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Integrable Random Structures

Objective

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 in collaboration with Corwin, Seppalainen and Zygouras. In particular, we have recently introduced a powerful combinatorial framework which underpins this connection. I have also explored this connection 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, which lies at the interface of probability, integrable systems, random matrices, statistical physics, automorphic forms, algebraic combinatorics and representation theory, will make novel contributions in all of these areas.

Field of science

  • /natural sciences/chemical sciences/polymer science

Call for proposal

ERC-2014-ADG
See other projects for this call

Funding Scheme

ERC-ADG - Advanced Grant

Host institution

UNIVERSITY COLLEGE DUBLIN, NATIONAL UNIVERSITY OF IRELAND, DUBLIN
Address
Belfield
4 Dublin
Ireland
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 287 861,18

Beneficiaries (3)

UNIVERSITY COLLEGE DUBLIN, NATIONAL UNIVERSITY OF IRELAND, DUBLIN
Ireland
EU contribution
€ 1 287 861,18
Address
Belfield
4 Dublin
Activity type
Higher or Secondary Education Establishments
THE UNIVERSITY OF WARWICK

Participation ended

United Kingdom
EU contribution
€ 116 376,51
Address
Kirby Corner Road - University House
CV4 8UW Coventry
Activity type
Higher or Secondary Education Establishments
UNIVERSITY OF BRISTOL

Participation ended

United Kingdom
EU contribution
€ 175 061,06
Address
Beacon House Queens Road
BS8 1QU Bristol
Activity type
Higher or Secondary Education Establishments