Local and global structure of uniform spanning trees

Asaf Nachmias

A spanning tree of a finite connected graph G is a connected subgraph of G touching every vertex and containing no cycles. We will consider uniformly drawn spanning trees of "high-dimensional" graphs and study their local behavior (for example, what is the typical number of leaves, vertices of degree k, or the number of occurrences of any fixed subtree) and their global structure (for example, its typical diameter or height seen from random vertex). Based on recent joint works with Noga Alon, Eleanor Archer, Peleg Michaeli, Yuval Peres, and Matan Shalev.

Prerequisites: We assume familiarity with the basic theory of electric networks, see for example, Chapter 2 of the Saint Flour Lecture Notes.