Real moments of the logarithmic derivative of characteristic polynomials in random matrix ensembles

TL;DR

This paper establishes new asymptotic formulas for real moments of the logarithmic derivative of characteristic polynomials in various random matrix ensembles, extending previous results beyond integer moments and employing novel proof techniques.

Contribution

It provides the first non-integer moment asymptotics for these derivatives across multiple ensembles using a new proof method that does not rely on prior integer moment results.

Findings

Asymptotics derived for real moments in unitary, orthogonal, and symplectic ensembles.

New proof techniques that differ from previous methods.

Extension of moment asymptotics to non-integer values.

Abstract

We prove asymptotics for real moments of the logarithmic derivative of characteristic polynomials evaluated at $1-\frac{a}{N}$ in unitary, even orthogonal, and symplectic ensembles, where $a>0$ and $a=o(1)$ as the size $N$ of the matrix goes to infinity. Previously, such asymptotics were known only for integer moments (in the unitary ensemble by the work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, and in orthogonal and symplectic ensembles by the work of Alvarez and Snaith), except that in the odd orthogonal ensemble real moments asymptotics were obtained by Alvarez, Bousseyroux and Snaith. Our proof is new and does not make use of the aforementioned integer moments results, and is different from the method in the work of Alvarez et al for the odd orthogonal ensemble.