Logarithmic operator intervals in the boundary theory of critical percolation

Abstract

We consider the sub-sector of the $c=0$ logarithmic conformal field theory (LCFT) generated by the boundary condition changing (bcc) operator in two dimensional critical percolation. This operator is the zero weight Kac operator $\phi_{1,2}$, identified with the growing hull of the SLE$_6$ process. We identify percolation configurations with the significant operators in the theory. We consider operators from the first four bcc operator fusions: the identity and bcc operator; the stress tensor and its logarithmic partner; the derivative of the bcc operator and its logarithmic partner; and the pre-logarithmic operator $\phi_{1,3}$. We construct several intervals in the percolation model, each associated to one of the LCFT operators we consider, allowing us to calculate crossing probabilities and expectation values of crossing cluster numbers. We review the Coulomb gas, which we use as a method of calculating these quantities when the number of bcc operator makes a direct solution to the system of differential equations intractable. Finally we discuss the case of the six-point correlation function, which applies to crossing probabilities between the sides of a conformal hexagon. Specifically we introduce an integral result that allows one to identify the probability that a single percolation cluster touches three alternating sides a hexagon with free boundaries. We give results of the numerical integration for the case of a regular hexagon.