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


Lambert, Gaultier; Paquette, Elliot (2023). Strong approximation of Gaussian β ensemble characteristic polynomials: The hyperbolic regime. Annals of Applied Probability, 33(1):549-612.

Abstract

We investigate the characteristic polynomials phi N of the Gaussian beta ensemble for general beta > 0 through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for phi 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 phi 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.

Abstract

We investigate the characteristic polynomials phi N of the Gaussian beta ensemble for general beta > 0 through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for phi 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 phi 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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Uncontrolled Keywords:Statistics, Probability and Uncertainty, Statistics and Probability Gaussian &beta-ensembles, product of random matrices, Gaussian multiplicative chaos LOG-CORRELATED FIELDS ; BETA-ENSEMBLES ; MAXIMUM ; ASYMPTOTICS ; EIGENVALUES ; RESPECT ; LIMIT ; SUMS
Language:English
Date:1 February 2023
Deposited On:10 Jan 2024 12:25
Last Modified:28 Jun 2024 03:37
Publisher:Institute of Mathematical Statistics
ISSN:1050-5164
Additional Information:Funding. 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.
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/22-aap1823
Project Information:
  • : FunderSNSF Ambizione Grant
  • : Grant IDS-71114-05-01
  • : Project Title
  • : FunderSimons Foundation travel Grant
  • : Grant ID638152
  • : Project Title
  • : FunderNSF grant
  • : Grant IDDMS:1441467
  • : Project Title
  • Content: Published Version
  • Language: English