Stochastic particle or spin systems are mathematical models of complex phenomena involving a large number of interrelated components. There are numerous examples within all areas of natural and social sciences, such as traffic flow on motorways or communication networks, opinion dynamics, spread of epidemics or fires, genetic evolution, reaction diffusion systems, crystal surface growth, financial markets, etc. The central question is to understand and predict emergent behaviour on macroscopic scales, as a result of the microscopic dynamics and interactions of individual components. Qualitative changes in this behaviour depending on the system parameters are known as collective phenomena or phase transitions and are of particular interest.

In this lecture we will focus on metastability -- a dynamical behaviour of a complex system that is related to the existence of multiple, well separated time scales. While at a short time scale, the system appears to be in a quasi-equilibrium (metastable state), at large time scales, it undergoes a phase transition. The understanding of quantitative aspects of dynamical phase transitions such as averaged transition times and spectral properties is one of the basic problem we are discussing in the lecture. Along the way, we will introduce analytical techniques such a discrete potential theory and martingale problems, and use them to identify in the limit of large system sizes precise formulas for metastable transition times.

#### Team

**Lecturer:**Martin Slowik#### Information

**Content of the lecture:**- Markov chains and boundary value problems
- discrete potential theory
- variational principles
- functional inequalities
- spectral properties of the associated generator
- applications