**Goals of the course**

This course will provide a solid knowledge about the mathematical foundation of statistical mechanics, which is the basis of many problems related to critical phenomena of complex systems. In particular we will introduce the notion of Gibbs measures and phase transitions. At the end of the course, the student will be able to understand and characterise critical phenomena in terms of extremal probability measures and their variational representations, including standard techniques for proving them (Peierls contour, cluster expansion).

**Content**

Statistical mechanics is a branch of physics aiming at understanding the laws of the macroscopic behaviour of

systems composed by many microscopic components. Critical phenomena, such as phase transitions, involve a

drastic change in the macroscopic state by tuning model parameters.

Critical phenomena are extremely universal far beyond physics, for instance in chemistry, biology or complex

systems. In this course we aim at giving a mathematical foundation for the study of many component systems

on the lattice and in the continuum space. Moreover, we would like to motivate the theory of Gibbs measures

starting from basic principles in classical mechanics.**In particular we will treat:**

- Gibbs ensembles and thermodynamic limits.
- Infinite volume Gibbs measures and DLR formalism in the lattice and in the continuum.
- Variational characterisation.
- Peierls contour method.
- Cluster expansion and polymer models.
- Piragov Sinai theory and its application to the Blume-Capel model.
- Particle Systems in Continuum: Gibbsian formalism for superstable interactions.
- Example: Lebowitz-Mazel-Presutti model.

**Prerequisites**

No prior knowledge in statistical physics is required. The student has to have a strong background in probability theory, stochastic processes and measure theory.

**Format**

Combination of lectures given by the lecturers with homework and short presentations by the students about

their individual project.

**Learning goals**

After completion of the course, the student is able to:

• convert material from part of a graduate-level textbook or a scientific paper into a coherent and comprehensible presentation for fellow students and mathematicians in general,

• choose appropriate means of communicating theoretical mathematics to fellow students and mathematicians, in written and oral form,

• explain specific topics from the content list of the seminar to fellow students, and put them in perspective as far as their relevance to wider mathematics is concerned,

• prepare notes for the audience based on (and, if necessary, expanding) their oral presentation.

**Rules about Homework/Exam****Evaluation**

The contribution of each student is twofold:

60% - individual project consisting of 40% written thesis and 20% oral presentation

40% - Homework assignments

Lecture Notes/Literature**References**

S. Friedli, Y. Velenik, Statistical Mechanics of Lattice Systems: A concrete mathematical introduction, https://www.unige.ch/math/folks/velenik/smbook/

Lecture notes of Prof. Sabine Jansen, Gibbsian Point Processes, http://www.mathematik.uni-muenchen.de/~jansen/gibbspp.pdf**Organizers**

Wioletta Ruszel and Cristian Spitoni

- Docent: Jochem Hoogendijk
- Docent: Wioletta Ruszel
- Docent: Cristian Spitoni