Weak and strong confinement in the Freud random matrix ensemble and gap probabilities

Claeys, Tom;Krasovsky, Igor;Minakov, Oleksandr
(2023) Communications in Mathematical Physics — Vol. 402, p. 833-894 (2023)

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  • Claeys, Tomorcid-logoUCLouvain
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  • Krasovsky, IgorImperial College London
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  • Minakov, OleksandrUCLouvain
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Abstract
The Freud ensemble of random matrices is the unitary invariant ensemble corresponding to the weight exp(−n|x|^β), β > 0, on the real line. We consider the local behaviour of eigenvalues near zero, which exhibits a transition in β. If β ≥ 1, it is described by the standard sine process. Below the critical value β = 1, it is described by a process depending on the value of β, and we determine the first two terms of the large gap probability in it. This so called weak confinement range 0 < β < 1 corresponds to the Freud weight with the indeterminate moment problem. We also find the multiplicative constant in the asymptotic expansion of the Freud multiple integral for β ≥ 1.
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Claeys, T., Krasovsky, I., & Minakov, O. (2023). Weak and strong confinement in the Freud random matrix ensemble and gap probabilities. Communications in Mathematical Physics, 402, 833-894. https://hdl.handle.net/2078.5/269426 (Original work published 2023)