Precise Eigenvalue Location for Random Regular Graphs

Explore a 44-minute lecture by Theo McKenzie of Stanford University on "Precise Eigenvalue Location for Random Regular Graphs," presented at IPAM's Free Entropy Theory and Random Matrices Workshop on February 27, 2025. Delve into the spectral analysis of random regular graphs, which serve as important models for sparse, well-connected networks with applications in theoretical computer science and statistical physics. Learn how, despite the challenging analysis caused by strong dependence between entries, precise information about the spectrum can be achieved, demonstrating that eigenvalues fluctuate within optimally small intervals and edge eigenvalues follow the distribution of the largest eigenvalue in a Gaussian Orthogonal Ensemble matrix. Discover the implications of this research, including the finding that most regular graphs are Ramanujan with optimally large spectral gaps. The presentation details a methodology involving tight analysis of the Green's function of the adjacency operator, specifically examining changes after random edge switches.