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

TL;DR

This paper investigates the gap probability generating function at the spectrum edge for orthogonal symmetry in random matrices, showing it can be expressed via Painlevé transcendents and rigorously derived through superimposed ensemble analysis.

Contribution

It provides a rigorous derivation of the gap probability generating function at the spectrum edge using superimposed ensembles and Painlevé transcendents for orthogonal symmetry.

Findings

The generating function can be expressed in terms of Painlevé transcendents.

Rigorous justification of the scaled limit in superimposed ensembles.

Explicit formulas for gap probabilities at spectrum edges.

Abstract

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.