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• W3095592040 abstract "Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance: $ bullet $ its value in $1$ (Keating-Snaith theorem), $ bullet $ the truncation of its Fourier series up to any fraction of its degree, $ bullet $ the computation of the relative volume of the Birkhoff polytope, $ bullet $ its products and ratios taken in different points, $ bullet $ the product of its iterated derivatives in different points, $ bullet $ functionals in relation with sums of divisor functions in $ mathbb{F}_q[X] $. $ bullet $ its mid-secular coefficients, $ bullet $ the of moments, etc. We revisit or compute for the first time the asymptotics of the integer moments of these last functionals and several others. The method we use is a very general one based on reproducing kernels, a symmetric function generalisation of some classical orthogonal polynomials interpreted as the Fourier transform of particular random variables and a local Central Limit Theorem for these random variables. We moreover provide an equivalent paradigm based on a randomisation of the mid-secular coefficients to rederive them all. These methodologies give a new and unified framework for all the considered limits and explain the apparition of Hankel determinants or Wronskians in the limiting functional." @default.
• W3095592040 created "2020-11-09" @default.
• W3095592040 creator A5048392115 @default.
• W3095592040 date "2020-11-04" @default.
• W3095592040 modified "2023-09-27" @default.
• W3095592040 title "A new approach to the characteristic polynomial of a random unitary matrix" @default.
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• W3095592040 hasPublicationYear "2020" @default.
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