Random spanning trees, Tel Aviv university, Fall 2014
Consider the uniform measure on the set of all spanning trees
in the box of side length n in the integer lattice. Take n to
infinity and consider the limiting measure. It is easy to see
that this is a measure on infinite subgraphs of Z^d that have
no cycles. Is it almost surely a tree, or is the resulting
limit almost surely a disconnected forest? We will prove a remarkable result
of Pemantle (1991) that the limiting measure is concentrated on trees if and only if d <= 4 and study basic properties of the limiting measure (how to draw such trees/forests directly, recurrence/transience, isoperimetric inequalities and more).
We will see these relate to various contemporary topics in discrete probability
(which we will not assume any knowledge of): random walks,
electrical networks, discrete potential theory, percolation,
determinantal processes and even some elementary geometric
group theory. We will assume only basic knowledge (first year undergrad courses) in probability, linear algebra and analysis. This course may be suitable to strong undergraduate students.