Abstract:

Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. Specifically, the spectral gap—the difference between the largest and second-largest eigenvalues—measures the expansion properties of a graph. In this talk, I will focus on random d-regular graphs. 

I will begin by providing background on the eigenvalues of random d-regular graphs and their connections to random matrix theory. In the second part of the talk, I will discuss our recent results on eigenvalue rigidity and edge universality for these graphs. Eigenvalue rigidity asserts that, with high probability, each eigenvalue concentrates around its classical location as predicted by the Kesten-McKay distribution. Edge universality states that the second-largest eigenvalue and the smallest eigenvalue of random d-regular graphs converge to the Tracy-Widom distribution from the Gaussian Orthogonal Ensemble. Consequently, approximately 69% of d-regular graphs are Ramanujan graphs. This work is based on joint work with Theo McKenzie and Horng-Tzer Yau.

Short Bio:

Jiaoyang Huang is an Associate Professor in the Department of Statistics and Data Science at the Wharton School of the University of Pennsylvania, with a secondary appointment in the Department of Mathematics. He received his Ph.D. in Mathematics from Harvard University in 2019, under the supervision of Professor Horng-Tzer Yau. Following his doctoral studies, he was a member at the Institute for Advanced Study from 2019 to 2020, and a Simons Junior Fellow at New York University from 2020 to 2022. His research focuses on random matrix theory, a mathematical framework widely used across modern science and engineering. He develops and applies this theory to study the universal behavior of sparse random graphs, interacting particle systems, and random growth models—problems with deep connections to graph theory, network theory, and mathematical physics. His work has been supported by the National Science Foundation and a Sloan Research Fellowship. He is also the recipient of the 2024 Bernoulli Society New Researcher Award and a 2022 finalist for the Blavatnik Regional Awards.