Anchored random clusters and SLE excursions

TL;DR

This paper reviews CFT techniques to compute SLE observables, including known and new formulas for cluster densities, using correlation functions involving degenerate boundary operators.

Contribution

It introduces a CFT-based method to derive both existing and novel SLE observables related to critical percolation and FK clusters.

Findings

Recovered Schramm's left-passage probability.

Derived SLE Green's functions.

Obtained new formulas for pivotal point densities.

Abstract

We provide a pedagogical review of CFT techniques to compute certain Schramm-Loewner Evolution (SLE) observables in the upper half-plane. The approach relies on the ability to express the observables as bulk-boundary correlation functions that involve degenerate boundary operators and, therefore, obey certain differential equations. In particular, we recover Schramm's left-passage probability for SLE, the SLE Green's functions, and the generalized densities of ``anchored'' critical percolation clusters first obtained by Kleban, Simmons, and Ziff. We also obtain new formulas corresponding to the densities of pivotal points between critical Fortuin-Kasteleyn (FK) clusters.