Quantitative Tracy-Widom laws for sparse random matrices

TL;DR

This paper establishes that the fluctuations of the largest eigenvalue of sparse random matrices, including Erdős-Rényi graphs, follow the Tracy-Widom law with a specific convergence rate, using advanced Green function comparison techniques.

Contribution

It proves Tracy-Widom fluctuations for the largest eigenvalue of sparse matrices in a new regime and develops algorithms for Green function comparison at fine spectral scales.

Findings

Largest eigenvalue fluctuations follow Tracy-Widom law

Convergence rate is almost O(N^{-1/3} + p^{-2} N^{-4/3})

Applicable to Erdős-Rényi graphs and similar sparse matrices

Abstract

We consider the fluctuations of the largest eigenvalue of sparse random matrices, the class of random matrices that includes the normalized adjacency matrices of the Erd\H{o}s-R\'enyi graph $G(N, p)$. We show that the fluctuations of the largest eigenvalue converge to the Tracy-Widom law at a rate almost $O(N^{-1/3 } + p^{-2} N^{-4/3})$ in the regime $p \gg N^{-2/3 }$. Our proof builds upon the Green function comparison method initiated by Erd\H{o}s, Yau, and Yin [22]. To show a Green function comparison theorem for fine spectral scales, we implement algorithms for symbolic computations involving averaged products of Green function entries.