Advances in Theoretical and Mathematical Physics

Volume 21 (2017)

Number 3

Universality for a class of random band matrices

Pages: 739 – 800

DOI: http://dx.doi.org/10.4310/ATMP.2017.v21.n3.a5

Authors

Paul Bourgade (Courant Institute, New York University, New York, N.Y., U.S.A.)

Laszlo Erdős (Institute of Science and Technology Austria, Klosterneuburg, Austria)

Horng-Tzer Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Jun Yin (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Abstract

We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $W \sim N$. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.

Keywords

universality, band matrices, Dyson–Brownian motion, quantum unique ergodicity

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The work of P.B. is partially supported by NSF grants DMS-1208859 and DMS-1513587.

The work of L.E. is partially supported by ERC Advanced Grant, RANMAT 338804.

The work of H.-T.Y is partially supported by NSF grant DMS-1307444, DMS-1606305 and a Simons Investigator award.

The work of J.Y. is partially supported by NSF Grant DMS-1207961. The major part of this research was conducted when all authors were visiting IAS and were also supported by the NSF Grant DMS-1128255.

Published 25 August 2017