Joint moments of derivatives of characteristic polynomials

Abstract

We present a simple technique to compute moments of derivatives of unitary characteristic polynomials. The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted Schur functions. The answer is finally given as a sum over partitions of functions of the contents. One can also obtain alternative expressions involving hypergeometric functions of matrix arguments.Résumé. Nous introduisons une nouvelle technique, en deux parties, pour calculer les moments de dérivées de polynômes caractéristiques. La première étape repose sur une idée de Bump et Gamburd et utilise l'orthonormalité des fonctions de Schur sur les groupes unitaires pour calculer des moyennes de polynômes caractéristiques de matrices aléatoires. La deuxième étape, qui est nécessaire pour passer aux dérivées, utilise une généralisation du théorème binomial due à Okounkov et Olshanski. Ce théorème livre les coefficients des séries de Taylor pour les fonctions de Schur sous la forme de "shifted Schur functions". La réponse finale est donnée sous forme de somme sur les partitions de fonctions des contenus. Nous obtenons aussi d'autres expressions en terme de fonctions hypergéométriques d'argument matriciel.

A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

“…where Φ ∞ (z), Φ 0 (z), and Φ 1 (z) are holomorphic and invertible in neighbourhoods of the respective points. Also, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) we see that, in our case, the formal monodromy exponents θ ∞ , θ 0 , and θ 1 are…”

“…The results obtained suggest that in general F N (h, k) grows like N k 2 +2h as N → ∞ and it is a key problem to prove this and then to evaluate the limit (1)(2)(3)(4) F (h, k) := lim N →∞ 1 N k 2 +2h F N (h, k). For h, k ∈ N, k > h − 1/2, an expression for F N (h, k) was obtained in [12] in terms of certain sums over partitions. A similar answer was also found in the case h = (2m − 1)/2, m ∈ N, k ∈ N, k > h − 1/2 in [46] (see [46] also for a survey on other related results).…”

“…Further evidence of a connection between number theory and random matrix theory was given when Keating and Snaith [32,33] used results for the characteristic polynomial of a random unitary matrix to formulate conjectures about moments of the zeta function that are supported by number-theoretic and numerical results (see, for instance, the review articles [30,45,34,31]). Since then, a number of more general results on the joint moments of the characteristic polynomial and its derivative have been proven and used to formulate conjectures about the joint moments of L-functions and their derivatives [7,8,9,10,12,23,24,46].…”

“…Equation (1)(2)(3)(4)(5)(6)(7)(8) is a special case of the σ-Painlevé V equation with three parameters given in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) (see Okamoto [40,41,42] and Jimbo and Miwa [27]). For further examples of solutions of Painlevé equations expressed as Wronskian determinants of generalised Laguerre polynomials or confluent hypergeometric functions see [17,29,36,39].…”

Large spaces between the zeros of the Riemann zeta-function and random matrix theory II

“…I first saw the explicit formula (5) in a preprint of Dehaye [1] (who attributes it to Hughes). In the case h = 3 the formula as I have written it in (4) requires adjustment to fit with (5) in that an extra factor κ 2 − 9 should be introduced in both the numerator and denominator.…”