Generalized Moments of Characteristic Polynomials of Random Matrices

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


A central theme of this thesis is random matrix theory and its connection to log-correlated fields. We present results on the statistical properties of characteristic polynomials of matrices from one of the three ‘classical’ compact matrix groups: unitary, U(N); symplectic Sp(2N); or orthogonal O(2N) (in particular SO(2N) ⊂ O(2N)). Take A ∈ G(N), amatrixfrom G(N) ∈{U(N),Sp(2N),SO(2N)}, then denote by PN(A,θ) = det(I −Aexp(−iθ)) its characteristic polynomial. The logarithm of |PN(A,θ)| displays logarithmic correlations. We focus too on connections; random matrix theory and its influence on number theory, probability, statistical physics, and combinatorics. Our main results concern the study of moments of moments of characteristic polynomials. We give kth moment, defined with respect to the matrix group average, of the random variable corresponding to the 2βth moment of PN(A,θ) with respect to the uniform measure dθ 2π, for all k,β ∈ N. We showthat these moments of moments are polynomials in N and give the respective degrees in each unitary, symplectic, and orthogonal case. For unitary matrices, this resolves a conjecture of Fyodorov and Keating [82] regarding the scaling of the moments with N as N →∞, for k,β ∈N. In the symplectic and orthogonal cases, we show that the leading order differs from the unitary case. Unifying all the moments of moments is the underlying integrable system. We here emphasise a connection with representation theory, giving a formulation of the moments in each case in terms of combinatorial counts. Additionally, we develop a branching model of the moments of moments and demonstrate that the Fyodorov and Keating conjecture extends to this setting. We also analyse mixed moments of unitary characteristic polynomials asymptotically, and relate the solution to a particular Painlevé differential equation. Additionally, the asymptotic behaviour of moments of logarithmic derivatives of unitary characteristic polynomials near the unit circle are determined.
Date of Award23 Jun 2020
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorJon Keating (Supervisor)

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