2019
DOI: 10.1007/s10955-019-02266-8
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Mesoscopic Linear Statistics of Wigner Matrices of Mixed Symmetry Class

Abstract: We prove a central limit theorem for the mesoscopic linear statistics of N × N Wigner matrices H satisfying E|Hij| 2 = 1/N andWe show that on all mesoscopic scales η (1/N η 1), the linear statistics of H have a sharp transition at 1 − σ ∼ η. As an application, we identify the mesoscopic linear statistics of Dyson's Brownian motion Ht started from a real symmetric Wigner matrix H0 at any nonnegative time t ∈ [0, ∞]. In particular, we obtain the transition from the central limit theorem for GOE to the one for GU… Show more

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Cited by 9 publications
(6 citation statements)
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“…for a centred random variable h. Unlike in previous works (e.g. [27], [32], [28], [9], [29], [26]), our object E G F * requires recursive cumulant expansions in order to control the error terms. The first expansion by (3.3) leads to the Schwinger-Dyson (or self-consistent) equation…”
Section: )mentioning
confidence: 99%
“…for a centred random variable h. Unlike in previous works (e.g. [27], [32], [28], [9], [29], [26]), our object E G F * requires recursive cumulant expansions in order to control the error terms. The first expansion by (3.3) leads to the Schwinger-Dyson (or self-consistent) equation…”
Section: )mentioning
confidence: 99%
“…CLT for linear statistics ( 1.6 ) for Wigner matrices H has been proven [ 1 , 3 , 13 , 26 , 28 30 , 33 , 34 , 36 , 38 , 41 , 47 , 49 ] for test functions of the form with some fixed reference point , scaling exponent and smooth function g with compact support, i.e for macroscopic ( ) and mesoscopic ( ) test functions living on a single scale . These proofs give optimal error terms for such functions but they were not optimized for dealing with functions that oscillate on a mesoscopic scale and have macroscopic support, like for some , .…”
Section: Introductionmentioning
confidence: 99%
“…In the case of mesoscopic linear statistics, one also has a central limit theorem without normalization in the CUE case (see [29]). The behavior of mesoscopic linear statistics of other random matrix ensembles has also been studied: the Gaussian Unitary Ensemble (see [15]), more general Wigner matrices (see [19,20]) and determinantal processes (see [23]), the Circular Beta Ensemble (see [25]), the thinned CUE, for which a random subset of the eigenvalues has been removed (see [8]). However, the smooth mesoscopic linear statistics of permutation matrices have not been previously studied.…”
Section: Introductionmentioning
confidence: 99%