Patterned Random Matrices: deviations from universality

TL;DR

This paper explores how structural constraints on real symmetric matrices cause deviations from the universal Wigner distribution in their level spacing statistics, with analytical results for small matrices and numerical evidence for larger sizes.

Contribution

It analytically derives level spacing distributions for three constrained matrix ensembles and demonstrates deviations from universality both analytically for small matrices and numerically for larger ones.

Findings

Reverse circulant matrices with zeros have slower-than-exponential spacing distribution.

Symmetric circulant matrices exhibit Poisson spacings.

Palindromic Toeplitz matrices show level repulsion but differ from Wigner distribution.

Abstract

We investigate the level spacing distribution for three ensembles of real symmetric matrices having additional structural constraint to reduce the number of independent entries to only $ (n+1)/2 $ in contrast to the $ n(n+1)/2 $ for a real symmetric matrix of size $ n \times n $. We derive all the results analytically exactly for the $ 3\times 3 $ matrices and show that spacing distribution display a range of behaviour based on the structural constraint. The spacing distribution of the ensemble of reverse circulant matrices with additional zeros is found to fall slower than exponential for larger spacing while that of symmetric circulant matrices has poisson spacings. The palindromic symmetric toeplitz matrices on the other hand show level repulsion but the distribution is significantly different from Wigner. The behaviour of spacings for all the three ensembles clearly show the departure from universal result of Wigner distribution for real symmetric matrices. The deviation from universality continues in large $ n $ cases as well, which we study numerically.