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Delocalization Transition for Critical Erdős-Rényi Graphs
Johannes Alt1, Raphael Ducatez1, Antti Knowles1
1Section of Mathematics, University of Geneva, Rue du Conseil-Général 7-9, 1205 Geneva, Switzerland.
Communications in Mathematical Physics
|November 1, 2021
Summary
We analyzed eigenvectors in critical Erdős-Rényi graphs, revealing a spectrum split into delocalized and semilocalized phases. This transition is sharp, with eigenvector localization changing abruptly.
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.
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