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

Abstract

In recent decades, determinants and Pfaffians were found for eigenvalue correlations of various random matrix ensembles. These structures simplify the average over a large number of ratios of characteristic polynomials to integrations over one and two characteristic polynomials only. Up to now it was thought that determinants occur for ensembles with Dyson index β = 2, whereas Pfaffians only for ensembles with β = 1, 4. We derive a non-trivial Pfaffian determinant for β = 2 random matrix ensembles which is similar to the one for β = 1, 4. Thus, it unveils a hidden universality of this structure. We also give a general relation between the orthogonal polynomials related to the determinantal structure and the skew-orthogonal polynomials corresponding to the Pfaffian. As a particular example, we consider the chiral unitary ensembles in great detail.