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.
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 lecture notes 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 arXiv:1804.04120 and arXiv:1512.08509 and forthcoming work with Perla Sousi.