Noncommutative statistical mechanics: probability at the confluence

Lead Research Organisation: Lancaster University
Department Name: Mathematics and Statistics

Abstract

The asymmetric simple exclusion process (ASEP) is a fundamental mathematical construction in statistical mechanics that combines the notion of stochastic motion with a physical requirement that matter cannot coexist in the same location. As such, it has wide-ranging applications, from traffic and fluid flow to biological processes. However, recent advances, particularly the emerging parallelism between ASEP-type processes and well-known families of orthogonal polynomials, hint at a hidden probabilistic phenomenon that promises to greatly expand the applicability of the ASEP.

The key is in a common computational artifact, referred to as the matrix ansatz. In this project, we explore a new perspective on the ASEP, based on the relationship between its matrix ansatz and noncommutative infinite-dimensional phenomena. In this manner, we aim to transform the interface between the 'classical' and noncommutative probability theories, arrive at a more fundamental understanding of the ASEP, and open up innovative approaches to a long-standing open question of mathematical analysis, the free group factors isomorphism problem.

Publications

10 25 50
 
Description "Positivity problems in combinatorics" (PI: Natasha Blitvic)
Amount £7,000 (GBP)
Organisation Heilbronn Institute for Mathematical Research 
Sector Academic/University
Country United Kingdom
Start 05/2022 
End 03/2023
 
Description Renewal of funding (for calendar year 2022) for collaborative travel between Lancaster and Universite de Franche-Comte (France): "Noncommutative Probability, Matrix Analysis and Quantum Groups"
Amount £2,190 (GBP)
Funding ID 608420427 
Organisation British Council 
Sector Charity/Non Profit
Country United Kingdom
Start 01/2021 
End 12/2022
 
Description Research Membership at the Mathematical Sciences Research Institute 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact The PI spent six weeks as an invited Research Member at the Mathematical Sciences Research Institute (Berkeley, California, USA) at the program "Universality and Integrability in Random Matrix Theory and Interacting Particle Systems". This was a unique opportunity to discuss with leading experts exploring the interface between random matrix theory and interacting particle systems, which provides the backdrop for this funded project. The PI delivered a seminar talk on an adjacent topic to an audience of experts, postdocs and postgraduate students, which sparked further discussions and has led to subsequent invitations.
Year(s) Of Engagement Activity 2021
URL https://www.msri.org/programs/328