# 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.

### 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 | 18/09/2017 | 31/03/2021 | 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'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. These results are forthcoming on the arXiv. We also compute compute the pointwise limit at the edge of the spectrum. We find that the limit is independent of m and N/L and agrees with what has been found for the Ginibre ensemble (see Edelman, Kostlan and Shub, 1994). This suggests it may be a universal quantity. |

Exploitation Route | The techniques developed to prove Forrester's conjecture certainly have application to tackle other problems in the field of products of random matrices. Furthermore, we also derive a range of results for the Laplace/Steepest descent method. This method is widely used in mathematics and physics and so these results could have wide application. |

Sectors | Other |