On the gap probability generating function at the spectrum edge in the case of orthogonal symmetry
Abstract
Description
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 Painlevé 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.
12 pages, to appear Contemporary Mathematics (Proceedings) `Integrable Systems, Random Matrices and Applications', edited by J. Baik, T. Kriecherbauer and K. McLaughlin
12 pages, to appear Contemporary Mathematics (Proceedings) `Integrable Systems, Random Matrices and Applications', edited by J. Baik, T. Kriecherbauer and K. McLaughlin