Ramanujan Property of Random Regular Graphs and Delocalization of Random Band Matrices

Explore advanced topics in random matrix theory through this 49-minute conference lecture that examines two significant recent developments in the field. Learn about the edge universality property for random d-regular graphs, where the speaker demonstrates how the distributions of extreme eigenvalues converge to the Tracy-Widom₁ distribution associated with the Gaussian Orthogonal Ensemble. Discover the proof that approximately 69% of d-regular graphs on N vertices are Ramanujan graphs, satisfying the condition that their second-largest eigenvalue absolute value is at most 2, which resolves conjectures by Sarnak and Miller-Novikoff-Sabelli. Examine the delocalization properties of eigenvectors in N×N Hermitian d-dimensional random band matrices with band width W, including the proof that all L²-normalized eigenvectors are delocalized in the bulk spectrum under appropriate conditions. Understand how eigenvalue statistics in these band matrices follow those of the Gaussian unitary ensemble, providing insights into the universal behavior of random matrix systems across different mathematical structures.