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

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 $β=2$ 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.