Evenness symmetry and inter-relationships between gap probabilities in random matrix theory
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Peter J Forrester
Abstract
Our interest is in the generating function EN,β(I;ξ;wβ) for the probabilities EN,β(n;I;wβ) that in a matrix ensemble with unitary (β = 2) or orthogonal (β = 1) symmetry, characterized by the weight wβ(λ) and having N eigenvalues, the interval I contains exactly n eigenvalues. Using a determinant formula for EN,2, a general quadratic identity is obtained which relates EN,2 in the case I and w2(x) even to a product of generating functions EN,2 with different I, w2(λ) and N, and for which the eigenvalues are positive. Also, generalizing some earlier calculations, the sum EN,1 (2n − 1; I; w1) + EN,1 (2n; I; w1) for N even, I = (−t, t) and w1 an even classical weight is shown to equal EN/2,2(n; (0, t2); w2) for w2 related to w1. Implications of these identities are discussed.
© Walter de Gruyter
Articles in the same Issue
- Evenness symmetry and inter-relationships between gap probabilities in random matrix theory
- Higher-order Sobolev and Poincaré inequalities in Orlicz spaces
- Characteristically nilpotent Lie algebras and symplectic structures
- Equalities in algebras of generalized functions
- A geometric study of generalized Neuwirth groups
- On C. T. C. Wall's suspension theorem
- On compactness of Sobolev embeddings in rearrangement-invariant spaces
- Extending rationally connected fibrations
Articles in the same Issue
- Evenness symmetry and inter-relationships between gap probabilities in random matrix theory
- Higher-order Sobolev and Poincaré inequalities in Orlicz spaces
- Characteristically nilpotent Lie algebras and symplectic structures
- Equalities in algebras of generalized functions
- A geometric study of generalized Neuwirth groups
- On C. T. C. Wall's suspension theorem
- On compactness of Sobolev embeddings in rearrangement-invariant spaces
- Extending rationally connected fibrations
