Trickle-Down in Localization Schemes and Applications
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914882469756928 |
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| author | Anari, Nima Koehler, Frederic Vuong, Thuy-Duong |
| author_facet | Anari, Nima Koehler, Frederic Vuong, Thuy-Duong |
| contents | Trickle-down is a phenomenon in high-dimensional expanders with many important applications -- for example, it is a key ingredient in various constructions of high-dimensional expanders or the proof of rapid mixing for the basis exchange walk on matroids and in the analysis of log-concave polynomials. We formulate a generalized trickle-down equation in the abstract context of linear-tilt localization schemes. Building on this generalization, we improve the best-known results for several Markov chain mixing or sampling problems -- for example, we improve the threshold up to which Glauber dynamics is known to mix rapidly in the Sherrington-Kirkpatrick spin glass model. Other applications of our framework include improved mixing results for the Langevin dynamics in the $O(N)$ model, and near-linear time sampling algorithms for the antiferromagnetic and fixed-magnetization Ising models on expanders. For the latter application, we use a new dynamics inspired by polarization, a technique from the theory of stable polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16104 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trickle-Down in Localization Schemes and Applications Anari, Nima Koehler, Frederic Vuong, Thuy-Duong Probability Data Structures and Algorithms Mathematical Physics Trickle-down is a phenomenon in high-dimensional expanders with many important applications -- for example, it is a key ingredient in various constructions of high-dimensional expanders or the proof of rapid mixing for the basis exchange walk on matroids and in the analysis of log-concave polynomials. We formulate a generalized trickle-down equation in the abstract context of linear-tilt localization schemes. Building on this generalization, we improve the best-known results for several Markov chain mixing or sampling problems -- for example, we improve the threshold up to which Glauber dynamics is known to mix rapidly in the Sherrington-Kirkpatrick spin glass model. Other applications of our framework include improved mixing results for the Langevin dynamics in the $O(N)$ model, and near-linear time sampling algorithms for the antiferromagnetic and fixed-magnetization Ising models on expanders. For the latter application, we use a new dynamics inspired by polarization, a technique from the theory of stable polynomials. |
| title | Trickle-Down in Localization Schemes and Applications |
| topic | Probability Data Structures and Algorithms Mathematical Physics |
| url | https://arxiv.org/abs/2407.16104 |