Report

Teaser, summary, work performed and final results

Periodic Reporting for period 3 - IntRanSt (Integrable Random Structures)

Teaser

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...

Summary

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

In this reporting period, the main focus has been on research related to objectives (1) and (3), with substantial progress. For objective (1), a careful development of the underlying combinatorial framework has been completed, and is currently being developed further in the context of some more general situations. 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.

With Jonathan Warren (University of Warwick) and PhD student Theodoros Assiotis (University of Warwick), 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. We formulate a class of diffusions on Gelfand-Tsetlin patterns which generalise Warren’s process, which corresponds to the Hermite polynomials, to the other classical orthogonal polynomials of Laguerre and Jacobi type. We show that these processes have transition densities of block-determinant type, and Markovian projections related to the classical ensembles of random matrix theory. We have written a paper on this work entitled `Interlacing diffusions’, which is publicly available on arxiv and has been submitted for publication.

With PDRA Fabio Deelan Cunden (appointed January 2017) and Francesco Mezzadri (Bristol), we have formulated and developed a matrix model for free fermions at finite temperature in the context of compact classical groups. This work was motivated by considerations regarding the KPZ scaling limit of random polymers as discussed in Objective 3 of the proposal, and is an ongoing research direction. This work has been reported in the paper `Free fermions and the classical compact groups\', now published in the Journal of Statistical Physics.

With PDRA Fabio Deelan Cunden, Francesco Mezzadri (Bristol) and Nick Simm (Warwick), we have recently discovered some remarkable new integrable structures in random matrix theory, and we have just finished writing a paper on this topic entitled `Moments of random matrices and hypergeometric orthogonal polynomials\', now publicly available on arxiv. In this work we show that moments of classical random matrix ensembles have a remarkable analytic structure (functional equations, critical zeros, etc) owing to novel representations in terms of hypergeometric orthogonal polynomials.

With P. Facchi, M. Ligabo` and P. Vivo, Fabio Deelan Cunden has written two papers: `Universality of the third-order phase transition in the constrained Coulomb gas\', published in the Journal of Statistical Physics, and `Universality of the weak pushed-to-pulled transition in systems with repulsive interactions\', available on arxiv.

Final results

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.

Website & more info

More info: http://www.maths.ucd.ie/.