Cluster densities at 2-D critical points in rectangular geometries

TL;DR

This paper calculates and verifies cluster densities at 2-D critical points in rectangles, using conformal field theory and simulations, providing explicit formulas and analyzing boundary effects for various models.

Contribution

It introduces explicit formulas for cluster densities conditioned on boundary contact in 2-D critical systems, extending previous crossing probability results and verifying them with simulations.

Findings

Derived explicit formulas for cluster densities in critical models.

Verified theoretical predictions with high-precision simulations.

Generalized crossing probability formulas to multiple models.

Abstract

Making use of a recent complete calculation of a chiral six-point correlation function C(z) in a rectangle we calculate various quantities of interest for percolation (SLE parameter \kappa = 6) and many other two-dimensional critical points. In particular, we specify the density at z of critical clusters conditioned to touch either or both vertical sides of the rectangle, with these sides 'wired,' i.e. constrained to be in a single cluster, and the horizontal sides free. These quantities probe the structure of various cluster configurations, including those that contribute to the crossing probability. We first examine the effects of boundary conditions on C for the critical O(n) loop models in both high and low density phases and for both Fortuin-Kasteleyn (FK) and spin clusters in the critical Q-state Potts models. A Coulomb gas analysis then allows us to calculate the cluster densities with various conditionings in terms of the known conformal blocks. Explicit formulas generalizing Cardy's horizontal crossing probability to these models (using previously known results) are also presented. These solutions are employed to generalize previous results demonstrating factorization of higher-order correlation functions to the critical systems mentioned. An explicit formula for the density of critical percolation clusters that cross a rectangle horizontally with free boundary conditions is also given. Simplifications of the hypergeometric functions in our solutions for various models are presented. High precision simulations verify these predictions for percolation and for the Q=2 and 3-state Potts models, including both FK and spin clusters. Our formula for the density of crossing clusters in percolation in open systems is also verified.