Uniqueness of Gibbs measures for continuous hardcore models
© 2019, Institute of Mathematical Statistics. We formulate a continuous version of the well-known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter λ > 0. In this version the state or "spin value" xu of any node u o...
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| Format: | Article |
| Language: | English |
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Institute of Mathematical Statistics
2021
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| Online Access: | https://hdl.handle.net/1721.1/134006 |
| _version_ | 1826211678132371456 |
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| author | Gamarnik, David Ramanan, Kavita |
| author2 | Sloan School of Management |
| author_facet | Sloan School of Management Gamarnik, David Ramanan, Kavita |
| author_sort | Gamarnik, David |
| collection | MIT |
| description | © 2019, Institute of Mathematical Statistics. We formulate a continuous version of the well-known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter λ > 0. In this version the state or "spin value" xu of any node u of the graph lies in the interval [0, 1], the hardcore constraint xu + xv ≤ 1 is satisfied for every edge (u, v) of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter λ > 0. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when λ = 1 and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case λ = 1. |
| first_indexed | 2024-09-23T15:09:47Z |
| format | Article |
| id | mit-1721.1/134006 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:09:47Z |
| publishDate | 2021 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/1340062023-10-05T19:53:43Z Uniqueness of Gibbs measures for continuous hardcore models Gamarnik, David Ramanan, Kavita Sloan School of Management © 2019, Institute of Mathematical Statistics. We formulate a continuous version of the well-known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter λ > 0. In this version the state or "spin value" xu of any node u of the graph lies in the interval [0, 1], the hardcore constraint xu + xv ≤ 1 is satisfied for every edge (u, v) of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter λ > 0. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when λ = 1 and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case λ = 1. 2021-10-27T19:57:36Z 2021-10-27T19:57:36Z 2019 2021-04-14T15:03:44Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/134006 en 10.1214/18-AOP1298 The Annals of Probability Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Gamarnik, David Ramanan, Kavita Uniqueness of Gibbs measures for continuous hardcore models |
| title | Uniqueness of Gibbs measures for continuous hardcore models |
| title_full | Uniqueness of Gibbs measures for continuous hardcore models |
| title_fullStr | Uniqueness of Gibbs measures for continuous hardcore models |
| title_full_unstemmed | Uniqueness of Gibbs measures for continuous hardcore models |
| title_short | Uniqueness of Gibbs measures for continuous hardcore models |
| title_sort | uniqueness of gibbs measures for continuous hardcore models |
| url | https://hdl.handle.net/1721.1/134006 |
| work_keys_str_mv | AT gamarnikdavid uniquenessofgibbsmeasuresforcontinuoushardcoremodels AT ramanankavita uniquenessofgibbsmeasuresforcontinuoushardcoremodels |