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Delocalization Transition for Critical Erdős–Rényi Graphs
We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph [Formula: see text] , where d is of order [Formula: see text] . We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized,...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer Berlin Heidelberg
2021
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550299/ https://www.ncbi.nlm.nih.gov/pubmed/34720130 http://dx.doi.org/10.1007/s00220-021-04167-y |
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author | Alt, Johannes Ducatez, Raphael Knowles, Antti |
author_facet | Alt, Johannes Ducatez, Raphael Knowles, Antti |
author_sort | Alt, Johannes |
collection | PubMed |
description | We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph [Formula: see text] , where d is of order [Formula: see text] . 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 [Formula: see text] of an eigenvector [Formula: see text] , defined through [Formula: see text] . Our results remain valid throughout the optimal regime [Formula: see text] . |
format | Online Article Text |
id | pubmed-8550299 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Springer Berlin Heidelberg |
record_format | MEDLINE/PubMed |
spelling | pubmed-85502992021-10-29 Delocalization Transition for Critical Erdős–Rényi Graphs Alt, Johannes Ducatez, Raphael Knowles, Antti Commun Math Phys Article We analyse the eigenvectors of the adjacency matrix of a critical Erdős–Rényi graph [Formula: see text] , where d is of order [Formula: see text] . 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 [Formula: see text] of an eigenvector [Formula: see text] , defined through [Formula: see text] . Our results remain valid throughout the optimal regime [Formula: see text] . Springer Berlin Heidelberg 2021-08-14 2021 /pmc/articles/PMC8550299/ /pubmed/34720130 http://dx.doi.org/10.1007/s00220-021-04167-y Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Alt, Johannes Ducatez, Raphael Knowles, Antti Delocalization Transition for Critical Erdős–Rényi Graphs |
title | Delocalization Transition for Critical Erdős–Rényi Graphs |
title_full | Delocalization Transition for Critical Erdős–Rényi Graphs |
title_fullStr | Delocalization Transition for Critical Erdős–Rényi Graphs |
title_full_unstemmed | Delocalization Transition for Critical Erdős–Rényi Graphs |
title_short | Delocalization Transition for Critical Erdős–Rényi Graphs |
title_sort | delocalization transition for critical erdős–rényi graphs |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550299/ https://www.ncbi.nlm.nih.gov/pubmed/34720130 http://dx.doi.org/10.1007/s00220-021-04167-y |
work_keys_str_mv | AT altjohannes delocalizationtransitionforcriticalerdosrenyigraphs AT ducatezraphael delocalizationtransitionforcriticalerdosrenyigraphs AT knowlesantti delocalizationtransitionforcriticalerdosrenyigraphs |