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Uniform spanning trees in high dimension 3

Banff International Research Station for Mathematical Innovation and Discovery

Uniform spanning trees have played an important role in modern probability theory as a non-trivial statistical mechanics model that is much more tractable than other (more physically relevant) models such as percolation and the Ising model. It also enjoys many connections with other topics in probability and beyond, including electrical networks, loop-erased random walk, dimers, sandpiles, l^2 Betti numbers, and so on. In this course, I will introduce the model and explain how we can understand its large-scale behaviour at and above the upper critical dimension d=4.</p>

In Lecture 1 I will discuss the main sampling algorithms for the UST and the connectivity/disconnectivity transition in dimension 4. Chapter 4 of my <a href="https://www.dropbox.com/s/pt0tyfdrffrmsp1/RW_and_USTs_Public.pdfdl=0" style="color:blue;">lecture notes</a> contains complementary material that will flesh out many of the details from this lecture. Lectures 2 and 3 will discuss scaling exponents in dimensions d>=4 and will be based primarily on the papers <a href="https://arxiv.org/abs/1804.04120" style="color:blue;"> arXiv:1804.04120</a> and <a href="https://arxiv.org/abs/1512.08509" style="color:blue;"> arXiv:1512.08509</a> and forthcoming work with Perla Sousi.</p>

Mathematics; Probability Theory

video/mp4

Author affiliation: University of Cambridge

2021-02-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77267

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/