Linear eigenvalue statistics of random matrices with a variance profile

Abstract

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centred, of a random matrix with a variance profile and the standard Gaussian random variable. The second order Poincaré inequality type result introduced in [12] is used to establish the bound. Using this bound we prove Central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We reestablish some existing results on fluctuations of linear eigenvalue statistics of some well known random matrix ensembles by choosing appropriate variance profiles.

Fluctuations Of Linear Spectral Statistics Of Deformed Wigner Matrices

“…Gaussian fluctuations with different scale, mean and variance also hold for linear spectral statistics of the form ş R ϕ z pxqµ WN pdxq when the entries of the Wigner matrix have an infinite fourth moment ( [BGM16]; see also [BGGM14] for the case of non square integrable entries, in which case Wigner's Theorem fails to hold [BAG08]). When entries of the Wigner matrix are not identically distributed in such a way that their variances differ (these matrices are called band matrices or sometimes Wigner matrices with variance profile), fluctuations of linear spectral statistics have also been described (see [AJS19] and references therein).…”

Single eigenvalue fluctuations of general Wigner-type matrices

“…Fluctuations for generalized Wigner matrices were studied by Li-Xu [45] and for some general classes of random matrices by Anderson-Zeitouni [7] and by Guionnet [34]. Fluctuations for some polynomial test functions for matrices with a variance profile were obtained by Adhikari-Jana-Saha [2]. Linear statistics of heavy-tailed and half-heavy-tailed matrices have been considered by Benaych-Georges and Maltsev [11], Benaych-Georges, Guionnet and Male [9], and Lodhia and Maltsev [46]; sparse matrices were considered by He in [36].…”

“…Assume tM ĺ r{2 and φp´1q " 0 and φp1q " 1. We have, ż r´r,rs2 ˆφpxq ´φpyq x ´y ˙2 dxdy " 2| logptq| `pK `1q2 Op| log | log t||q (5.190) Proof. Note that under our assumptions, }φ 1 } 1 ĺ CK.…”

Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices

“…Gaussian fluctuations with different scale, mean and variance also hold for linear spectral statistics when the entries of the Wigner matrix have an infinite fourth moment ( [BGM16]; see also [BGGM14] for the case of non square integrable entries, in which case Wigner's Theorem fails to hold [BAG08]). When entries of the Wigner matrix are not identically distributed in such a way that their variances differ (these matrices are called band matrices or sometimes Wigner matrices with variance profile), fluctuations of linear spectral statistics have also been described (see [AJS19] and references therein). Fluctuations of linear spectral statistics were also investigated at the mesoscopic scale.…”