Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well-defined scaling limit for every $n\ge 2$. Moreover, the limiting functions $P_n(x_1,\text{…},x_n)$ transform covariantly under Möbius transformations of the plane as well as under local conformal maps, that is, they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and $P_n(s{x}_1,\text{…},s{x}_n)=s^{-5n/48}{P}_n(x_1,\text{…},x_n)$ for any $s>0$. This implies that $P_2(x_1,x_2)=C_2{\parallel x_1-x_2\parallel }^{-5/24}$ and $P_3(x_1,x_2,x_3)=C_3\parallel x_1-x_2{\parallel }^{-5/48}\parallel x_1-x_3{\parallel }^{-5/48}{\parallel x_2-x_3\parallel }^{-5/48}$, for some constants C₂ and C₃.

中文翻译：

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二维临界渗流中连接概率和场的共形协方差

将渗透拟合到共形场论框架中需要表明连接概率具有共形不变的缩放限制。对于三角晶格上的临界点渗流，我们证明了n 个顶点属于同一疏散簇的概率对于每个点都有明确定义的缩放限制$n\ge 2$。此外，限制函数${磷}_n（X_1,\text{……},X_n）$在平面的莫比乌斯变换以及局部共形映射下进行协变变换，也就是说，它们的行为类似于共形场论中主算子的相关函数。特别是，它们在平移、旋转和反转下是不变的，并且${磷}_n（s{X}_1,\text{……},s{X}_n）=s^{-5n/48}{磷}_n（X_1,\text{……},X_n）$对于任何$s>0$。这意味着${磷}_2（X_1,X_2）=C_2{\parallel X_1-X_2\parallel }^{-5/24}$和${磷}_3（X_1,X_2,X_3）=C_3\parallel X_1-X_2{\parallel }^{-5/48}\parallel X_1-X_3{\parallel }^{-5/48}{\parallel X_2-X_3\parallel }^{-5/48}$，对于一些常数C ₂和C ₃。