Uploaded September 2026 | Updated September 2026, 1 week ago
Jiaoyang Huang (University of Pennsylvania)
https://simons.berkeley.edu/talks/jiaoyang-huang-university-pennsylvania-2026-09-03
Joint Boot Camp: Spectral Theory Beyond Graphs + Pseudorandomness and High-Dimensional Expansion
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 begin by providing background on the eigenvalues of random d-regular graphs and their connections to random matrix theory. Then I will discuss our 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. Finally I will present a streamlined framework for proving the convergence to the random matrix statistics based on a microscopic version of the loop equations. These characterizations provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble under consideration. In many models, these equations follow from local laws and integration by parts.
Jiaoyang Huang (University of Pennsylvania)
https://simons.berkeley.edu/talks/jiaoyang-huang-university-pennsylvania-2026-09-03
Joint Boot Camp: Spectral Theory Beyond Graphs + Pseudorandomness and High-Dimensional Expansion
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 begin by providing background on the eigenvalues of random d-regular graphs and their connections to random matrix theory. Then I will discuss our 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. Finally I will present a streamlined framework for proving the convergence to the random matrix statistics based on a microscopic version of the loop equations. These characterizations provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble under consideration. In many models, these equations follow from local laws and integration by parts.
