Diameter of uniform spanning trees on random weighted graphs

Abstract

For any edge weight distribution, we consider the uniform spanning tree (UST) on finite graphs with i.i.d. random edge weights. We show that, for bounded degree expander graphs and finite boxes of ${\mathbb{Z}}^d$, the diameter of the UST is of order $n^{1/2+o(1)}$ with high probability, where $n$ is the number of vertices.

25 pages, 1 figure. To appear in Annales de l'Institut Henri Poincaré - Probabilités et Statistiques