Hard and soft edge spacing distributions for random matrix ensembles with orthogonal and symplectic symmetry

TL;DR

This paper derives exact formulas for the distribution of gaps at the spectrum edges of random matrix ensembles with orthogonal and symplectic symmetry, using Painlevé transcendents and Fredholm determinants.

Contribution

It provides new exact evaluations of edge gap probabilities for these ensembles, linking symmetry types through inter-relations and advanced special functions.

Findings

Exact gap probabilities expressed via Painlevé transcendents.

Derived formulas for orthogonal and symplectic ensembles.

Enhanced understanding of spectrum edge behaviors.

Abstract

Inter-relations between random matrix ensembles with different symmetry types provide inter-relations between generating functions for the gap probabilites at the spectrum edge. Combining these in the scaled limit with the exact evaluation of the gap probabilities for certain superimposed ensembles with orthogonal symmetry allows for the exact evaluation of the gap probabilities at the hard and soft spectrum edges in the cases of orthogonal and symplectic symmetry. These exact evaluations are given in terms of Painlev\'e transcendents, and in terms of Fredholm determinants.