Abstract
We investigate the characteristic polynomials of the Gaussian β-ensemble for general through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for 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 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.
Funding Statement
G.L. research is supported by the SNSF Ambizione Grant S-71114-05-01. E.P. supported by Simons Foundation travel Grant 638152.
We acknowledge support from the Park City Mathematics Institute 2017, at which this program was begun, and in particular acknowledge NSF grant DMS:1441467.
Acknowledgments
We would like to thank Diane Holcomb, conversations with whom helped launched this project. We also thank the anonymous referees for their careful reading of the manuscript which has greatly improved its quality.
Citation
Gaultier Lambert. Elliot Paquette. "Strong approximation of Gaussian β ensemble characteristic polynomials: The hyperbolic regime." Ann. Appl. Probab. 33 (1) 549 - 612, February 2023. https://doi.org/10.1214/22-AAP1823
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