# Probability Theory

The group Probability Theory is responsible for research and teaching in probability theory.

Interacting stochastic systems consist of a large number of interacting random components.

These components interact with each other and with their environment. Even when the interaction is local, such systems typically exhibit a complex global behavior, with a long-range dependence resulting in anomalous fluctuations and phase transitions. To mathematically understand these systems requires the use of powerful probabilistic ideas and techniques. The challenge is to introduce simple models, which serve as paradigms, and to unravel the complex random spatial structures arising in these models. Statistical physics and ergodic theory provide the conceptual ideas, while probability theory provides the mathematical language and framework. The important challenge is to give a precise mathematical treatment of the physics and that arises from the underlying complexity.

Much of the knowledge about interacting stochastic systems that has been built up over the past decades is currently making its way into population genetics and complex networks. One of the tasks is to help facilitate this cross-fertilization and to address concrete questions at the interface.

Research at Leiden in Probability Theory concentrates on interacting stochastic systems (disordered systems, percolation, random polymers, metastability, sandpiles), ergodic properties of random processes (dynamical Gibbs-non-Gibbs transitions, hidden Markov chains), complex networks (structure, organisation, function) and population genetics (resampling, mutation, migration). Key tools are large deviation theory, stochastic analysis, variational calculus and combinatorics. There is an interesting link between algebraic dynamical systems and solvable models of statistical physics. It turns out that entropies of apparently different systems often coincide, and that this `mere' coincidence is not accidental. Research aims at providing an explanation for this phenomenon. A powerful combinatorial technique to study high-dimensional systems is the 'lace expansion'. Research aims at obtaining a rigorous understanding of phase transitions in high dimensions, including diffusion on critical network structures.

Research at Leiden in Operations Research concentrates on Markov chains, Markov decision processes and Markov games, with applications to problems in stochastic networks. One of the main issues concerns stability. How can stability be checked? If stable, then how fast does the network reach its stationary distribution? If unstable, then what does the quasi-stationary distribution look like? How can efficient algorithms be developed to control the network according to certain pre-set optimization criteria? Are these algorithms amenable to practical implementation? What can one say about the structure of optimal policies? Which type of customer should be prioritised to optimise network performance? These questions can be studied within the framework of Markov chain theory. Often the situation arises where there are conflicting interests, for instance, maximizing server efficiency while minimizing customer dissatisfaction. This can be handled with the help of Markov game models.