Title: Spectral gap in random regular graphs and hypergraphs
Abstract: Random graphs and hypergraphs have been used for decades to model large-scale networks, from biological, to electrical, and to social. Various random graphs (and their not-so-random properties) have been connected to algorithms solving problems from community detection to matrix completion, coding theory, and various other statistics / machine learning fundamental questions; in the past decade, this research area has expanded to include random hypergraphs. One of these special properties is the spectral gap for graph-associated matrices; roughly speaking, it means that the main eigenvalue(s) are well-separated from the bulk and it guarantees strong connectivity properties. This talk will take a look at the spectra of adjacency / Laplacian matrices for some random regular models, explain how we know that the spectral gap is there, and connect spectral properties to the aforementioned applications. It will cover joint work with Gerandy Brito, Kameron Decker Harris, and Yizhe Zhu.
Ioana Dumitriu is a Professor of Mathematics at The University of California, San Diego.