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Watch this 55-minute lecture by Jiaoyang Huang from the University of Pennsylvania, presented at Harvard CMSA's Program on Classical, quantum, and probabilistic integrable systems. Explore the fascinating relationship between Ramanujan properties and edge universality in random regular graphs, with particular focus on their extremal eigenvalues. Learn about spectral gaps in graphs and their significance in theoretical computer science and combinatorics. The lecture begins with foundational background on eigenvalues of random d-regular graphs and their connections to random matrix theory, then progresses to recent research findings on eigenvalue rigidity and edge universality. Discover how these properties relate to the Kesten-McKay distribution and the Tracy-Widom distribution from the Gaussian Orthogonal Ensemble, leading to the conclusion that approximately 69% of d-regular graphs are Ramanujan graphs. This presentation covers joint research work with Theo McKenzie and Horng-Tzer Yau.
