Supersymmetry approach to Wishart correlation matrices: Exact results

TL;DR

This paper derives exact formulas for the eigenvalue distribution of real and complex Wishart matrices using supersymmetry, providing new insights especially for the real case where no explicit results existed before.

Contribution

It introduces a supersymmetry-based method to compute the one-point function for real Wishart matrices, filling a gap in the theoretical understanding.

Findings

Derived the first explicit one-point function for real Wishart matrices.

Expressed the real case as a twofold integral involving Jack polynomials.

Validated the formulas through numerical simulations.

Abstract

We calculate the `one-point function', meaning the marginal probability density function for any single eigenvalue, of real and complex Wishart correlation matrices. No explicit expression had been obtained for the real case so far. We succeed in doing so by using supersymmetry techniques to express the one-point function of real Wishart correlation matrices as a twofold integral. The result can be viewed as a resummation of a series of Jack polynomials in a non-trivial case. We illustrate our formula by numerical simulations. We also rederive a known expression for the one-point function of complex Wishart correlation matrices.