Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter

TL;DR

This paper introduces parameter-dependent matrix ensembles derived from superpositions and decimations of classical orthogonal ensembles, providing new correlation functions and revealing novel functional forms in the process.

Contribution

It extends classical matrix ensemble identities to include parameters, creating new interpolating ensembles and deriving their correlation functions with novel polynomial constructions.

Findings

Parameter-dependent ensembles interpolate between classical ensembles.

New correlation functions are computed for finite and scaled limits.

Results differ from previously known functional forms for orthogonal and symplectic cases.

Abstract

A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously.