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Area of Science:

  • Graph theory
  • Random matrix theory
  • Statistical physics

Background:

  • Erdős-Rényi graphs are fundamental models in random graph theory.
  • Understanding eigenvector localization is crucial for analyzing complex systems.

Purpose of the Study:

  • To analyze the behavior of eigenvectors in critical Erdős-Rényi graphs.
  • To characterize the spectral properties and localization patterns of these eigenvectors.

Main Methods:

  • Analysis of adjacency matrix eigenvectors for critical Erdős-Rényi graphs.
  • Investigation of spectral splitting and eigenvector localization.
  • Definition and analysis of the localization exponent.

Main Results:

  • The spectrum splits into a delocalized middle phase and a semilocalized edge phase.
  • Eigenvectors in the semilocalized phase are concentrated around resonant vertices with exponential decay.
  • A sharp transition between phases is observed, indicated by a discontinuity in the localization exponent.

Conclusions:

  • Critical Erdős-Rényi graphs exhibit distinct eigenvector localization regimes.
  • The findings provide insights into the spectral properties of random graphs.
  • The results are valid for the optimal regime of graph parameters.