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# Talk: Martin Barlow

## Quenched and Annealed Heat Kernels on the Uniform Spanning Tree

### Martin Barlow

#### University of British Columbia, Canada

The uniform spanning tree (UST) on ${Z}^{2}$ was constructed by Pemantle in 1991 as the limit of the UST on finite boxes $\left[-n,n{\right]}^{2}$. In this talk I will discuss the form of the heat kernel (i.e. random walk transition probability) on this random graph. I will compare the bounds for the UST with those obtained earlier for supercritical percolation.

This is joint work with Takashi Kumagai and David Croydon.