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

Abstract

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.

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

“…We employ a modification of Andréief's integration theorem derived in [49] and find a determinantal point process…”

“…Then the limit ε → 0 in combination with taking the imaginary part yields Dirac delta functions such that one of the integrals over r 1 and r 2 can be exactly performed. The remaining integral can be easily performed, too, by employing the idea of [49] where the term 1/ det(bΛ − r1 1 p ) can be written as a quotient of a Vandermonde determinant and a Cauchy-Vandermonde determinant. The degeneracy of the spectrum is not that academic.…”

Surprising Pfaffian factorizations in random matrix theory with Dyson index β = 2

“…[34,35] circumvents the integration theorem by Dyson and Metha [24,25,4,48]. Moreover the approach of Refs.…”

“…For example a similar situation arises in the transition from GUE to GOE or GSE [42,34]. If the ensemble is purely a GUE then then the eigenvalue correlations can be cast into determinants whereas the smallest interaction with a GOE or a GSE yields a Pfaffian.…”

“…In Refs. [34,35] the general conditions where derived to find these structures. Thus all these particular random matrix ensembles were unified in one procedure to derive these structures.…”