Delocalization Transition for Critical Erdős–Rényi Graphs
| Autor: | Johannes Alt, Raphael Ducatez, Antti Knowles |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Physics
010102 general mathematics Spectrum (functional analysis) Order (ring theory) Statistical and Nonlinear Physics Disjoint sets Function (mathematics) 01 natural sciences Article Combinatorics Delocalized electron 0103 physical sciences Exponent Adjacency matrix 0101 mathematics 010306 general physics Mathematical Physics Eigenvalues and eigenvectors |
| Zdroj: | Communications in Mathematical Physics |
| ISSN: | 1432-0916 0010-3616 |
| Popis: | We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph $${\mathbb {G}}(N,d/N)$$ G ( N , d / N ) , where d is of order $$\log N$$ log N . We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent $$\gamma (\varvec{\mathrm {w}})$$ γ ( w ) of an eigenvector $$\varvec{\mathrm {w}}$$ w , defined through $$\Vert \varvec{\mathrm {w}} \Vert _\infty / \Vert \varvec{\mathrm {w}} \Vert _2 = N^{-\gamma (\varvec{\mathrm {w}})}$$ ‖ w ‖ ∞ / ‖ w ‖ 2 = N - γ ( w ) . Our results remain valid throughout the optimal regime $$\sqrt{\log N} \ll d \leqslant O(\log N)$$ log N ≪ d ⩽ O ( log N ) . |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
| Nepřihlášeným uživatelům se plný text nezobrazuje | K zobrazení výsledku je třeba se přihlásit. |