A Localization–Delocalization Transition for Nonhomogeneous Random Matrices

Laura Shou, Ramon van Handel

Research output: Contribution to journalArticlepeer-review


We consider N×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≫logN the random matrix possesses a delocalized approximate top eigenvector, while when d≪logN 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.

Original languageEnglish (US)
Article number26
JournalJournal of Statistical Physics
Issue number2
StatePublished - Feb 2024

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • 60B20
  • Approximate eigenvectors
  • Localization–delocalization transition
  • Nonhomogeneous random matrices


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