2015

Mesoscopic linear statistics of Wigner matrices

Lodhia, A., Simm, N. J.

Understand

We study linear spectral statistics of $N \times N$ Wigner random matrices $\mathcal{H}$ on mesoscopic scales.

Under mild assumptions on the matrix entries of $\mathcal{H}$, we prove that after centering and normalizing, the trace of the resolvent $\mathrm{Tr}(\mathcal{H}-z)^{-1}$ converges to a stationary Gaussian process as $N \to \infty$ on scales $N^{-1/3} \ll \mathrm{Im}(z) \ll 1$ and explicitly compute the covariance structure.
The limit process is related to certain regularizations of fractional Brownian motion and logarithmically correlated fields appearing in \cite{FKS13}.
Finally, we extend our results to general mesoscopic linear statistics and prove that the limiting covariance is given by the $H^{1/2}$-norm of the test functions.

Built on

Statistical Theory of the Energy Levels of Complex Systems. IV
F.J. Dyson and M.L. Mehta · 1963
Earlier work this paper cites.
Fractional Brownian Motions, Fractional Noises and Applications
B. B. Mandelbrot and J. W. van Ness · 1968
Earlier work this paper cites.
A limit theorem for measurable random processes and its applications
L. S. Grinblat · 1976
Earlier work this paper cites.
On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in L p E L^{E}_{p}
Heinz Cremers and Dieter Kadelka · 1986
Earlier work this paper cites.
Universality of the correlations between eigenvalues of large random matrices
E. Brézin and A. Zee · 1993
Earlier work this paper cites.
The functional calculus
E. B. Davies · 1995
Earlier work this paper cites.
Asymptotic properties of large random matrices with independent entries
A.M. Khorunzhy, B.A. Khoruzhenko, and L.A. Pastur · 1996
Earlier work this paper cites.
On fluctuations of eigenvalues of random Hermitian matrices
K. Johansson · 1998
Earlier work this paper cites.
Asymptotic distribution of smoothed eigenvalue density. I. Gaussian random matrices
A. Boutet de Monvel and A. Khorunzhy · 1999
Earlier work this paper cites.
Asymptotic distribution of smoothed eigenvalue density. II. Wigner random matrices
A. Boutet de Monvel and A. Khorunzhy · 1999
Earlier work this paper cites.
The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities
A. Soshnikov · 2000
Earlier work this paper cites.
On the Characteristic Polynomial of a Random Unitary Matrix
C. P. Hughes, J. P. Keating, and N. O’Connell · 2001
Earlier work this paper cites.
Theory and Applications of Long-Range Dependence
P. Doukhan, G. Oppenheim, and M. Taqqu · 2003
Earlier work this paper cites.
CLT for linear spectral statistics of large-dimensional sample covariance matrices
Z.D. Bai and J.W. Silverstein · 2004
Earlier work this paper cites.
Random Matrices
M. L. Mehta · 2004
Earlier work this paper cites.
On the convergence of the spectral empirical process of Wigner matrices
Z. D. Bai and J. Yao · 2005
Earlier work this paper cites.
Limiting laws of linear eigenvalue statistics for Hermitian matrix models
L. Pastur · 2006
Earlier work this paper cites.
Gaussian fluctuations for non-Hermitian random matrix ensembles
B. Rider and J.W. Silverstein · 2006
Earlier work this paper cites.
The noise in the Circular Law and the Gaussian free field
B. Rider and B. Virág · 2007
Earlier work this paper cites.

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Then

The Local Semicircle Law for a General Class of Random Matrices
L. Erdös, A. Knowles, H-T. Yau, and J. Yin · 2013
Later among the works it cites.
Averaging Fluctuations in Resolvents of Random Band Matrices
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Original
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Mark Rudelson and Roman Vershynin · 2013
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A contour line of the continuum Gaussian free field
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Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
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CLT for spectra of submatrices of Wigner random matrices
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Fixed energy universality for generalized Wigner matrices
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Yan V. Fyodorov and Jonathan P. Keating · 2014
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Fractional gaussian fields: a survey
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