Correlations in random cluster model at $q=1$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915386272776192 |
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| author | Nguyen, Son Pylyavskyy, Pavlo |
| author_facet | Nguyen, Son Pylyavskyy, Pavlo |
| contents | Let $μ$ be a measure that samples a subset of a finite ground set, and let $\mathcal{A}_e$ be the event that element $e$ is sampled. The measure $μ$ is negatively correlated if for any pair of elements $e, f$ one has $μ(\mathcal{A}_e \cap \mathcal{A}_f) - μ(\mathcal{A}_e) μ(\mathcal{A}_f) \leq 0$. A measure is positively correlated if the direction of the inequality is reversed. For the random cluster model on graphs positive correlation between edges is known for $q \geq 1$ due to the FKG inequality, while the negative correlation is only conjectured for $0 \leq q \leq 1$. The main result of this paper is to give a combinatorial formula for the difference in question at $q=1$. Previously, such a formula was known in the uniform spanning tree case, which is a limit of the random cluster model at $q=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Correlations in random cluster model at $q=1$ Nguyen, Son Pylyavskyy, Pavlo Combinatorics Mathematical Physics Probability Let $μ$ be a measure that samples a subset of a finite ground set, and let $\mathcal{A}_e$ be the event that element $e$ is sampled. The measure $μ$ is negatively correlated if for any pair of elements $e, f$ one has $μ(\mathcal{A}_e \cap \mathcal{A}_f) - μ(\mathcal{A}_e) μ(\mathcal{A}_f) \leq 0$. A measure is positively correlated if the direction of the inequality is reversed. For the random cluster model on graphs positive correlation between edges is known for $q \geq 1$ due to the FKG inequality, while the negative correlation is only conjectured for $0 \leq q \leq 1$. The main result of this paper is to give a combinatorial formula for the difference in question at $q=1$. Previously, such a formula was known in the uniform spanning tree case, which is a limit of the random cluster model at $q=0$. |
| title | Correlations in random cluster model at $q=1$ |
| topic | Combinatorics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2507.09520 |