Models of random spanning trees
This work provides foundational theoretical tools for analyzing random MST, a widely used but mathematically understudied object, benefiting researchers in probability and combinatorial optimization.
The paper develops mathematical tools for the quantitative study of random minimum-weight spanning trees (MST), which are less explored than uniform spanning trees (UST), despite MST being widely used in practice. The analysis covers i.i.d. weights and product measures with arbitrary distributions.
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.