Conformal Covariance of Connection Probabilities and Fields in 2D Critical Percolation

TL;DR

This paper proves that connection probabilities in 2D critical percolation have conformally invariant scaling limits, behaving like correlation functions of primary operators, and introduces a related spin model with similar properties.

Contribution

It establishes the conformal covariance of connection probabilities and correlation functions in 2D critical percolation, linking percolation to conformal field theory.

Findings

Connection probabilities have well-defined conformally invariant scaling limits.

Two-point and three-point functions follow specific power-law behaviors.

The associated magnetization field has a well-defined scaling limit with conformal covariance.

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 $n \geq 2$. Moreover, the limiting functions $P_n(x_1,\ldots,x_n)$ transform covariantly under M\"obius 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 M\"obius 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.