Gibbs measures appear in statistical physics as the natural candidates for modelling the equilibrium states of physical systems with large number of components. They are characterized by local “rules”, similar to what we know from the study of Markov chains. In particular, they form a natural generalization of Markov chains in which a time index is replaced by a space index. Gibbs measures resulted from attempts to put into general probabilistic setting a very specific model (called the Ising model) designed to capture the magnetic properties of ferromagnetic materials. They now appear in other areas such as economics (as models for studying the stabilization of an economy consisting of dependent agents), sociology (for modelling the polarization phenomena in society and social networks), biology etc. During the course we will discuss the conditions that a given class of local “rules” need to satisfy in other to admit a Gibbs measure. Phase transition occurs if there is more than one Gibbs measure associated with the class. We will exhibit this for the Ising model. Condition for Uniqueness of Gibbs measures admitted by a given class will also be discussed. We will also discuss what happens when a Gibbs measure is transformed.

**Literature**

A.C.D. van Enter and W.Th.F. den Hollander: Interacting particle systems and Gibbs measures, MRI masterclass notes, 1992—1993, and A.C.D. van Enter, R. Fernández and A. D. Sokal: Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations: Scope and Limitations of Gibbsian Theory, J.Stat.Phys. , 72 , pp. 879—1167, 1993

**Pre-requisites**

A basic knowledge of propability theory and analysis (topology) is required. A knowledge of notions such as conditional expectations and Markov chains will be essential.

**Examination**

Oral examination (1-2 hours)