Derivation of determinantal structures for random matrix ensembles in a new way

TL;DR

The paper introduces a novel method for calculating averages over ratios of characteristic polynomials in random matrix ensembles, using a supersymmetry-inspired approach without actual superspaces, unifying determinantal structures and deriving k-point correlation functions.

Contribution

It presents a new approach called 'supersymmetry without supersymmetry' for analyzing random matrix ensembles, unifying determinantal structures and deriving correlation functions.

Findings

Derived determinantal structures for various ensembles.

Provided a new expression for k-point correlation functions.

Unified treatment of different random matrix ensembles.

Abstract

There are several methods to treat ensembles of random matrices in symmetric spaces, circular matrices, chiral matrices and others. Orthogonal polynomials and the supersymmetry method are particular powerful techniques. Here, we present a new approach to calculate averages over ratios of characteristic polynomials. At first sight paradoxically, one can coin our approach "supersymmetry without supersymmetry" because we use structures from supersymmetry without actually mapping onto superspaces. We address two kinds of integrals which cover a wide range of applications for random matrix ensembles. For probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way. As a new application we derive an expression for the k-point correlation function of an arbitrary rotation invariant probability density over the Hermitian matrices in the presence of an external field.