# Log Correlations and Random Matrices

## Periodic Reporting for period 3 - LogCorRM (Log Correlations and Random Matrices)

Reporting period: 2019-09-01 to 2021-02-28

A matrix is a set of numbers treated as a single entity. Matrices arise in many contexts in mathematics and its applications; for example, they are used to represent data, they play a foundational role in quantum mechanics, and they are central to descriptions of networks. For over 100 years mathematicians and statisticians have been interested in the properties of matrices constructed from random sets of numbers. These are important in analysing large complex datasets, complex quantum systems, and large complex networks, for example.

We are interested in certain extreme, rare phenomena that are important in many applications of random matrix theory. Researchers have been trying to understand these phenomena for many years, but without success because we did not have the right way to characterise them. Recently, a series of new ideas have opened the door to analysing the phenomena in question. These new ideas link the random matrix problem to a general area of mathematical research concerned with random geometrical and dynamics processes having certain specific correlations, in which these has been considerable progress over the past decade. This allows is to take general ideas, methods and results and adapt them to apply to the extreme, rare phenomena that we want to understand in the context of random matrices.

Our overall objectives are to calculate certain quantities, called the 'moments of the moments of the characteristic polynomials' of the random matrices. These characterise the extreme, rare phenomena that we want to understand. The formulae we hope to derive can then be applied to many other areas of science and the social science where matrices are used to describe complex data sets or complex physical systems. We believe these moments of moments exhibit behaviour that is analogous to that associated with phase transitions in physical systems.

Remarkably, the phenomena that arise in the context of random matrices, which we hope to understand, seem to be closely related to similar phenomena that occur in number theory. The Riemann zeta-function is a mathematical object that encodes information about the distribution of the prime numbers. It is one of a class of similar functions called L-functions. It is the subject of one of the greatest challenges in mathematics: the Riemann Hypothesis. The Riemann zeta-function, and the other L-functions, exhibit extreme, rare phenomena (in this case, unusually large values) which are surprisingly similar to those we study for random matrices. We seek to understand this better, in the hope of gaining new insights into various foundational questions in number theory that arise in the study of the Riemann zeta-function and other L-functions. In particular, we want to understand the 'moments of the moments' of the Riemann zeta-function and other L-functions and to investigate whether these exhibit phase-transition-like behaviour.
We have made progress in deriving formulae for the 'moments of the moments of the characteristic polynomials' of the random matrices of interest to us. This has involved incorporating new ideas from several other areas of mathematics, uncovering new connections, and has led to new insights into the theory of random matrices. It has involved proving formulae previously believed, but not known, to be true. We expect the new ideas introduced to feed through into applications of random matrix theory in the future.

We have also started to apply the new understanding achieved to investigate the corresponding phenomena in number theory. We have established formulae for the moments of the moments of the Riemann zeta-function and certain other L-functions. This has led to significant progress in characterising the rare, extremely large values these functions take. We anticipate that this will open up new avenues of research in number theory, for example concerning the distribution of the prime numbers.
We believe that the new connections we have established between the phenomena we seek to understand and other areas of mathematics, and the formulae we have derived as a result for the moments of the moments, represent significant progress. In particular, some of the new connections we have established indicate links between areas of mathematics that were not previously expected.