On the gap probability generating function at the spectrum edge in the case of orthogonal symmetry

Details On the gap probability generating function at the spectrum edge in the case of orthogonal symmetry On the gap probability generating function at the spectrum edge in the case of orthogonal symmetry

2007-08-12

arxiv.org/abs/0708.1589v1

Physics

Mathematical Physics

12 pages, to appear Contemporary Mathematics (Proceedings) `Integrable Systems, Random Matrices and Applications', edited by J

Scientific paper

The gap probability generating function has as its coefficients the probability of an interval containing exactly $k$ eigenvalues. For scaled random matrices with orthogonal symmetry, and the interval at the hard or soft spectrum edge, the gap probability generating functions have the special property that they can be evaluated in terms of Painlev\'e transcendents. The derivation of these results makes use of formulas for the same generating function in certain scaled, superimposed ensembles expressed in terms of its correlation functions. It is shown that by a judicious choice of the superimposed ensembles, the scaled limit necessary to derive these formulas can be rigorously justified by a straight forward analysis.

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