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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,...

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Autores principales: Alt, Johannes, Ducatez, Raphael, Knowles, Antti
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer Berlin Heidelberg 2021
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] .
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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
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