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2012

Averaging Fluctuations in Resolvents of Random Band Matrices

Erdos, Laszlo, Knowles, Antti, Yau, Horng-Tzer

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We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint.

We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries.
Our results generalize the previous results of [5,16,17] which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models.
Our bounds apply to random band matrices, and improve previous estimates from order 2 to order 4 in the cases relevant for applications.

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L. Erdős, B. Schlein, and H.T. Yau, Local semicircle law and complete delocalization for Wigner random matrices , Comm. Math. Phys. 287
2009
Earlier work this paper cites.
L. Erdős, S. Péché, J.A. Ramirez, B. Schlein, and H.T. Yau, Bulk universality for Wigner matrices , Comm. Pure Appl. Math. 63
2010
Earlier work this paper cites.

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L. Erdős, J. Ramirez, B. Schlein, T. Tao, V. Vu, and H.T. Yau, Bulk universality for Wigner hermitian matrices with subexponential decay , Math. Res. Lett. 17
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L. Erdős, J. Ramirez, B. Schlein, and H.T. Yau, Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation , Electr. J. Prob. 15
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L. Erdős, A. Knowles, H.T. Yau, and J. Yin, Delocalization and diffusion profile for random band matrices , Preprint arXiv:1205.5669
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L. Erdős, H.T. Yau, and J. Yin, Bulk universality for generalized Wigner matrices , Preprint arXiv:1001.3453
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N.S. Pillai and J. Yin, Universality of covariance matrices , Preprint arXiv:1110.2501
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Then

L. Erdős and A. Knowles, Quantum diffusion and delocalization for band matrices with general distribution , Ann. H. Poincaré 12
2011
Later among the works it cites.
L. Erdős, B. Schlein, H.T. Yau, and J. Yin, The local relaxation flow approach to universality of the local statistics of random matrices , Ann. Inst. Henri Poincaré (B) 48
2012
Closest in time.

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