Emma Bailey
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Research

My research is grounded in random matrix theory, though I am very interested in the connections between RMT and number theory, probability, combinatorics and beyond.

My PhD was titled Generalized moments of characteristic polynomials of random matrices. There exists a philosophy (see Montgomery-Dyson, Keating-Snaith, Katz-Sarnak etc) that characteristic polynomials of certain types of matrices share properties with certain number theoretic functions, for example the Riemann zeta function. My work is often inspired by trying to further our understanding of this apparent connection.

In particular, my recent research has focussed on 'moments of moments' - averaging characteristic polynomials over two natural spaces. This perspective reveals connections between integrable systems, Painlevé, combinatorics, and branching structures.

Below is a list of my publications.

  • Moments of moments and branching random walks
    Joint with J. P. Keating
    link
  • On the moments of the moments of ζ(1/2+it)
    Joint with J. P. Keating
    link
  • On the moments of the moments of the characteristic polynomials of Haar distributed symplectic and orthogonal matrices
    Joint with T. Assiotis, J. P. Keating
    To appear in Annales de l'Institut Henri Poincaré D
    link
  • Mixed moments of characteristic polynomials of random unitary matrices
    Joint with S. Bettin, G. Blower, B. Conrey, A. Prokhorov, M. Rubinstein, N. Snaith
    Journal of Mathematical Physics 60.8 083509, (2019)
    link, Editor's pick
  • On the moments of the moments of the characteristic polynomials of random unitary matrices
    Joint with J. P. Keating
    Commun. Math. Phys. 371 689--726, (2019)
    link