Discrete Circular Beta Ensembles
نویسنده
چکیده
Let μ be a measure with support on the unit circle and n ≥ 1, β > 0. The associated circular β ensemble involves a probability distribution of the form P β (μ; t1, t2, ..., tn) = C |V (t1, t2, ..., tn)| β dμ (t1) ...dμ (tn) , where C is a normalization constant, and V (t1, t2, ..., tn) = ∏ 1≤i 0 and n ≥ 2. The β-ensemble with temperature 1/β, associated with the measure μ, involves a probability distribution on Γn of the form P β (μ; t1, t2, ..., tn) = 1 Zn |V (t1, t2, ..., tn)| dμ (t1) ...dμ (tn) , (1.1) where (1.2) V (t1, t2, ...tn) = ∏ 1≤i<j≤n (tj − ti) = det [ tj−1 i ] 1≤i,j≤n and (1.3) Zn = ∫ ... ∫ |V (t1, t2, ...tn)| dμ (t1) ...dμ (tn) . These ensembles arise in analysing random unitary (β = 2), orthogonal (β = 1), and symplectic matrices (β = 4) in mathematical physics [1], [7], Date : October 30, 2012. 1991 Mathematics Subject Classification. Primary 41A10, 41A17, 42C99; Secondary 33C45.
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