Repeat times and a two-weight UST model

TL;DR

This paper analyzes a random weighted uniform spanning tree model on complete graphs, revealing that for large gamma, the tree diameter scales as n^{1/3} log n, using Erdős-Rényi component estimates.

Contribution

It introduces a novel weighted UST model with specific edge weight distributions and derives the typical diameter behavior for large gamma values.

Findings

Tree diameter scales as n^{1/3} log n for large gamma.

Uses estimates on repeat times in Erdős-Rényi graphs.

Provides concentration bounds on component diameters.

Abstract

We study a model of random weighted uniform spanning trees on the complete graph with $n$ vertices, where each edge is assigned a weight of $n^{1+\gamma}$ with probability $1/n$ and $1$ otherwise. Whenever $\gamma$ is large enough, we prove that the diameter of the resulting tree is typically of order $n^{1/3} \log n$, up to a $\log \log n$ correction. Our approach uses estimates on repeat times for selecting components in a critical Erd\H{o}s-R\'enyi graph, as well as concentration bounds on the sums of diameters of these components.