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

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 Painlevé transcendents, and in terms of Fredholm determinants.

14 pages

References in corpus (5)

• Level-Spacing Distributions and the Airy Kernel
• Classical skew orthogonal polynomials and random matrices
• A determinantal formula for the GOE Tracy-Widom distribution
• Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles
• Relationships between $\tau$-function and Fredholm determinant expressions for gap probabilities in random matrix theory