论文标题
连接概率和田野的共形协方差
Conformal Covariance of Connection Probabilities and Fields in 2D Critical Percolation
论文作者
论文摘要
将渗透拟合到共形场理论框架中需要表明连接概率具有不变的缩放限制。对于三角形晶格上的关键站点渗透，我们证明$ n $ Vertices属于同一开放群集的可能性具有每$ N \ geq 2 $的定义明确的缩放限制。此外，限制函数$ p_n（x_1，\ ldots，x_n）$在平面的莫比乌斯变换下以及在局部保形图下的ibiiant transformiant，即它们的行为就像同轴野外理论中主操作员的相关功能。特别是，它们在翻译，旋转和反转下是不变的，以及$ p_n（sx_1，\ ldots，sx_n）= s^{ - 5n/48} p_n（x_1，x_1，\ ldots，x_n）$ for Any $ s> 0 $。这意味着$ p_ {2}（x_1，x_2）= c_2 \ vert x_1-x_2 \ vert^{ - 5/24} $和$ p_3（x_1，x_2，x_2，x_3）= c_3 \ c_3 \ vert x_1-x_1-x_1-x_2 x_2 _2 \ vert^{ - 5/5/5/48} x- \ vert^{ - 5/48} \ vert x_2-x_3 \ vert^{ - 5/48} $，对于某些常数$ C_2 $和$ C_3 $。 我们还定义了一个站点稀释的自旋模型，其$ n $ - 点相关功能$ \ mathrm {c} _ {n} $可以用渗透连接概率来表示，因此，具有与函数$ p_n $相同的属性，具有明确定义的缩放极限。特别是，$ \ mathrm {c} _ {2}（x_1，x_2）= p_ {2}（x_1，x_2）$。我们证明，与此自旋模型相关的磁化场在适当的分布空间中具有明确定义的缩放限制。限制场在Möbius变换下与指数（缩放维度）$ 5/48 $转换。对磁化场四点函数的启发式分析表明，存在缩放尺寸$ 5/4 $的附加保形场，这计算了渗透四臂事件的数量，并且可以用所谓的“四级运算符”的共同场理论来识别。
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 $n \geq 2$. Moreover, the limiting functions $P_n(x_1,\ldots,x_n)$ transform covariantly under Möbius transformations of the plane as well as under local conformal maps, i.e., they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and $P_n(sx_1,\ldots,sx_n)=s^{-5n/48}P_n(x_1,\ldots,x_n)$ for any $s>0$. This implies that $P_{2}(x_1,x_2)=C_2 \Vert x_1-x_2 \Vert^{-5/24}$ and $P_3(x_1,x_2,x_3) = C_3 \Vert x_1-x_2 \Vert^{-5/48} \Vert x_1-x_3 \Vert^{-5/48} \Vert x_2-x_3 \Vert^{-5/48}$, for some constants $C_2$ and $C_3$. We also define a site-diluted spin model whose $n$-point correlation functions $\mathrm{C}_{n}$ can be expressed in terms of percolation connection probabilities and, as a consequence, have a well-defined scaling limit with the same properties as the functions $P_n$. In particular, $\mathrm{C}_{2}(x_1,x_2)=P_{2}(x_1,x_2)$. We prove that the magnetization field associated with this spin model has a well-defined scaling limit in an appropriate space of distributions. The limiting field transforms covariantly under Möbius transformations with exponent (scaling dimension) $5/48$. A heuristic analysis of the four-point function of the magnetization field suggests the presence of an additional conformal field of scaling dimension $5/4$, which counts the number of percolation four-arm events and can be identified with the so-called "four-leg operator'' of conformal field theory.
