Connection Probabilities of Multiple FK-Ising Interfaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916350627151872 |
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| author | Feng, Yu Peltola, Eveliina Wu, Hao |
| author_facet | Feng, Yu Peltola, Eveliina Wu, Hao |
| contents | We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight $q \in [1,4)$. Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of $q$ than the FK-Ising model ($q=2$). Given the convergence of interfaces, the conjectural formulas for other values of $q$ could be verified similarly with relatively minor technical work. The limit interfaces are variants of $\mathrm{SLE}_κ$ curves (with $κ= 16/3$ for $q=2$). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all $q \in [1,4)$, thus providing further evidence of the expected CFT description of these models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_08800 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Connection Probabilities of Multiple FK-Ising Interfaces Feng, Yu Peltola, Eveliina Wu, Hao Probability Mathematical Physics 82B20, 60J67, 60K35 We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight $q \in [1,4)$. Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of $q$ than the FK-Ising model ($q=2$). Given the convergence of interfaces, the conjectural formulas for other values of $q$ could be verified similarly with relatively minor technical work. The limit interfaces are variants of $\mathrm{SLE}_κ$ curves (with $κ= 16/3$ for $q=2$). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all $q \in [1,4)$, thus providing further evidence of the expected CFT description of these models. |
| title | Connection Probabilities of Multiple FK-Ising Interfaces |
| topic | Probability Mathematical Physics 82B20, 60J67, 60K35 |
| url | https://arxiv.org/abs/2205.08800 |