The k-Point Random Matrix Kernels Obtained from One-Point Supermatrix Models

TL;DR

This paper reveals that kernels for k-point correlations in Gaussian Random Matrix Ensembles can be directly derived from supermatrix models for one-point functions, linking spectral determinants to correlation structures.

Contribution

It demonstrates that the generating functions of one-point functions contain all information needed to obtain the kernels for k-point correlations, a novel insight in random matrix theory.

Findings

Kernels are obtained from supermatrix models for one-point functions.

One-point generating functions encode all k-point correlation information.

Links established between spectral determinants and correlation kernels.

Abstract

The k-point correlation functions of the Gaussian Random Matrix Ensembles are certain determinants of functions which depend on only two arguments. They are referred to as kernels, since they are the building blocks of all correlations. We show that the kernels are obtained, for arbitrary level number, directly from supermatrix models for one-point functions. More precisely, the generating functions of the one-point functions are equivalent to the kernels. This is surprising, because it implies that already the one-point generating function holds essential information about the k-point correlations. This also establishes a link to the averaged ratios of spectral determinants, i.e. of characteristic polynomials.