Lecturer contact information:
Description:
Many systems consisting of a large number of interacting components undergo a phase transition,
that is, an abrupt change in qualitative behaviour as a parameter crosses a critical value.
Often the first step in understanding such phenomena is to study a simplified model, where the
geometry of the underlying space plays little or no role; for example, the model placed on a complete
graph or a regular tree. This approach is useful due to the phenomenon that the behaviour of the
simplified model is typically also reflected on lattices in Euclidean space of sufficiently high
dimension d. However, rigorously establishing the latter is often challenging. This
course will explain methods that can be used to do this, in a few example systems. We will
discuss the example of loop-erased random walk in d>4, where a direct probabilistic
analysis is possible. We will also discuss self-avoiding walk in d>4 and percolation in d>6.
Here the technique of the lace-expansion developed by Brydges, Spencer, Hara, Slade, and others
has been the most powerful tool. There are also very recent breakthroughs by Duminil-Copin and Panis that
avoid lace-expansion. This course will give insight into both approaches.
We will also touch on other examples and open problems along the way.
Pre-requisites:
measure-theoretic probability; basic properties of
random walk on Zd such as recurrence/transience; familiarity with the extinction/survival
transition for branching processes.
Assessment:
Students wanting to take the course for credit should contact the lecturer for details.
Lecture times:Fridays 10:00-12:00, starting 2 May 2025 for 8 weeks, ending 20 June 2025 on MS Teams; see TCC webpage.