TCC Course: Statistical Physics Models in High Dimensions

Lecturer contact information:

  • Email:
  • Phone: (01225) 384264
  • 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, where the technique of lace-expansion developed by Brydges, Spencer, Hara, Slade, and others has been successful to a large extent. 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.

    Lectures are on Mondays 10:00-12:00, starting 16 January 2023, for 8 weeks, on MS Teams; see TCC webpage.