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Gaussian and Chiral β-Ensembles, which generalise well-known orthogonal (β=1), unitary (β=2), and symplectic (β=4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like {β,N,n}⇔{4/β,n,N} for all β>0, where N and n respectively denote the number of eigenvalues and products of characteristic polynomials. At the edge of the spectrum, matrix integrals...
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