[Paper Review] Local law and complete eigenvector delocalization for supercritical Erd\H{o}s-R\'enyi graphs.

Yukun He, Antti Knowles+1 more|arXiv (Cornell University)|Aug 28, 2018

Random Matrices and Applications15 references3 citations

TL;DR

This paper establishes a local law and complete eigenvector delocalization for the adjacency matrix of supercritical Erdōs-Rényi random graphs down to the critical scale $pN = C\log N$, where $C$ is a universal constant. The authors introduce a novel family of multilinear large deviation estimates for sparse random vectors that balance $\ell^2$ and $\ell^\infty$ norms with combinatorial factors, enabling strong concentration bounds essential for proving the local law and eigenvector delocalization in the very sparse regime, thus extending previous results that required $pN \gtrsim (\log N)^6$. The key contribution is the first local law and complete delocalization result all the way down to the critical connectivity threshold.

ABSTRACT

We prove a local law for the adjacency matrix of the Erd\H{o}s-R\'enyi graph $G(N, p)$ in the supercritical regime $ pN \geq C\log N$ where $G(N,p)$ has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from [11] by extending them all the way down to the critical scale $pN = O(\log N)$. A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed $\ell^2$ and $\ell^\infty$ norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale $pN = O(\log N)$. These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.

Motivation & Objective

• To establish a local law for the adjacency matrix of the Erdōs-Rényi random graph $G(N,p)$ in the supercritical regime $pN \geq C\log N$, extending previous results that required $pN \gtrsim (\log N)^6$.
• To prove complete eigenvector delocalization, showing $\|u_i\|_\infty \lesssim N^{-1/2}$ with high probability, down to the critical scale $pN = C\log N$.
• To develop a new family of multilinear large deviation estimates for sparse random vectors that optimally balance $\ell^2$ and $\ell^\infty$ norms of coefficients with combinatorial factors, enabling concentration bounds in the very sparse regime.
• To demonstrate that both the local law and eigenvector delocalization fail in the subcritical regime $pN \leq (1-\varepsilon)\log N$, where isolated vertices lead to localized eigenvectors and a macroscopic number of zero eigenvalues.

Proposed method

• Derive a self-consistent equation for the Green's function $G(z) = (A - z)^{-1}$ using the Schur complement formula, where $A$ is the rescaled adjacency matrix of $G(N,p)$.
• Use a bootstrap argument in the spectral scale $\eta$ to propagate local law estimates from larger to smaller scales down to $\eta \sim N^{-1}$.
• Introduce a new family of multilinear large deviation bounds for sparse random vectors, controlling expressions like $\sum_{i \neq j} a_{ij} X_i X_j$ in terms of mixed $\ell^2$ and $\ell^\infty$ norms of the coefficients $a_{ij}$, with explicit dependence on $pN$.
• Apply these bounds with $r = \log N$ to control error probabilities via Chebyshev's inequality, achieving $N^{-D}$ error bounds for any fixed $D > 0$ necessary for high-probability statements.
• Use a union bound over $N^2$ entries of the Green's function, combined with the large deviation bounds, to control the maximum deviation $\max_{i,j} |G_{ij}(z) - \delta_{ij} m(z)|$.
• Establish the local law by showing $\max_{i,j} |G_{ij}(z) - \delta_{ij} m(z)| \lesssim \zeta$ with high probability, where $\zeta$ is a scale-dependent error term depending on $\eta$, $pN$, and $N$.

Experimental results

Research questions

• RQ1 Can a local law for the adjacency matrix of the Erdōs-Rényi graph be established all the way down to the critical scale $pN = C\log N$?
• RQ2 Is complete eigenvector delocalization, i.e., $\|u_i\|_\infty \lesssim N^{-1/2}$, valid for the supercritical Erdōs-Rényi graph at the critical threshold $pN = C\log N$?
• RQ3 Can large deviation estimates for sparse random vectors be refined to handle the very sparse regime $pN = O(\log N)$, where standard bounds fail?
• RQ4 Why do the local law and eigenvector delocalization fail in the subcritical regime $pN \leq (1-\varepsilon)\log N$?
• RQ5 What is the optimal dependence of the local law error bound on the sparsity parameter $pN$?

Key findings

• The local law holds for the rescaled adjacency matrix $A$ down to the critical scale $pN = C\log N$, with $\max_{i,j} |G_{ij}(z) - \delta_{ij} m(z)| \lesssim \zeta$ with high probability for all $\eta \gg N^{-1}$, where $\zeta$ is a scale-dependent error term.
• Complete eigenvector delocalization is proven: $\max_i \|u_i\|_\infty \lesssim N^{-1/2}$ with high probability in the supercritical regime $pN \geq C\log N$.
• The local law and eigenvector delocalization fail in the subcritical regime $pN \leq (1-\varepsilon)\log N$, where isolated vertices cause localized eigenvectors with $\|u_i\|_\infty = 1$ and a macroscopic number of zero eigenvalues.
• The authors construct a new family of multilinear large deviation bounds for sparse random vectors, which control $\left\| \sum_{i \neq j} a_{ij} X_i X_j \right\|_r$ in terms of $\gamma = \left( \max_i \frac{1}{N} \sum_j |a_{ij}|^2 \right)^{1/2}$ and $\psi = \max_{i,j} \frac{|a_{ij}|}{\sqrt{pN}}$, achieving optimal dependence on $pN$ down to $O(\log N)$.
• The proof achieves high-probability bounds with error probability $N^{-D}$ for any fixed $D > 0$ by using $r = \log N$ in the large deviation estimates and a union bound over $N^2$ entries.
• The result is optimal in the sense that the assumption $pN \geq C\log N$ cannot be improved, as the local law and delocalization fail for $pN \leq (1-\varepsilon)\log N$.