Two-Dimensional One Component Plasma and Non-Hermitian Random Matrices
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
The joint probability density function of the eigenvalues of Non-Hermitian Matrices has the same form of the Boltzmann factor of a two-dimensional plasma of Coulomb charges, 2D-OCP. This statistical mechanics fluid model has appeared in several areas of physics and mathematics. Indeed, the logarithmic repulsion of the charges occurs as interaction between vortices and dislocations in systems such as superconductors, superfluids, rotating Bose-Einstein condensates. There is also an analogy between the 2D-OCP and the Laughlin trial wave function in the theory of fractional quantum Hall effect. The project will explore the connection between Random Matrix Theory and the 2D-OCP to gain insight into plasmas of Coulomb charges.
Organisations
People |
ORCID iD |
| Francesco Mezzadri (Primary Supervisor) | |
| Alex Little (Student) |
Publications
Little A
(2022)
On the number of real eigenvalues of a product of truncated orthogonal random matrices
in Electronic Journal of Probability
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509619/1 | 01/10/2016 | 30/09/2021 | |||
| 1941742 | Studentship | EP/N509619/1 | 01/10/2017 | 30/09/2021 | Alex Little |
| Description | My research has focused on trying to prove a conjecture by Forrester, Ipsen and Kumar (2017). This conjecture relates to products of truncated orthogonal matrices, the number of their real eigenvalues and their distribution along the real line. A truncated orthogonal matrix is an NxN upper left corner of an (N+L)x(N+L) Haar distributed orthogonal matrix. If one then takes a product of m independently sampled truncated orthogonal matrices, Forrester et al.'s conjecture then gives a form for the average number and distribution along the real line of the real eigenvalues of this product. This is in the regime N,L -> infty and N/L fixed. We prove this conjecture (in collaboration with Nicholas Simm and Francesco Mezzadri) and these results have been published in the Electronic Journal Probability (see Electron. J. Probab. 27: 1-32 (2022). DOI: 10.1214/21-EJP732). My research has also elucidated the relationship between non-intersecting Brownian motion, free fermions and Yang-Mills theory. This correspondence appears in the papers of Forrester, Majumdar and Schehr (2011) and Cunden, Mezzadri and O'Connell (2018). In my work this correspondence is extended to all compact Lie groups, and given additional assumptions the results of the aforementioned papers are recovered in a simple way. This work appears in my thesis and is forthcoming on the arXiv. |
| Exploitation Route | The methods used to prove the conjecture of Forrester, Ipsen and Kumar are applicable to a broad class of real asymmetric ensembles and their products. Furthermore the proof involved deriving sharp bounds on generalised hypergeometric functions, something which has broad applicability. My results on the relationship between random matrix theory and the theory of compact Lie groups could be extended to consider non-compact groups. This may offer a Lie theoretic perspective on the work of Tribe and Zaboronski (2012) on the Ginibre process. |
| Sectors | Other |