# Benjamin C. Wallace

## Learning seminar on statistical physics

**Overview**

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:

- Foundations (Jan 24-Feb 28)
- Phase transitions (Feb 28-Apr 10)
- Critical phenomena (Apr 18-May 2)

**Schedule**

- Jan 24
- Introduction to equilibrium statistical mechanics
- Topics: ensembles, relation to quantum theory, spin systems
- Notes
- Jan 31
- Gibbs states for the classical Ising model
- Topics: definition, reduction of existence proof
- Notes
- Feb 14
- Correlation inequalities and existence of Gibbs states
- Topics: Holley/FKG inequalities, existence, extremality and other properties
- Notes
- Feb 28
- Pressure and magnetization
- Topics: existence, properties, characterization of phase transition
- Notes
- Mar 7
- High-temperature regime
- Topics: high-temperature expansion, uniqueness, 1-dimensional model
- Notes
- Mar 14
- Low-temperature regime and polymer models
- Topics: low-temperature expansion, Peierls argument, phase transition, polymer models
- Notes
- Mar 21
- General cluster expansion
- Topics: formal expansion, convergence criterion
- Notes
- Mar 28
- Application of cluster expansion
- Topics: Ising model in a large magnetic field
- Notes
- Apr 10
- Mayer and virial expansions
- Topics: lattice gas, expansions
- Notes
- Apr 18
- Random walk representations
- Topics: Gaussian fields, BFS representation
- Notes
- May 2
- Final remarks on critical phenomena
- Topics: Gaussian upper bounds, critical exponents, mean-field behaviour
- Notes

**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

**References**

- Bau16: R. Bauerschmidt, Ferromagnetic spin systems, 2016.
- BDGS10: R. Bauerschmidt, H. Duminil-Copin, J. Goodman, and G. Slade, Lectures on self-avoiding walks, 2010.
- FV17: S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, 2017.
- Geo11: H.-O. Georgii, Gibbs measures and phase transitions, 2011.
- GHM99: H.-O. Georgii, O. Häggström, and C. Maes, The random geometry of equilibrium phases, 1999.
- GJ87: J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, 1987.
- PS16: R. Peled, Y. Spinka, Lecture Notes on the Spin and Loop O(n) models, 2016.
- Sim93: B. Simon, The Statistical Mechanics of Lattice Gases, Vol. 1, 1993.

- D. Brydges, A short course on cluster expansions, 1986.
- D. Ueltschi, Cluster expansions and correlation functions, 2005.

- D. Brydges, J. Imbrie, and G. Slade, Functional integral representations for self-avoiding walk, 2009.
- H. Duminil-Copin, Graphical representations of lattice spin models, 2016.
- H. Duminil-Copin, Lectures on the Ising and Potts models on the hypercubic lattice, 2017.
- R. Fernández, J. Fröhlich, and A. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory, 1992.

- R. Bauerschmidt, D. Brydges, and G. Slade, Renormalisation group analysis of 4D spin models and self-avoiding walk, 2016.
- D. Brydges, Lectures on the Renormalisation Group, 2009.