Strong approximation of Gaussian beta-ensemble characteristic polynomials: the hyperbolic regime

We investigate the characteristic polynomials $\varphi_N$ of the Gaussian $\beta$-ensemble for general $\beta>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for $\varphi_N$ in terms of a Gaussian log--correlated field in order to ultimately deduce some of its fine asymptotic properties. We distinguish between different types of transfer matrices and analyze completely the hyperbolic regime of the recurrence. As a result, we obtain a new coupling between $\varphi_N(z)$ and a Gaussian analytic function with an error which is uniform for $z \in \mathbb{C}$ separated 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 arXiv:2009.05003. 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.