Integrable stochastic models of non-intersecting paths, interfaces and diffusion in random media – IntStoch

Complex random systems with many degrees of freedom, or constituted of many interacting particles, often display a universal behaviour at large scale. This means that scaling exponents, and even the precise statistics of random fluctuations of macroscopic quantities, are largely independent of the specifics of the model and the details of the interaction rules. Hence, within one universality class, a lot of information can be obtained from the study of one integrable, i.e. exactly solvable, toy-model in the class. The field of integrable probability is the art of designing and studying such models, using a wide range of tools from stochastic analysis, algebraic combinatorics, random matrix theory, representation theory and quantum integrable systems. In particular, it fueled important progress in recent years about the Kardar-Parisi-Zhang (KPZ) universality class, a class of models describing random interface growth and fluctuations of random paths in random media.
This project aims at developing new frameworks to deal with models in the KPZ class, that are important to compare theoretical predictions with experimental settings. Applications will include the study of fluctuations in systems of interacting particles with boundaries and the estimation of hitting times of extreme diffusions in random media. Another objective is to bring the tools of integrable probability outside the KPZ class, to study random surfaces associated to non-intersecting diffusions in random media and non-commutative analogues of models in the KPZ class. Along the way, we will generalize several of the beautiful mathematical structures that underlie stochastic integrability.