2014/08/10 by Thomas Löbbe, Löbbe, Thomas
Mathematics · #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1408.2218
For two lacunary sequences (Mn,1)n\≥ 2,(Mn,2)n\≥ 0 and\nsuitable functions f we introduce random matrix ensembles with\n n Xn,n'=f(Mn+n',1x1,M|n-n'|,2x2). We prove weak\nconvergence of the mean empirical eigenvalue distribution towards the\nsemicircle law under some further number theoretic properties of the sequence\n(Mn,1)n\≥ 1. Furthermore we give examples to show that even in this\nparticular class of random matrix ensembles the asymptotic behaviour of the\nspectrum becomes delicate. We prove that the empirical spectral distribution\ndoes not converge to the semicircle law in general even if the correlation of\ntwo entries decays exponentially in the distance. For\nf(x1,x2)=1/\√(2)\⋅(\cos(2\π(x1+x2))+\cos(4\π(x1+x2))) and\nMn,1=2n we show that the mean empirical spectral distribution does not\nconverge to semicircle law while for any sequence (Mn,1)n\≥ 1 with\nMn+1,1/Mn,1\→\∞ for n\→\∞ and any periodic function f of\nfinite total variation in the sense of Hardy and Krause with mean zero and unit\nvariance the mean spectral distribution converges to the semicircle law.\n