Fisher-Hartwig asymptotics and log-correlated fields in random matrix theory

Thesis

This thesis is concerned with establishing and studying connections between random matrices and log-correlated fields. This is done with the help of formulae, including some newly established ones, for the asymptotics of Toeplitz, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities.

In Chapter 1, we give an introduction to the mathematical objects that we are interested in. In particular we explain the relations between the characteristic polynomial of random matrices, log-correlated fields, Gaussian multiplicative chaos, the moments of moments, and Toeplitz and Toeplitz+Hankel deter- minants with Fisher-Hartwig singularities. Chapter 1 is based on joint work with Jon Keating [FK21], Tom Claeys and Jon Keating [CFK23], and Isao Sauzedde [FS22].

In Chapter 2, based on joint work with Jon Keating [FK21], we establish formulae for the asymptotics of Toeplitz, and Toeplitz+Hankel determinants with two complex conjugate pairs of merging Fisher-Hartwig singularities. We prove those formulae using Riemann-Hilbert techniques, which are heavily inspired by the ones used by Deift, Its, and Krasovsky [DIK11], and Claeys and Krasovsky [CK15].

In Chapter 3, based on joint work with Jon Keating [FK21], we complete the connection between the classical compact groups and Gaussian multiplicative chaos, by showing that analogously to the case of the unitary group first estab- lished by Webb [Web15], the characteristic polynomial of random orthogonal and symplectic matrices, when properly normalized, converges to a Gaussian multiplicative chaos measure on the unit circle.

In Chapter 4, based on joint work with Tom Claeys and Jon Keating [CFK23], we compute the asymptotics of the moments of moments of random orthogonal and symplectic matrices, which can be expressed in terms of integrals over Toeplitz+Hankel determinants. The phase transitions we observe are in stark contrast to the ones proven for the unitary group by Fahs [Fah21].

In Chapter 5, based on joint work with Isao Sauzedde [FS22], we establish convergence in Sobolev spaces, of the logarithm of the characteristic polynomial of unitary Brownian motion to the Gaussian free field on the cylinder, thus proving the dynamical analogue of the classical stationary result by Hughes, Keating and O’Connell [HKO01].

Authors: Forkel, J

• DOI: 10.5287/ora-4rqm8ywg2
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Gaussian Multiplicative Chaos; Riemann-Hilbert Problems; Log-Correlated Fields; Toeplitz and Toeplitz+Hankel Determinants; Mathematical Physics; Fisher-Hartwig Singularities; Random Matrix Theory; Complex Analysis
• Subjects: Random Matrix Theory; Mathematical Physics; Complex Analysis