Edge Universality of Sparse Random Matrices

2022/06/14 by Jiaoyang Huang, Horng‐Tzer Yau, Huang, Jiaoyang +1 · 3 citations
Mathematics · #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2206.06580

Abstract

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd\H os-Rényi graph G(N,p). Recently, it was shown by Lee, up to an explicit random shift, the optimal rigidity of extreme eigenvalues holds, provided the averaged degree grows with the size of the graph, pN>N^ε. We prove in the same regime, (i) Optimal rigidity holds for all eigenvalues with respect to an explicit random measure. (ii) Up to an explicit random shift, the fluctuations of the extreme eigenvalues are given the Tracy-Widom distribution.

Cited by

• Fluctuations of functions of sparse Erdős-Rényi graphs
• Quantitative Tracy-Widom laws for sparse random matrices
• Sharp Square Root Bounds for Edge Eigenvector Universality in Sparse Random Regular Graphs

Related

• Local law and Tracy-Widom limit for sparse random matrices
• Edge universality of sparse Erdős-Rényi digraphs
• Higher order fluctuations of extremal eigenvalues of sparse random matrices
• Transition from Tracy-Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős-Rényi graphs
• Optimal Eigenvalue Rigidity of Random Regular Graphs