Benjamin C. Wallace

Learning seminar on statistical physics

We will study rigorous methods for understanding phase transitions and critical phenomena. We will restrict our attention to classical models of statistical physics, but many of these methods can be applied in quantum statistical mechanics and quantum field theory. Our emphasis will be on those methods that are approximate but apply to rather general classes of systems (as opposed to exactly solvable models), which frequently involve geometric representations of the model under consideration. Primary examples are the cluster expansion and random walk representation for spin systems.

The topics of the seminar can be divided into the following three rough and partly overlapping categories:
  1. Foundations (Jan 24-Feb 28)
  2. Phase transitions (Feb 28-Apr 10)
  3. Critical phenomena (Apr 18-May 2)
The notes from the seminar are available here.

Jan 24
Introduction to equilibrium statistical mechanics
Topics: ensembles, relation to quantum theory, spin systems
Jan 31
Gibbs states for the classical Ising model
Topics: definition, reduction of existence proof
Feb 14
Correlation inequalities and existence of Gibbs states
Topics: Holley/FKG inequalities, existence, extremality and other properties
Feb 28
Pressure and magnetization
Topics: existence, properties, characterization of phase transition
Mar 7
High-temperature regime
Topics: high-temperature expansion, uniqueness, 1-dimensional model
Mar 14
Low-temperature regime and polymer models
Topics: low-temperature expansion, Peierls argument, phase transition, polymer models
Mar 21
General cluster expansion
Topics: formal expansion, convergence criterion
Mar 28
Application of cluster expansion
Topics: Ising model in a large magnetic field
Apr 10
Mayer and virial expansions
Topics: lattice gas, expansions
Apr 18
Random walk representations
Topics: Gaussian fields, BFS representation
May 2
Final remarks on critical phenomena
Topics: Gaussian upper bounds, critical exponents, mean-field behaviour
Other topics
Topics we did not have time to cover, but could be of interest to anyone who has been following, include:
spin systems with continuous symmetry (Mermin-Wagner theorem, infrared bound, reflection positivity), Lee-Yang circle theorem, solvable models (e.g. 1-dimensional and mean-field Ising model, spherical model), long-range models, random current representation and application to continuity of Ising phase transition, supersymmetry and Grassmannian integral representation of self-avoiding walk, cycle/loop representations of quantum spin systems, renormalisation group methods

Cluster expansions: Representations of walks and spin systems: Renormalisation group:
Left: two-dimensional self-avoiding walk with 1 million steps. Right: interface curve in the critical two-dimensional Ising model with Dobrushin boundary conditions.