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A recursion formula for moments of derivatives of random matrix polynomials

Altuğ, SA; Bettin, S; Petrow, I; Rishikesh; Whitehead, I; (2014) A recursion formula for moments of derivatives of random matrix polynomials. The Quarterly Journal of Mathematics , 65 (4) pp. 1111-1125. 10.1093/qmath/hat054. Green open access

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Abstract

We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O−(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of L-functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants that allows the rapid computation of the (k+1)st constant in terms of the kth and (k−1)st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of τ-functions associated with Painlevé differential equations.

Type: Article
Title: A recursion formula for moments of derivatives of random matrix polynomials
Open access status: An open access version is available from UCL Discovery
DOI: 10.1093/qmath/hat054
Publisher version: https://doi.org/10.1093/qmath/hat054
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10084832
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