Asymptotics for averages over classical orthogonal ensembles

Claeys, Tom;Glesner, Gabriel;Minakov, Oleksandr;Yang, Meng
(2021) International Mathematics Research Notices — Vol. rnaa354, n° rnaa354, p. rnaa354 (2021)

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Authors
  • Claeys, Tomorcid-logoUCLouvain
    Author
  • Glesner, GabrielUCLouvain
    Author
  • Minakov, OleksandrUCLouvain
    Author
  • Yang, MengUCLouvain
    Author
Abstract
We study averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haar distributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with Fisher-Hartwig singularities in cases where some of the singularities merge together, and for symbols with a gap or an emerging gap. We obtain these asymptotics by relying on known analogous results in the unitary group and on asymptotics for associated orthogonal polynomials on the unit circle. As consequences of our results, we derive asymptotics for gap probabilities in the Circular Orthogonal and Symplectic Ensembles, and an upper bound for the global eigenvalue rigidity in the orthogonal ensembles.
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Claeys, T., Glesner, G., Minakov, O., & Yang, M. (2021). Asymptotics for averages over classical orthogonal ensembles. International Mathematics Research Notices, rnaa354(rnaa354), rnaa354. https://doi.org/10.1093/imrn/rnaa354 (Original work published 2021)