Observables of random spanning trees in random environment

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.