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IntRanSt Report Summary

Project ID: 669306
Funded under: H2020-EU.1.1.

Periodic Reporting for period 2 - IntRanSt (Integrable Random Structures)

Reporting period: 2016-10-01 to 2017-08-31

Summary of the context and overall objectives of the project

The last few years have seen significant advances in the discovery and development of integrable models in probability, especially in the context of random polymers and the Kardar-Parisi-Zhang (KPZ) equation. Among these are the semi-discrete (O’Connell-Yor) and log-gamma (Seppalainen) random polymer models. Both of these models can be understood via a remarkable connection between the geometric RSK correspondence (a geometric lifting, or de-tropicalization, of the classical RSK correspondence) and the quantum Toda lattice, the eigenfunctions of which are known as Whittaker functions. This connection was discovered by the PI and further developed with his collaborators. In particular,we have recently introduced a powerful combinatorial framework which underpins this connection. The PI has also explored these ideas from an integrable systems point of view, revealing a very precise relation between classical, quantum and stochastic integrability in the context of the Toda lattice and some other integrable systems.

The main objectives of this proposal are:

(1) to further develop the combinatorial framework in several directions which, in particular, will yield a wider family of integrable models,
(2) to clarify and extend the relation between classical, quantum and stochastic integrability to a wider setting, and
(3) to study thermodynamic and KPZ scaling limits of Whittaker functions (and associated measures) and their applications.

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

"In this reporting period, the main focus has been on objectives (1) and (3), with substantial progress. For objective (1), a careful development of the underlying combinatorial framework has been completed, as a first step towards the main goals of this objective. For objective (3), we have focussed on thermodynamic scaling limits and, in particular, have developed a detailed understanding of the zero-temperature and semi-classical cases. These investigations are ongoing. In the meantime, with Jonathan Warren (University of Warwick) and our jointly supervised PhD student Theodoros Assiotis, we have obtained some new results in the zero-temperature setting, which furthers our understanding of the role of classical orthogonal polynomials in the this setting. This work has been made publicly available on arXiv with the title ""Interlacing diffusions"" and will soon be submitted to a journal."

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.
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