Monday, July 29, 2013

1109.5109 (Mario Kieburg)

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

Mario Kieburg
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.
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