Surprising Pfaffian factorizations in Random Matrix Theory with Dyson index $\beta =2$

Abstract

In the past 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 $\beta =2$ whereas Pfaffians only for ensembles with $\beta =1,4$. We derive a non-trivial Pfaffian determinant for $\beta =2$ random matrix ensembles which is similar to the one for $\beta =1,4$. 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.

Comments: 23 pages; PACS: 02.10.Yn, 02.50.-r, 05.90.+m, 12.38.-t