### Scaling limit of high-dimensional uniform spanning trees

Probability Seminar

29th October 2021, 3:30 pm – 4:30 pm

Fry Building, 2.04 (also on Zoom)

A spanning tree of a finite connected graph G is a connected subgraph of G that touches every vertex and contains no cycles. In this talk we will consider uniformly drawn spanning trees on families of ``high-dimensional'' graphs, and show that, under appropriate rescaling, they converge in distribution as metric-measure spaces to Aldous' Brownian CRT. This extends an earlier result of Peres-Revelle (2004) who previously showed a form of finite-dimensional convergence. As a consequence of our result, we deduce that certain functionals of the uniform spanning trees also converge to their continuum analogues on the CRT, including the height, the diameter and the law of a simple random walk. Based on joint work with Asaf Nachmias and Matan Shalev.

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