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

TL;DR

This paper studies how constraining certain elements of large Gaussian-uniform random matrices affects their spectral fluctuation properties, finding that small constraints do not alter the universal fluctuation behavior.

Contribution

It demonstrates that spectral fluctuation measures remain unchanged under certain constraints, extending the understanding of universality in random matrix theory.

Findings

Spectral fluctuation measures are preserved under small constraints.

A critical constraint level exists beyond which spectral properties change.

Universal fluctuation behavior persists below the critical constraint.

Abstract

We investigate the spectral fluctuation properties of constrained ensembles of random matrices (defined by the condition that a number N(Q) of matrix elements vanish identically; that condition is imposed in unitarily invariant form) in the limit of large matrix dimension. We show that as long as N(Q) is smaller than a critical value (at which the quadratic level repulsion of the Gaussian unitary ensemble of random matrices may be destroyed) all spectral fluctuation measures have the same form as for the Gaussian unitary ensemble.