Level Repulsion in Constrained Gaussian Random-Matrix Ensembles

TL;DR

This paper introduces new constrained Gaussian random-matrix ensembles, analyzes their spectral properties, and establishes conditions under which level repulsion persists despite constraints, extending to different symmetry classes.

Contribution

It defines new constrained ensembles, derives conditions for level repulsion to persist, and generalizes the Fourier theorem to relate spectral properties under constraints.

Findings

Level repulsion persists at short distances in constrained ensembles.

Constraints can lift level repulsion only in the limit of strict enforcement.

DGUE interpolates between GUE and constrained 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.