Unimodular Galton-Watson Treeings and Irregular Random Graphs - Almost Ramanujan

Explore the relationship between spectral radii in sparse graph limit theory through this 34-minute lecture by László Márton Tóth. Delve into the comparison of graphings and unimodular random graphs (URGs) as representations of graph sequence limits. Examine how the spectral radius of the random walk operator differs between these two objects, with graphings providing global information and URGs offering local, componentwise insights. Learn about Fraczyk's example demonstrating potential disparities between these invariants. Discover the proof that Unimodular Galton-Watson trees exhibit identical spectral radii for both representations. Uncover the implications of this finding for irregular random graphs sampled with the configuration model, leading to the conclusion that they are almost Ramanujan. Gain insights into this ongoing research, conducted in collaboration with Charles Bordenave, which extends Friedman's second eigenvalue theorem beyond regular graphs.