Abstract
We construct the distributionP(S) of nearest-neighbor level spacings for the orthogonal, unitary, and symplectic ensembles of (Hermitian and unitary) random matrices in the limit of large dimension. The Taylor expansion ofP(S) aroundS=0 is given explicitly to arbitrarily high orders. By employing a diagonal Padé approximation we interpolate between the small-S behavior given by the Taylor expansion and the rigorously known asymptotic form at largeS.
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Actually, (2.6) requires l1≠l 2≠...≠l n . When some of thel 1 coin-cide such thatk different values are assumed, 1≦k≦n, with multiplicitiesn 1,n 2,...,n k and ∑n i =n, the factor 1/n! must be replaced by 1/n 1!n 2!...n k! For the Taylor coefficients,E 1, given in (2.11 or 14), the distinction in question is irrelevant, due to the appearance of another factorn 1!...n k !/n! accompanying the summation over the configurations {l i }
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Dietz, B., Haake, F. Taylor and Padé analysis of the level spacing distributions of random-matrix ensembles. Z. Physik B - Condensed Matter 80, 153–158 (1990). https://doi.org/10.1007/BF01390663
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DOI: https://doi.org/10.1007/BF01390663