Scientific Background

This project aims to encourage the collaboration between researchers who, motivated by the study of a variety of problems concerning integrability in the broad sense (quantum integrability, integrable differential equations, integrable probabilities) are led to the analysis of certain quantities that are expressed as determinants of structured matrices (e.g. Toeplitz, Hankel), or Fredholm determinants associated with “integrable” operators. The characterization of such objects can be reduced to the study of a Riemann-Hilbert problem. The latter consists in the construction of a holomorphic function on the complex plane, with values in the scalars, matrices or operators, having a fixed asymptotic behavior at infinity and prescribed jumps on a set of curves. Typically, for various reasons underlying the relevant physical problem (highlighting a universal behavior, limiting behavior, validity of an approximation or verification of heuristic predictions, etc.) one is interested in the equations and the asymptotic behavior, in one or more parameters, associated with the object described by the solution of the underlying Riemann–Hilbert problem.

In recent years there have been quite spectacular advances in this field, with results that deserve to be generalized and put into a unitary conceptual framework. Among these advances are the Lax formalism for integro-differential equations generalizing the Painlevé equations, the development of asymptotic analysis techniques for operator-valued Riemann-Hilbert problems, the relationships between random growth models (Kardar-Parisi-Zhang (KPZ) equation and its fixed point, ASEP (Asymmetric Exclusion Processes) and soliton equations, applications to Gaussian multiplicative chaos.

The applications that we aim at concern mainly determinantal point processes, one-dimensional spatial random growth models (Kardar-Parisi-Zhang equation, ASEP-type models), quantum integrable models. By combining ideas from random matrix theory, asymptotic analysis of differential equations and statistical physics, we propose to develop a systematic and unitary approach to the study of these models