Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges

TL;DR

This paper derives universal formulas for eigenvalue correlations during orthogonal-unitary and symplectic-unitary transitions in random matrices, focusing on hard and soft spectral edges, with implications for understanding spectral universality.

Contribution

It provides explicit quaternion determinant expressions for parameter-dependent eigenvalue distributions and connects behaviors at hard and soft edges in random matrix ensembles.

Findings

Derived quaternion determinant formulas for eigenvalue distributions.

Computed limiting behaviors at spectral edges for large matrices.

Identified universal asymptotics of two-point correlations.

Abstract

For the orthogonal-unitary and symplectic-unitary transitions in random matrix theory, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion determinant. For the parameter dependent Gaussian and Laguerre ensembles the matrix elements of the determinant are expressed in terms of corresponding skew-orthogonal polynomials, and their limiting value for infinite matrix dimension are computed in the vicinity of the soft and hard edges respectively. A connection formula relating the distributions at the hard and soft edge is obtained, and a universal asymptotic behaviour of the two point correlation is identified.