Exact and explicit evaluation of Brezin-Hikami kernels

TL;DR

This paper derives exact, explicit formulas for the Brezin-Hikami kernels in Hermitian random matrix ensembles, enabling precise analysis of energy correlations with broad parameter applicability.

Contribution

It provides the first explicit, exact formulas for these kernels, expanding the analytical tools available for studying Hermitian random matrix energy correlations.

Findings

Exact formulas for the kernels are derived.

Analytical and graphical representations of physical quantities are provided.

Results are valid for arbitrary parameter values.

Abstract

We present exact and explicit form of the kernels $hat{K}(x, y)$ appearing in the theory of energy correlations in the ensembles of Hermitian random matrices with Gaussian probability distribution, see E. Brezin and S. Hikami, Phys. Rev. E 57, 4140 and E 58, 7176 (1998). In obtaining this result we have exploited the analogy with the method of producing exact forms of two-sided, symmetric Levy stable laws, presented by us recently. This result is valid for arbitrary values of parameters in question. We furnish analytical and graphical representations of physical quantities calculated from $hat{K}(x, y)$'s.