# MA4L2 Content

**Content: **Statistical mechanics describes physical systems with a huge number of particles.

In physics, the goal is to describe macroscopic phenomena in terms of microscopic models and to give a meaning to notions such as temperature or entropy. Mathematically, it can be viewed as the study of random variables with spatial dependence. Models of statistical mechanics form the background for recent advances in probability theory and stochastic analysis, such as SLE and the theory of regularity structures. So, they form an important background for understanding these topics of modern mathematics.

The module will give a thorough mathematical introduction to the Ising model and to the gaussian free field on regular graphs, and to the theory of infinite volume Gibbs measures.

**Aims: **To familiarise students with statistical mechanicsmodels, phase transitions, and critical behaviour.

**Objectives: **By the end of the module students should be able to:

- Apply basic ideas of phase transitions and critical behaviour to lattice systems of statistical mechanics
- Understand the theory of infinite volume Gibbs measures
- Understand how large complex systems at equilibrium can be described from microscopic rules
- Have understood basic ideas of phase transitions and critical behaviour in the case of the Ising model and the

gaussian free field; they will have mastered the theory of infinite volume Gibbs measures.

**Books: **We will mainly follow Chapters 3, 6, 7 of the new introductory textbook:

Sacha Friedli and Yvan Velenik, Equilibrium Statistical Mechanics of Classical Lattice Systems: a Concrete Introduction. Available at

http://www.unige.ch/math/folks/velenik/smbook/index.html

Interested students can also look into:

David Ruelle, Statistical Mechanics: Rigorous Results, World Scientific, 1999.

James Sethna: Statistical Mechanics: Entropy, Order Parameters and Complexity Oxford Master Series in Physics, 2006.