A random matrix decimation procedure relating β = 2/(r + 1) to β = 2(r + 1)

摘要

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case r = 1 of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as β-ensembles. The inter-relations give that the joint distribution of every (r + 1)st eigenvalue in certain β-ensembles with β = 2/(r + 1) is equal to that of another β-ensemble with β = 2(r + 1). The proof requires generalizing a conditional probability density function due to Dixon and Anderson.