论文标题
多个FK-ing界面的连接概率
Connection Probabilities of Multiple FK-Ising Interfaces
论文作者
论文摘要
我们发现一般的边界与边界连接概率和临界平面FK-sister模型中多个接口的缩放限制，从而验证了物理文献的预测。我们还使用库仑气体积分来讨论猜想的公式，以在[1,4）$中使用cluster-weight $ q \ in Cluster Planar Random-Cluster模型中的相应数量。到目前为止，包括我们的收敛性证明依赖于离散的复杂分析技术，并且超出了$ q $的其他值（$ q = 2 $）。鉴于接口的融合，可以通过相对较小的技术工作类似地验证其他值的猜测公式。极限接口是$ \ mathrm {sle}_κ$ curves的变体（$κ= 16/3 $ for $ q = 2 $）。他们的分区功能（赋予连接概率）也满足了在共同场理论（CFT）中预测的相关函数的属性，这些属性预计将描述关键随机群集模型的缩放限制。我们在[1,4）$中验证了所有$ q \的这些属性，从而提供了进一步的证据，证明了这些模型的预期CFT描述。
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.
