[Paper Review] Fluctuations of extreme eigenvalues of sparse Erd\H{o}s-R\'enyi graphs

Yukun He, Antti Knowles|arXiv (Cornell University)|May 5, 2020

Random Matrices and Applications19 references4 citations

TL;DR

This paper establishes that for sparse Erdős-Rényi random graphs with edge probability $ Np \gg N^\varepsilon $, the extreme eigenvalues (excluding the Perron-Frobenius eigenvalue) exhibit asymptotically Gaussian fluctuations governed by a single random variable representing the total degree of the graph. The result extends previous Tracy-Widom fluctuations down to the optimal scaling $ Np \gg N^\varepsilon $, using a novel rigidity bound of order $ N^{-1/2-\varepsilon}(Np)^{-1/2} $ that avoids $ (Np)^{-1} $ expansions.

ABSTRACT

We consider a class of sparse random matrices which includes the adjacency matrix of the Erd\\H{o}s-R\\'enyi graph $\\mathcal{G}(N,p)$. We show that if $N^{\\varepsilon} \\leq Np \\leq N^{1/3-\\varepsilon}$ then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result [19] on the fluctuations of the extreme eigenvalues from $Np \\geq N^{2/9 + \\varepsilon}$ down to the optimal scale $Np \\geq N^{\\varepsilon}$. The main technical achievement of our proof is a rigidity bound of accuracy $N^{-1/2-\\varepsilon} \\, (Np)^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from [9,19,24]. Our result is the last missing piece, added to [8, 12, 19, 24], of a complete description of the eigenvalue fluctuations of sparse random matrices for $Np \\geq N^{\\varepsilon}$.

Motivation & Objective

• To describe the asymptotic distribution of extreme eigenvalues of sparse random matrices in the regime $ Np \gg N^\varepsilon $, extending beyond previous results.
• To resolve the last missing piece in the complete characterization of eigenvalue fluctuations for sparse random matrices down to the optimal scale.
• To establish that all nontrivial eigenvalues away from zero have asymptotically Gaussian fluctuations, governed by a single random variable—the total degree of the graph.
• To develop a new rigidity bound for extreme eigenvalues that avoids the $ (Np)^{-1} $-expansion techniques used in prior works.

Proposed method

• Introduces a general class of sparse random matrices satisfying moment and tail conditions on entries, including the normalized adjacency matrix of $ \mathcal{G}(N,p) $.
• Defines a key random variable $ \mathcal{Z} = \frac{1}{N}\operatorname{Tr}H^2 - 1 $, which governs the eigenvalue fluctuations and converges to a standard normal distribution.
• Establishes a rigidity bound of accuracy $ N^{-1/2-\varepsilon}(Np)^{-1/2} $ for extreme eigenvalues, avoiding reliance on $ (Np)^{-1} $ expansions from earlier works.
• Uses a self-consistent equation approach and resolvent analysis to relate the Stieltjes transform of the empirical measure to the perturbation parameter $ \mathcal{Z} $.
• Applies a localization argument to show that eigenvalue fluctuations are driven entirely by $ \mathcal{Z} $, with corrections bounded by $ O_{\prec}(1/(\sqrt{N}q^3)) $.
• Employs a perturbative expansion of the semicircle law and edge behavior to relate eigenvalue locations to the rescaled parameter $ \mathcal{Z} $.

Experimental results

Research questions

• RQ1 Do extreme eigenvalues of sparse Erdős-Rényi graphs exhibit Gaussian fluctuations down to $ Np \gg N^\varepsilon $, beyond the previously known $ Np \gg N^{2/9} $ regime?
• RQ2 Is the fluctuation of the second-largest eigenvalue governed by a single random variable, and what is its interpretation in terms of graph structure?
• RQ3 Can a rigidity bound for extreme eigenvalues be established without relying on $ (Np)^{-1} $-order expansions, enabling sharper control in the sparse regime?
• RQ4 How does the eigenvalue distribution near the edge of the semicircle law deform under the influence of the total degree random variable $ \mathcal{Z} $?
• RQ5 What is the precise scaling of eigenvalue fluctuations in the regime $ 1 \ll Np \ll N^{1/3} $, and does it match the Gaussian limit?

Key findings

• For $ Np \gg N^\varepsilon $, the fluctuations of the second-largest eigenvalue $ \lambda_{N-1} $ are asymptotically Gaussian, with $ \sqrt{\frac{N^2 p}{2}}(\lambda_{N-1} - \mathbb{E}\lambda_{N-1}) \xrightarrow{d} \mathcal{N}(0,1) $, extending the range from $ Np \gg N^{2/9} $ to $ Np \gg N^\varepsilon $.
• The eigenvalue fluctuations are fully governed by a single random variable $ \mathcal{Z} $, which corresponds to the normalized total degree of the graph and converges to a standard normal distribution.
• A new rigidity bound of accuracy $ N^{-1/2-\varepsilon}(Np)^{-1/2} $ is established for extreme eigenvalues, avoiding the $ (Np)^{-1} $-expansion used in EKYY1, LS1, and HLY.
• The result completes the picture of eigenvalue fluctuations for sparse random matrices in the regime $ Np \gg N^\varepsilon $, filling the last missing piece after EKYY2, LS1, HLY, and H19.
• The edge of the semicircle law is deformed by $ \mathcal{Z} $, and the eigenvalue location $ \gamma_i $ satisfies $ \gamma_i - \gamma_{0,i}(1 + \mathcal{Z}/2) = O_{\prec}(1/(\sqrt{N}q^3)) $, confirming the leading-order shift.
• The proof technique avoids $ (Np)^{-1} $ expansions, relying instead on a refined analysis of the Stieltjes transform and self-consistent equations, enabling the optimal scaling.