the minimum-weight spanning tree (MST). Though MST is a workhorse in applications, the mathematical properties of random MST are far less explored than those of UST. In this paper we develop tools for the quantitative study of random MST. We consider the standard case that the weights are drawn i.i.d. from a single distribution on the real numbers, as well as successive generalizations that lead to \emph{product measures}, where the weights are independently drawn from arbitrary distributions.