Conformal covariance of connection probabilities and fields in 2D critical percolation

Abstract

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.

We also define a site-diluted spin model whose n-point correlation functions 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, $C_2(x_1,x_2)=P_2(x_1,x_2)$. 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.