The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph
Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-04-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.316/ |
| _version_ | 1797651472849043456 |
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| author | McKenzie, Theo |
| author_facet | McKenzie, Theo |
| author_sort | McKenzie, Theo |
| collection | DOAJ |
| description | Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite. |
| first_indexed | 2024-03-11T16:16:18Z |
| format | Article |
| id | doaj.art-6a92276ae8e4446ca32109989b31b678 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:18Z |
| publishDate | 2022-04-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-6a92276ae8e4446ca32109989b31b6782023-10-24T14:19:45ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-04-01360G439940810.5802/crmath.31610.5802/crmath.316The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graphMcKenzie, Theo0Evans Hall, University of California, Berkeley, CA, USAAnantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.316/ |
| spellingShingle | McKenzie, Theo The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph Comptes Rendus. Mathématique |
| title | The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph |
| title_full | The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph |
| title_fullStr | The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph |
| title_full_unstemmed | The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph |
| title_short | The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph |
| title_sort | necessity of conditions for graph quantum ergodicity and cartesian products with an infinite graph |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.316/ |
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