Decay of correlations and zeros for the hard-core model
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914406810517504 |
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| author | Peters, Han Reppekus, Josias Regts, Guus |
| author_facet | Peters, Han Reppekus, Josias Regts, Guus |
| contents | In a recent paper the last author proved that absence of complex zeros of the partition function of the hard-core model near a parameter $λ>0$ implies a form of correlation decay called strong spacial mixing. In this paper we investigate the reverse implication. We introduce a strengthening of strong spatial mixing that we call very strong spatial mixing (VSSM).
Our main result is that if VSSM holds at a parameter $λ>0$ for a family of graphs, this implies that the partition function has no zeros near that parameter for each graph in the family. We also demonstrate that a closely related variant of very strong spatial mixing does not imply zero-freeness. As a consequence of our main result, we moreover obtain that VSSM implies spectral independence. Our proof relies on transforming the problem to the analysis of an induced non-autonomous dynamical system given by Möbius transformations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17858 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Decay of correlations and zeros for the hard-core model Peters, Han Reppekus, Josias Regts, Guus Probability Mathematical Physics Combinatorics Complex Variables In a recent paper the last author proved that absence of complex zeros of the partition function of the hard-core model near a parameter $λ>0$ implies a form of correlation decay called strong spacial mixing. In this paper we investigate the reverse implication. We introduce a strengthening of strong spatial mixing that we call very strong spatial mixing (VSSM). Our main result is that if VSSM holds at a parameter $λ>0$ for a family of graphs, this implies that the partition function has no zeros near that parameter for each graph in the family. We also demonstrate that a closely related variant of very strong spatial mixing does not imply zero-freeness. As a consequence of our main result, we moreover obtain that VSSM implies spectral independence. Our proof relies on transforming the problem to the analysis of an induced non-autonomous dynamical system given by Möbius transformations. |
| title | Decay of correlations and zeros for the hard-core model |
| topic | Probability Mathematical Physics Combinatorics Complex Variables |
| url | https://arxiv.org/abs/2603.17858 |