Limits of uniform spanning trees and forests

par Eleanor Archer (Université Paris Dauphine)

jeudi 3 déc. 2026, 11:00 → 12:15 Europe/Paris

Auditorium 3 (Toulouse School of Economics)

Abstract : Given a finite, connected graph G=(V,E), the uniform spanning tree is the subgraph chosen uniformly from the set of spanning trees of G. This a fundamental object in probability theory, with origins dating back to the works of Kirchhoff in 1876.

An old result of Grimmett (1980), along with recent results of Nachmias and Peres (2022), and Kusz (2025), show that uniform spanning trees admit a universal local limit. More recently, a generalisation of the uniform spanning tree, known as the massive spanning forest, was introduced. It was shown by D'Achille, Enriquez and Melotti (2026) that on the complete graph with n vertices, these forests also admit a (local) limit as n goes to infinity. This limit can fall into three regimes depending on the growth of a certain tuning parameter.

I will start by giving a brief introduction to the topic of uniform spanning trees and their local limits. I will then discuss the generalisation to massive spanning forests, and explain a recent result, obtained in collaboration with Breki Pálsson (University of Iceland), in which we obtain universality of the local limit, meaning that we obtain the same limit on any growing sequence of regular graphs (with degrees tending to infinity).