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arXiv · 1212.1389

A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials

S. Ali Altug·Sandro Bettin·Ian Petrow·Rishikesh·Ian Whitehead
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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 which allows the rapid computation of the (k+1)-st constant in terms of the k-th and (k-1)-st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of \tau-functions associated with Painlev\'e differential equations.

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S. Ali Altug, Sandro Bettin, Ian Petrow, Rishikesh, Ian Whitehead. 2012-12-06. A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials. https://doi.org/10.1093/qmath%2Fhat054

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