Generalized resolvent estimates for random band matrices

Fan Yang
University of California, Los Angeles (UCLA)
Department of Mathematics

Let H denote a general class of N by N symmetric 1-dimensional random band matrices with band width W much smaller than N. With a mean-field reduction method, Bourgade, Yau and Yin (2018) proved that the bulk universality and eigenvector delocalization hold for H with W>>N^{3/4} under some mild assumptions on the distributions of the matrix entries. The two key inputs for the proof are the quantum unique ergodicity property and a precise local estimate on the generalized resolvents (i.e. the local law) of band matrices. In this talk, I will talk about why it is crucial to have a sharp estimate for the local law. Then I will describe the main ideas in proving the local law for band matrices with W>>N^{3/4}. For W<

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