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By M. Aizenman (Chief Editor)

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Due˜nez The presence of the Euclidean factor S 1 (which comes from the subset of scalar multiples of the identity matrix within the ensemble) is rather convenient and natural. If we were to define “irreducible” circular ensembles analogously to Dyson’s circular ensembles, except requiring that they consist of matrices with unit determinant, then the spaces so obtained would be irreducible symmetric spaces of the compact type (i. , the factors S 1 would disappear from (16)). However, the measure on eigenvalues would no longer be translationally invariant (under transformations of the form → + (t, .

This is not the case for symmetric spaces of non-compact type. To clarify the difference, we analyze the example of the classical Gaussian matrix ensembles, which also fit within the framework of the theory of symmetric spaces (the construction is analogous to that of the circular ensembles): GUE ≈ SL(N, C)/SU (N ) × E 1 , GOE ≈ SL(N, R)/SO(N ) × E 1 , GSE ≈ SU ∗ (2N )/U Sp(2N ) × E 1 . (17) Finding the probability measure on eigenvalues also reduces to a factorization of measures dµ(H ) = dHaar(k)dν(a) in the sense of (5), where K is still the group of invariance (orthogonal, unitary, symplectic) of the ensemble’s measure, but where A E N is now a Euclidean space, which in the case of these ensembles consists of real diagonal matrices which can be parametrized by N -tuples = (λ1 , .

Note that here a, b > (A,B) as in (137), the −1 corresponds to A, B > −2. With cN as in (138) and ψN = ψN summation formula in this case reads (a,b) SR4 (x, y) = 1 1 − x 2 (A,B) K (x, y) − c2R−1 ψ2R (y)δψ2R−1 (x), 1 − y 2 2R,2 2 1 2 (156) where the operator δ acts by 1 δf (x) = (157) f (t)dt. x The formula (156) only holds verbatim when a > 0 (that is, A, B > −1), since the (A,B) integral defining δψN is divergent for A ≤ −1. However, we note that the skew orthogonal polynomials of the second kind are analytic functions of the parameters a, b > −1 (corresponding to A, B > −2), hence the kernel KN4 is an analytic function on a, b > −1.

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