Level repulsion in constrained Gaussian random-matrix ensembles

Abstract

Introducing sets of constraints, we define new classes of random-matrix ensembles, the constrained Gaussian unitary (CGUE) and the deformed Gaussian unitary (DGUE) ensembles. The latter interpolate between the GUE and the CGUE. We derive a sufficient condition for GUE-type level repulsion to persist in the presence of constraints. For special classes of constraints, we extend this approach to the orthogonal and to the symplectic ensembles. A generalized Fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones. We find that in the DGUEs, level repulsion always prevails at a sufficiently short distance and may be lifted only in the limit of strictly enforced constraints.

Spectral fluctuation properties of constrained unitary ensembles of Gaussian-distributed random matrices

“…The authors of Ref. [1] considered not only the CGUE but in addition also what they called "Deformed Gaussian Ensembles". Here the delta functions in Eq.…”

“…In Ref. [1] it was shown perturbatively that ρ(E) and ρ P (E) differ by a term of order N Q /N 2 . Therefore, we expect the two level densities to differ significantly when N Q ∼ N 2 .…”

“…In particular, the following sufficient condition for level repulsion was given. The quadratic level repulsion characteristic of the GUE (the ensemble of Gaussian-distributed unitary random matrices) also prevails for the CGUE provided that the number N Q of constraints does not exceed a critical value, N Q < N crit Q (1) with N crit Q defined in Eq. (16) below.…”

“…Some spectral properties of the CGUE were exhibited in Ref. [1]. In particular, the following sufficient condition for level repulsion was given.…”

“…To make the paper self-contained, we collect in Section 2 some definitions and results given in Ref. [1]. In Section 3 we slightly modify the definition of the constrained ensembles so as to remove a singularity.…”