Spectral Distribution of Non-independent Random Matrix Ensembles induced by Lacunary Systems

TL;DR

This paper investigates the spectral distribution of certain non-independent random matrix ensembles generated by lacunary sequences, establishing conditions under which the eigenvalue distribution converges to the semicircle law and providing counterexamples where it does not.

Contribution

It introduces a new class of random matrix ensembles based on lacunary sequences and analyzes their spectral distribution, revealing conditions for convergence to the semicircle law and illustrating delicate asymptotic behaviors.

Findings

Weak convergence to semicircle law under specific number theoretic conditions.

Counterexamples where spectral distribution does not converge to semicircle law.

Convergence depends on the growth rate of lacunary sequences and properties of the function f.

Abstract

For two lacunary sequences $(M_{n,1})_{n\geq 2},(M_{n,2})_{n\geq 0}$ and suitable functions $f$ we introduce random matrix ensembles with \begin{equation*} X_{n,n'}=f(M_{n+n',1}x_1,M_{|n-n'|,2}x_2). \end{equation*} We prove weak convergence of the mean empirical eigenvalue distribution towards the semicircle law under some further number theoretic properties of the sequence $(M_{n,1})_{n\geq 1}$. Furthermore we give examples to show that even in this particular class of random matrix ensembles the asymptotic behaviour of the spectrum becomes delicate. We prove that the empirical spectral distribution does not converge to the semicircle law in general even if the correlation of two entries decays exponentially in the distance. For $f(x_1,x_2)=1/\sqrt{2}\cdot(\cos(2\pi(x_1+x_2))+\cos(4\pi(x_1+x_2)))$ and $M_{n,1}=2^n$ we show that the mean empirical spectral distribution does not converge to semicircle law while for any sequence $(M_{n,1})_{n\geq 1}$ with $M_{n+1,1}/M_{n,1}\to\infty$ for $n\to\infty$ and any periodic function $f$ of finite total variation in the sense of Hardy and Krause with mean zero and unit variance the mean spectral distribution converges to the semicircle law.