Monday, August 22, 2022 at 10:00 a.m. (sharp)

PhD Candidate: Lucas Ashbury-Bridgwood

Supervisor: Balint Virag

Thesis title: Random Canonical Products and the Secular Function of the Stochastic Airy Operator

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Secular functions of random matrices and their limits are of recent interest in random matrix theory. Such functions are entire with zeros the spectra of the corresponding operators. For example, the general beta ensembles, extending the joint eigenvalue law of classical random matrix ensembles, have a universal soft edge limit upon rescaling called the Airy beta point process. This process also arises as eigenvalues of a random operator called the stochastic Airy operator. It is proven here that secular functions of the general beta ensembles converge in distribution to that of the stochastic Airy operator. Furthermore, this convergence is realized in the context of regularized determinants of operators. This is done by proving new asymptotics of the Airy process and rigidity estimates of the general beta ensembles and establishing this convergence for more general random sequences. These results extend the currently known case for the Gaussian ensembles in Lambert and Paquette (2020). Growth asymptotics are proven for the secular function of the stochastic Airy operator, and as an application some open questions in Lambert and Paquette (2020) are answered. By applying and extending the work in Valkó and Virág (2020) in the bulk case, the secular function is proven to be a unique limiting solution of an ordinary differential equation. Additionally, new convergence laws for discrete matrix models limiting to the stochastic Airy operator are proven, including convergence of the derivatives of eigenfunctions.

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A copy of the thesis can be found here: thesis lucas ashbury-bridgwood

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