A note on moments of derivatives of characteristic polynomials

Abstract

International audience 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. 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.

Derivative Moments for Characteristic Polynomials from the CUE

“…In order to put our results into context, we first recall a result of Dehaye [25,26]. To do this it will be necessary to fix some notations regarding combinatorics of partitions.…”

“…We have re-written Dehaye's result using our notation. In fact he considers quantities related to C N and C (see for example equations (10) and (11) of [26]), which are defined similarly to (2.5) and (2.6), but without the restriction on the number of parts of λ in the summation. The presence of the factor [k] λ mean that his and our quantities coincide for p 2k (but could be different for p > 2k).…”

“…Hughes [24] used random matrix theory to calculate F (h, k) for h = 1, 2, 3 and Dehaye [25,26] has derived formulae for F (h, k) for all h ∈ N in terms of sums over partitions (see section 2 below for a precise statement). Using their results, the values F (1, 1) = 1/12, F (1, 2) = 1/720 and F (2, 2) = 1/6720 can be calculated.…”

Upper bounds for fractional joint moments of the Riemann zeta function

“…The arithmetic factor is a well understood product over primes. The random matrix factor has many different expressions including combinatorial sums [13,14,20], a multiple contour integral in the case h = k [12], and a determinant of Bessel functions [2,12]. For h, k not necessarily equal, the random matrix factor can be solved for finite N and is related to the solution of a Painlevé V type differential equation [4].…”

A new approach to the characteristic polynomial of a random unitary matrix

“…The study of the characteristic polynomial became then an independent subject of its own, with specific methods at the junction of several fields, such as: analysis of Toeplitz determinants [155,156] and orthogonal polynomials on the unit circle [170,171,58,145], mathematical physics [175] and supersymmetry [80,103,114], algebra and integrable systems [1,2,245], algebraic combinatorics [38,88,108], representation theory and symmetric functions [66,87,100], probability theory [23,54,56,72], Weingarten calculus [75,191,248], (free) Itô calculus [39,65,84,161,196,244], etc. The conjunction of such a diversity of methods shows the richness of the topic and the variety of strategies one can use to perform computations on Z U N (x).…”