Observables of random spanning trees in random environment

TL;DR

This thesis investigates the properties of random spanning trees in random environments across various graphs, analyzing observables as functions of disorder strength and comparing them to classical spanning tree models.

Contribution

It introduces the study of RSTRE across different graph families, providing new insights into diameter and local observables, including novel results on Euclidean lattices and supercritical random graphs.

Findings

Diameter results align with previous studies

New insights into local observables

Diameter of unweighted UST on supercritical graphs

Abstract

In this thesis, we study a new disordered system called random spanning tree in random environment (RSTRE) across different families of graphs with varying disorder distributions. We examine several observables as functions of the disorder strength (inverse temperature) $\beta \geq 0$, and compare their values to the extreme cases $\beta = 0$ and $\beta \rightarrow \infty$, which correspond to the uniform spanning tree (UST) and the minimum spanning tree (MST), respectively. The results concerning the diameter are in line with those of arXiv:2311.01808 and arXiv:2410.16830, while the findings on local observables are based on arXiv:2410.16836. This thesis also includes new material on the RSTRE in the Euclidean infinite lattice, as well as a novel result on the diameter of the unweighted UST on a slightly supercritical random graph.