[Paper Review] No-gaps delocalization for general random matrices

Mark Rudelson, Roman Vershynin|arXiv (Cornell University)|Jun 12, 2015

Random Matrices and Applications34 references4 citations

TL;DR

This paper establishes no-gaps delocalization for eigenvectors of general random matrices, proving that any subset of coordinates of size at least $\varepsilon n$ carries a non-negligible fraction of the $\ell_2$ norm of the eigenvector with high probability. The authors develop a geometric approach using invertibility of random matrices, small ball probabilities, and least common denominators to extend delocalization beyond Gaussian and Wigner ensembles to matrices with independent or symmetric entries, including complex matrices with independent real and imaginary parts.

ABSTRACT

We prove that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its $\ell_2$ norm. Our results pertain to a wide class of random matrices, including matrices with independent entries, symmetric and skew-symmetric matrices, as well as some other naturally arising ensembles. The matrices can be real and complex; in the latter case we assume that the real and imaginary parts of the entries are independent.

Motivation & Objective

• To establish that eigenvectors of general random matrices are delocalized in the sense that no subset of coordinates of size $\varepsilon n$ carries a negligible $\ell_2$ mass.
• To extend delocalization results beyond Gaussian and Wigner matrices to a broad class of random matrices, including those with independent, symmetric, or skew-symmetric entries.
• To address the universality phenomenon in random matrix theory by showing that eigenvector delocalization—specifically no-gaps delocalization—holds for non-Gaussian, non-invariant ensembles.
• To provide a geometric framework based on invertibility and distance estimates to control the structure of kernel vectors and eigenvectors.
• To unify and generalize previous results on eigenvector delocalization by introducing a robust method applicable to both continuous and discrete distributions.

Proposed method

• Reduces the problem of eigenvector delocalization to the invertibility of random matrices restricted to subspaces, using geometric and probabilistic techniques.
• Employs small ball probability estimates via the least common denominator (LCD) to control the likelihood that a random vector is close to a fixed subspace.
• Introduces a net argument over vectors with bounded LCD and controlled real-imaginary correlations to handle the discrete case.
• Uses weak $L^p$ estimates and the weak triangle inequality to bound the sum of inverse distances from random vectors to subspaces.
• Applies a conditioning argument to handle complex matrices by fixing the imaginary part and treating real and imaginary parts independently.
• Combines results on distances between random vectors and subspaces with kernel structure analysis to derive invertibility bounds for general distributions.

Experimental results

Research questions

• RQ1 Can no-gaps delocalization be established for eigenvectors of general random matrices beyond the Gaussian and Wigner cases?
• RQ2 What is the minimal assumption on the distribution of matrix entries under which eigenvectors remain delocalized in the sense of having no large gaps in $\ell_2$ mass?
• RQ3 How can the invertibility of random matrices be controlled uniformly over subspaces to ensure that no eigenvector concentrates on any subset of coordinates?
• RQ4 To what extent can the small ball probability method be adapted to handle non-continuous, discrete, or dependent matrix entries?
• RQ5 Can the geometric approach based on distances and kernels be extended to complex matrices with independent real and imaginary parts?

Key findings

• For any $\varepsilon \in (0,1)$, with high probability, every eigenvector $v$ of a general random matrix satisfies $\left(\sum_{j \in J} |v_j|^2\right)^{1/2} \geq \phi(\varepsilon)\|v\|_2$ for all subsets $J \subset [n]$ of size at least $\varepsilon n$, where $\phi(\varepsilon)$ is a positive function depending only on $\varepsilon$.
• The authors prove that the smallest singular value of a random matrix restricted to a subspace of dimension $\varepsilon n$ is bounded below with high probability, via a small ball probability estimate with exponent $\varepsilon n$.
• For general distributions, the probability that the smallest singular value is less than $t\sqrt{n}$ decays as $\left[C \varepsilon^{-1.4} t^{0.45}\right]^{\varepsilon n}$, showing strong concentration of mass.
• The method applies to matrices with independent entries, symmetric and skew-symmetric matrices, and complex matrices with independent real and imaginary parts.
• The result holds uniformly over all eigenvectors, not just those in the bulk of the spectrum, and extends to matrices with discrete or non-identically distributed entries under mild moment conditions.
• The proof establishes that the kernel of a random matrix is incompressible with high probability, which is key to ruling out mass concentration on small sets of coordinates.