Browsing by Author "Bothner, Thomas"

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    On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo–Miwa–Ueno differential
    (Elsevier, 2019-03) Bothner, Thomas; Its, Alexander; Prokhorov, Andrei; Mathematical Sciences, School of Science
    Several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s [45]. Recent advances in the theory of tau functions [41], based on earlier works of B. Malgrange and M. Bertola, have allowed to extend the original Jimbo–Miwa–Ueno differential form to a 1-form closed on the full space of extended monodromy data of the underlying Lax pairs. This in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions, including the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymptotics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided each eigenvalue particle has been removed independently with probability 1-γ ∈ (0, 1⌋.
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    The sine process under the influence of a varying potential
    (AIP, 2018-09) Bothner, Thomas; Deift, Percy; Its, Alexander; Krasovsky, Igor; Mathematical Sciences, School of Science
    We review the authors’ recent work where we obtain the uniform large s asymptotics for the Fredholm determinant D(s,γ)≔det(I−γKs↾L2(−1,1)), 0 ≤ γ ≤ 1. The operator Ks acts with kernel Ks(x, y) = sin(s(x − y))/(π(x − y)), and D(s, γ) appears for instance in Dyson’s model of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned Gaussian unitary ensemble.