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Integrable Random Structures

Periodic Reporting for period 5 - IntRanSt (Integrable Random Structures)

Reporting period: 2020-09-01 to 2021-09-30

This project aims to develop and study integrable models in probability. Examples of such models include interacting particle systems, random polymers, random matrices and various combinatorial models related to the theory of symmetric functions and Young tableaux. A common theme is some underlying algebraic structure which accounts for the integrability of the model in question. The project has a strong emphasis on models of representation-theoretic origin, particularly in the context of the theory of Young tableaux and its vast generalisations.

The proposed research will be of interest to a growing community of researchers working at the interface of probability, integrable systems, algebra, analysis and mathematical physics, including researchers working on integrable systems, random matrix theory, combinatorics, representation theory, tropical geometry, interacting particle systems, random polymers and the KPZ equation. As it is a very topical research area at the interface of several different areas, this research will also be of interest to a wider community of researchers in the mathematical sciences. It will also generate new research directions for younger researchers, especially the PDRAs appointed to the program, in this rapidly expanding field.

This project will support the European research community’s drive to be among the prime generators of top-quality research. The nature of pure mathematical research is that it is traditionally somewhat removed from direct commercial usage. However, we might expect that the quality and cutting-edge nature of this proposal will ultimately have far wider implications which are harder to predict at the moment. We would expect there to be a significant impact, but only over the longer term. The fields of probability, analysis, integrable systems, representation theory and statistical physics play a fundamental part in our understanding of the world and have had proven impact on every branch of science, engineering and social sciences. Probability in particular is used everywhere: medicine, the weather, demographics, economics, the stock market, communications, cell biology, particle physics, climate change, etc. This project will provide leadership to the scientific base of this subject and ensure continued impact.
This project is based on a remarkable connection, discovered by the PI and his collaborators, between a birational version of the celebrated Robinson-Schensted-Knuth correspondence and GL(N)-Whittaker functions, and some related topics and applications in the wider field of integrable probability. The resulting theory has been substantially developed within the context of this project, including extensions to models with alternative inputs, additional symmetries and non-commutative versions. Some of the other achievements of the project include: a new approach to the study of moments of random matrices, connections between random matrices and SLE, progress on random sorting networks, scaling limits for Whittaker measures, integrable aspects of free fermions, applications to polymer localisation, universality of phase transitions for log-gases, joint moments of characteristic polynomials of random matrices.
This is a rapidly developing field with interactions across several areas of mathematics and has already had a significant impact, as evidenced by the growing number of scientific meetings devoted to these topics. The research carried out in this project is at the cutting edge of these developments, and therefore the expected potential impact is high.