A spectral condition for spectral gap: fast mixing in high-temperature Ising models
Abstract We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincaré inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than 1. The inequality...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2022
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| Online Access: | https://hdl.handle.net/1721.1/141917 |
| _version_ | 1826207132908781568 |
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| author | Eldan, Ronen Koehler, Frederic Zeitouni, Ofer |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Eldan, Ronen Koehler, Frederic Zeitouni, Ofer |
| author_sort | Eldan, Ronen |
| collection | MIT |
| description | Abstract
We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincaré inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than 1. The inequality implies a control on the mixing time of the Glauber dynamics. Our techniques rely on a localization procedure which establishes a structural result, stating that Ising measures may be decomposed into a mixture of measures with quadratic potentials of rank one, and provides a framework for proving concentration bounds for high temperature Ising models. |
| first_indexed | 2024-09-23T13:44:56Z |
| format | Article |
| id | mit-1721.1/141917 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:44:56Z |
| publishDate | 2022 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1419172023-02-03T20:40:39Z A spectral condition for spectral gap: fast mixing in high-temperature Ising models Eldan, Ronen Koehler, Frederic Zeitouni, Ofer Massachusetts Institute of Technology. Department of Mathematics Abstract We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincaré inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than 1. The inequality implies a control on the mixing time of the Glauber dynamics. Our techniques rely on a localization procedure which establishes a structural result, stating that Ising measures may be decomposed into a mixture of measures with quadratic potentials of rank one, and provides a framework for proving concentration bounds for high temperature Ising models. 2022-04-19T12:51:07Z 2022-04-19T12:51:07Z 2021-08-20 2022-04-17T03:37:24Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/141917 Eldan, Ronen, Koehler, Frederic and Zeitouni, Ofer. 2021. "A spectral condition for spectral gap: fast mixing in high-temperature Ising models." en https://doi.org/10.1007/s00440-021-01085-x Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Eldan, Ronen Koehler, Frederic Zeitouni, Ofer A spectral condition for spectral gap: fast mixing in high-temperature Ising models |
| title | A spectral condition for spectral gap: fast mixing in high-temperature Ising models |
| title_full | A spectral condition for spectral gap: fast mixing in high-temperature Ising models |
| title_fullStr | A spectral condition for spectral gap: fast mixing in high-temperature Ising models |
| title_full_unstemmed | A spectral condition for spectral gap: fast mixing in high-temperature Ising models |
| title_short | A spectral condition for spectral gap: fast mixing in high-temperature Ising models |
| title_sort | spectral condition for spectral gap fast mixing in high temperature ising models |
| url | https://hdl.handle.net/1721.1/141917 |
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