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Fan Yang: Delocalization of random band matrices

Time: Tue 2022-04-05 15.15 - 16.15

Location: Zoom, meeting ID: 698 3346 0369

Lecturer: Fan Yang (University of Pennsylvania)

Abstract

We consider a general class of random band matrices H on the d-dimensional lattice of linear size L. Its entries \(h_{xy}\) are independent random variables up to symmetry \(H=H^*\) and are negligible if \(|x-y|\) exceeds the band width W. It is conjectured that a sharp localization-delocalization transition of bulk eigenvectors occurs at a critical band width \(W=L^a\), and the critical exponent is \(a=\frac{1}{2}\) in dimension 1 and \(a=0\) in dimensions 2 and higher. In this talk, I will discuss about some recent progresses about the delocalized phase of random band matrices. In particular, we show that the critical exponents is at most \(\frac{3}{4}\) in dimension 1, at most \(\frac{2}{d+2}\) in dimensions 3 to 7, and equal to 0 in dimensions 8 and higher. Based on joint work with Bourgade, Yau and Yin.