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A localization-delocalization transition for nonhomogeneous random matrices

Laura Shou, Ramon van Handel

TL;DR

This work analyzes $N\times N$ self-adjoint Gaussian matrices with an arbitrary $d$-regular sparsity pattern and reveals a universal localization-delocalization transition for approximate top eigenvectors at the spectral edge. The authors introduce $(L,\kappa)$-delocalization and prove a dichotomy: for $d\ll\log N$ all $(1-\varepsilon)$-approximate top eigenvectors are localized, while for $d\gg\log N$ there exists a $(1-o(1))$-approximate top eigenvector that is delocalized, with the delocalization constrained by $\nu \lesssim \frac{c\kappa^2}{\log(e/\kappa)}$. The delocalization result leverages a subspace perspective: if the top eigenvectors span a subspace whose projection is sufficiently flat (diagonal entries of the projection are small), then a delocalized vector exists, which is established via $\ell^q$ bounds and a probabilistic construction. In the large-$d$ regime, resolvent techniques and a mesoscopic semicircle law show that the edge eigenspace satisfies the required isotropy, enabling the construction of a delocalized approximate top eigenvector. Overall, the paper demonstrates a robust, pattern-independent transition for approximate eigenvectors, distinct from the localization properties of exact eigenvectors and with implications for nonhomogeneous random matrices.

Abstract

We consider $N\times N$ self-adjoint Gaussian random matrices defined by an arbitrary deterministic sparsity pattern with $d$ nonzero entries per row. We show that such random matrices exhibit a canonical localization-delocalization transition near the edge of the spectrum: when $d\gg\log N$ the random matrix possesses a delocalized approximate top eigenvector, while when $d\ll\log N$ any approximate top eigenvector is localized. The key feature of this phenomenon is that it is universal with respect to the sparsity pattern, in contrast to the delocalization properties of exact eigenvectors which are sensitive to the specific sparsity pattern of the random matrix.

A localization-delocalization transition for nonhomogeneous random matrices

TL;DR

This work analyzes self-adjoint Gaussian matrices with an arbitrary -regular sparsity pattern and reveals a universal localization-delocalization transition for approximate top eigenvectors at the spectral edge. The authors introduce -delocalization and prove a dichotomy: for all -approximate top eigenvectors are localized, while for there exists a -approximate top eigenvector that is delocalized, with the delocalization constrained by . The delocalization result leverages a subspace perspective: if the top eigenvectors span a subspace whose projection is sufficiently flat (diagonal entries of the projection are small), then a delocalized vector exists, which is established via bounds and a probabilistic construction. In the large- regime, resolvent techniques and a mesoscopic semicircle law show that the edge eigenspace satisfies the required isotropy, enabling the construction of a delocalized approximate top eigenvector. Overall, the paper demonstrates a robust, pattern-independent transition for approximate eigenvectors, distinct from the localization properties of exact eigenvectors and with implications for nonhomogeneous random matrices.

Abstract

We consider self-adjoint Gaussian random matrices defined by an arbitrary deterministic sparsity pattern with nonzero entries per row. We show that such random matrices exhibit a canonical localization-delocalization transition near the edge of the spectrum: when the random matrix possesses a delocalized approximate top eigenvector, while when any approximate top eigenvector is localized. The key feature of this phenomenon is that it is universal with respect to the sparsity pattern, in contrast to the delocalization properties of exact eigenvectors which are sensitive to the specific sparsity pattern of the random matrix.
Paper Structure (8 sections, 14 theorems, 55 equations, 1 figure)

This paper contains 8 sections, 14 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sparse Gaussian matrices
  3. Main result
  4. Outline
  5. Proof of localization for d<<log N
  6. Delocalization of uniform random vectors in a subspace
  7. Proof of delocalization for d>>log N
  8. Delocalization of uniform random vectors in the sphere

Key Result

Theorem 1.2

The following hold whenever $d,N\to\infty$.

Figures (1)

  • Figure 1: Band matrix, block matrix, and their direct sum

Theorems & Definitions (32)

  • Example 1.1
  • Theorem 1.2: Eigenvalues
  • proof
  • Remark 1.3
  • Definition 1.4: Delocalization
  • Definition 1.5: Approximate top eigenvector
  • Theorem 1.6: Localization-delocalization transition
  • Theorem 2.1: Gaussian concentration
  • Corollary 2.2
  • proof
  • ...and 22 more