Repeat times and a two-weight UST model

TL;DR

We study the diameter of a random weighted uniform spanning tree on the complete graph with edge weights $w(e)=n^{1+\gamma}$ with probability $1/n$ and $1$ otherwise, focusing on the large-disorder regime $\gamma \ge 5$. The core approach links the model to critical Erdős–Rényi graphs, employing a Brownian-excursion description of repeat-times and concentration arguments for sums of component diameters, together with a contraction/Spatial Markov decomposition of the tree. We establish a matching lower and upper bound showing the averaged diameter scales as $n^{1/3}$ up to a $\log n$ factor, with an extra $\log\log n$ correction in the upper bound, i.e., $\operatorname{diam}(\mathcal{T}) \asymp n^{1/3} \log n$ up to logs in the large-$\gamma$ regime. The results also contrast this regime with $\gamma<0$, where the diameter behaves like that of the unweighted UST ($\asymp \sqrt{n}$), and they discuss the challenging intermediate regime $0\le \gamma <5$ with conjectures and partial bounds.

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 $n^{1+γ}$ with probability $1/n$ and $1$ otherwise. Whenever $γ$ 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 Erdős-Rényi graph, as well as concentration bounds on the sums of diameters of these components.

Theorems & Definitions (45)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3: Theorem 1.3 in NP08
• Lemma 2.4
• Lemma 2.5
• Remark 2.6
• proof : Proof of Lemma \ref{['L:collection_RG_facts']}