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

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.

arXiv commentfixed some minor inaccuracies in the introduction; main results unchanged