Strong approximation of Gaussian β ensemble characteristic polynomials: The hyperbolic regime

Abstract

We investigate the characteristic polynomials ${\mathit{\phi }}_{\mathit{N}}$ of the Gaussian β-ensemble for general $\mathit{\beta }>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for ${\mathit{\phi }}_{\mathit{N}}$ in terms of a Gaussian log-correlated field. We distinguish between different types of transfer matrices and analyze completely the hyperbolic part of the recurrence. As a result, we obtain a new coupling between ${\mathit{\phi }}_{\mathit{N}}$ and a Gaussian analytic function with an error which is uniform away from the support of the semicircle law. We use this as input to give the almost sure scaling limit of the characteristic polynomial at the edge in (Lambert and Paquette (2020)). This is also required to obtain analogous strong approximations inside of the bulk of the semicircle law. Our analysis relies on moderate deviation estimates for the product of transfer matrices and this approach might also be useful in different contexts.