Extreme value statistics of characteristic polynomials of random matrices and the Riemann zeta-function

Lead Research Organisation: University of Bristol
Department Name: Mathematics


The goal is to understand fluctuations in the extreme values of the characteristic
polynomials of random matrices, and of the Riemann zeta-function on its critical
line. This is a very active area of research with many deep and interesting unsolved
problems. The aim will be to use techniques relating to the theory of symmetric
polynomials to deduce exact formula for the moments of the moments of the
characteristic polynomials of random unitary matrices. We will also explore
approaches based on integrable systems. Ultimately, we will seek to understand the
behaviour in the large-matrx limit. We will use the random matrix results to explore
the corresponding statistics for the Riemann zeta-function on its critical line and will
compare the results with those coming from a randomised Euler product model.


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Bailey E (2019) Mixed moments of characteristic polynomials of random unitary matrices in Journal of Mathematical Physics

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Bailey E (2019) On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices in Communications in Mathematical Physics

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509619/1 01/10/2016 30/09/2021
1792464 Studentship EP/N509619/1 01/10/2016 30/07/2020 Emma Bailey
Description We have partially resolved a conjecture of Fyodorov and Keating concerning averages of unitary characteristic polynomials. For integer moment parameters we have established the leading order of the moments, as well as a polynomial structure in N, the matrix size. This is joint work with J. P. Keating.
Additionally, with T. Assiotis and J. P. Keating, we have proved analogous results for symplectic and orthogonal polynomials. All of the above results have connections to various moments of L-functions.
Finally, together with six co-authors, we have extended the community's understand of averages of mixed and log-derivative moments of unitary polynomials.
Exploitation Route Two of the above three projects mentioned have been published. Already they have been cited and used within fields other than my `home' field of random matrix theory.
Sectors Education,Other