# Asymptotics of correlation functions for random matrices

Lead Research Organisation:
Imperial College London

Department Name: Dept of Mathematics

### Abstract

Random matrices is an area of analysis and probability which was discovered in pioneering works of Wigner and Dyson.

Initially considered to describe the behaviour of many particle systems with unknown interactions via modelling these

interactions by random variables, random matrices have since found a multitude of applications

in physics, mathematics, and engineering. The main subject, still containing many unresolved problems, is the study of

correlations between eigenvalues of random matrices when the size of the matrix is large.

A typical such object is a probability that a given interval of the real line contains no eigenvalues of the random matrices.

It is briefly called gap probability. This probability is expressed for many interesting ensembles of

random matrices as a Hankel or Toeplitz determinant. The question is to find explicit expansions of this object for the case

of a large gap. This question is very easy to formulate, but obtaining the solution is a highly nontrivial task involving

development of appropriate methods. Very efficient ones rely on a so called Riemann-Hilbert approach, where the question

is reduced to a problem for functions of complex variable.

For some basic ensembles of random matrices, this question has been solved, but there are very important ones where

the problem remains open. The purpose of the project is to address this question in one or two such open cases.

Riemann-Hilbert steepest descent techniques used in this project have been initiated by Deift and Zhou 20 years ago, and have been developing ever since. They proved to be probably the most powerful tool of asymptotic analysis and helped to solve many important problems. Solution of a new problem using this approach also provides a development of the method itself, as new tricks have to be designed to overcome new obstacles.

Thus, in addition to solving a new problem, we also contribute to the development of an important method.

Initially considered to describe the behaviour of many particle systems with unknown interactions via modelling these

interactions by random variables, random matrices have since found a multitude of applications

in physics, mathematics, and engineering. The main subject, still containing many unresolved problems, is the study of

correlations between eigenvalues of random matrices when the size of the matrix is large.

A typical such object is a probability that a given interval of the real line contains no eigenvalues of the random matrices.

It is briefly called gap probability. This probability is expressed for many interesting ensembles of

random matrices as a Hankel or Toeplitz determinant. The question is to find explicit expansions of this object for the case

of a large gap. This question is very easy to formulate, but obtaining the solution is a highly nontrivial task involving

development of appropriate methods. Very efficient ones rely on a so called Riemann-Hilbert approach, where the question

is reduced to a problem for functions of complex variable.

For some basic ensembles of random matrices, this question has been solved, but there are very important ones where

the problem remains open. The purpose of the project is to address this question in one or two such open cases.

Riemann-Hilbert steepest descent techniques used in this project have been initiated by Deift and Zhou 20 years ago, and have been developing ever since. They proved to be probably the most powerful tool of asymptotic analysis and helped to solve many important problems. Solution of a new problem using this approach also provides a development of the method itself, as new tricks have to be designed to overcome new obstacles.

Thus, in addition to solving a new problem, we also contribute to the development of an important method.

## People |
## ORCID iD |

Igor Krasovsky (Primary Supervisor) | |

Theo-Harris Maroudas (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509486/1 | 01/10/2016 | 30/09/2021 | |||

1832015 | Studentship | EP/N509486/1 | 01/10/2016 | 31/03/2020 | Theo-Harris Maroudas |

Description | Our work is in the context of the so-called edge scaling limit of the largest eigenvalue in the Gaussian Unitary Ensemble of random matrices. Drawing from related results of P. Deift, A. Its and I. Krasovsky in this area, we proved an asymptotic formula describing the probability that two large intervals of the real line are free from eigenvalues. The formula itself resembles the known result on the Airy-kernel determinant, in the sense that the terms are of the same order of magnitude. Currently our result is up to the multiplicative constant, but we aim to find its exact value before the end of the award. In addition, we extended the steepest descent method presented by P. Deift, A. Its and X. Zhou to our setting. Their exposition of this method provided an analogous formula in the so-called bulk scaling limit. These contribute to the study of large gap probabilities initiated by fundamental works of F. Dyson in the bulk scaling, and later C. Tracy and H. Widom in the edge scaling. |

Exploitation Route | It is of interest to study the transition asymptotics from the well studied one interval case to the two interval case. This may present a sequence of transitions through well known functions, as is known to happen e.g. in the bulk scaling limit. Further, since we have successfully analysed the two interval case, our work provides a basis to study the case of any number of intervals. Each number of intervals will also contain the difficult problem of finding the multiplicative constant. More generally, the presentation of a new such formula motivates the study of analogous problems involving other kernels which are of importance in random matrix theory. In particular our result suggests that a similar analysis may be successfully undertaken in the case of kernels from the same family as the Airy. |

Sectors | Education |