﻿ YMSC Probability Seminar | Extreme eigenvalues of random regular graphs-Qiuzhen College,Tsinghua University

# YMSC Probability Seminar | Extreme eigenvalues of random regular graphs

Time：Thursday 16:00-17:00 June 13, 2024

Venue：双清综合楼C548

Organizer：吴昊，杨帆，姜建平，顾陈琳

Speaker：黄骄阳 Jiaoyang Huang University of Pennsylvania

Abstract

Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. Specifically, the spectral gap—the gap between the first and second largest eigenvalues—measures the expanding property of the graph. In this talk, I will focus on random $d$-regular graphs, for which the largest eigenvalue is $d$.

I'll first explain some conjectures on the extremal eigenvalue distributions of adjacency matrices of random $d$-regular graphs. In the second part of the talk, I will discuss a new proof of Alon's second eigenvalue conjecture, which asserts that with high probability, the second eigenvalue of a random $d$-regular graph concentrates around $2\sqrt{d-1}$. Our proof shows that the fluctuations of these extreme eigenvalues are bounded by $N^{−2/3+\varepsilon}$, where $\varepsilon>0$ can be arbitrarily small. This gives the same order of fluctuation as the eigenvalues of matrices from the Gaussian Orthogonal Ensemble. This work is based on joint research with Theo McKenzie and Horng-Tzer Yau.

Speaker

I am currently an Assistant Professor of Statistics and Data Science at University of Pennsylvania. Before that, I was a postdoc at Courant Institute of Mathematical Sciences of New York University, and a Junior Fellow at the Simons Society of Fellows from 2020 to 2022. And I was a member in the Institute for Advanced Study (IAS) for the 2019-2020 academic year. I got my Ph.D. degree in mathematics from Harvard University under the supervision of Professor Horng-Tzer Yau.

DATEJune 12, 2024
SHARE
Related News
• 0

#### Universality of extreme eigenvalues of a large non-Hermitian random matrix

Abstract：We will report on recent progress regarding the universality of the extreme eigenvalues of a large random matrix with i.i.d. entries. Beyond the radius of the celebrated circular law, we will establish a precise three-term asymptotic expansion for the largest eigenvalue (in modulus) with an optimal error term. Based on this result, we will further show that the properly normalized lar...

• 1

#### Extreme gap problems for random matrices

AbstractIn the talk, I will give a brief review of known results on the extreme gap problems (smallest and largest gaps of the eigenvalues) of various random matrix ensembles. Then I will present our recent work about the smallest gap of the Gaussian symplectic ensemble. This completes the picture of the small gap problem of classical Gaussian β ensembles for β=1, 2, 4. Our analysis can poten...