Local limits of random spanning trees in random environment

TL;DR

This paper investigates the local limits and edge overlap of random spanning trees in a weighted environment on complete graphs, revealing phase transitions depending on the parameter eta.

Contribution

It characterizes the phase transition in local limits and edge overlap of RSTRE as eta varies with respect to the graph size n.

Findings

Edge overlap is approximately eta for small eta

Edge overlap approaches n for large eta

Local limit transitions from uniform spanning tree to minimum spanning tree around eta  n

Abstract

We study the edge overlap and local limit of the random spanning tree in random environment (RSTRE) on the complete graph with $n$ vertices and weights given by $\exp(-\beta \omega_e)$ for $\omega_e$ uniformly distributed on $[0,1]$. We show that for $\beta$ growing with $\beta = o(n/\log n)$, the edge overlap is $(1+o(1)) \beta$, while for $\beta$ much larger than $n \log^2 n$, the edge overlap is $(1-o(1))n$. Furthermore, there is a transition of the local limit around $\beta = n$. When $\beta = o(n/ \log n)$ the RSTRE locally converges to the same limit as the uniform spanning tree, whereas for $\beta$ larger than $n \log^\lambda n$, where $\lambda = \lambda(n) \rightarrow \infty$ arbitrarily slowly, the local limit of the RSTRE is the same as that of the minimum spanning tree.