Gaussian fluctuations for non-Hermitian random matrix ensembles

By Brian Rider, J. Silverstein. 2005

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Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4+ɛ) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$ and letting N→∞, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type XN(f)=∑k=1Nf(λk) where λ1, λ2, …, λN denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533–605], where the analogous result for random sample covariance matrices is established.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2b18b5bf70fef06d39af8544ae0affa872f10e45

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Citations in the corpus

The Noise in the Circular Law and the Gaussian Free Field (2006, result): “This should be compared with the result of Rider and Silverstein [18], where the same central limit theorem is proved for random matrices with more general than NC(0, 1) entries, but where the test functions need to be analytic throughout |z| ≤ 4.”
Fluctuations of eigenvalues and second order Poincaré inequalities (2007, methodology): “This was proved for polynomial f by Jonsson [36], and for a much larger class of functions in Bai and Silverstein [6].”
Fluctuations of eigenvalues of random normal matrices (2008, background): “6 of [20]. Cf. also [ 19 ]. It should also be noted that Th. 1.8, as well as the more general Th. 8.2 below, follow from the well-known "physical” arguments due to Wiegmann et al. See e.g.”
Around the circular law (2011, background): “Note that the square of the smallest singular value sn(n −1/2G)2 of the Complex Ginibre Ensemble follows an exponential law [40] and this result is asymptotically universal [148]; • Gap probabilities and Voronöı cells (see [3, 62] for the Ginibre Ensemble); • Linear statistics of μX (for some result”
Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, background): “We remark that for macroscopic statistics 1 n ∑n i=1 F (λi/ √ n) with F fixed and analytic, such extensions (without the need for matching moments beyond the second moment) were already established in [40].”
Circular law theorem for random Markov matrices (2008, unknown): “Another very interesting open problem is the fluctuation of the global and extremal spectrum for which one can find some partial answers in [ 43 ], [36], [42], [38].”