English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For two lacunary sequences $(M_{n,1})_{n\geq 2},(M_{n,2})_{n\geq 0}$ andsuitable 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 weakconvergence of the mean empirical eigenvalue distribution towards thesemicircle 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 thisparticular class of random matrix ensembles the asymptotic behaviour of thespectrum becomes delicate. We prove that the empirical spectral distributiondoes not converge to the semicircle law in general even if the correlation oftwo 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 notconverge 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 $f$ offinite total variation in the sense of Hardy and Krause with mean zero and unitvariance the mean spectral distribution converges to the semicircle law.

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