Jacob J. H. Simmons

Anchored Critical Percolation Clusters and 2D Electrostatics

We consider the densities of clusters, at the percolation point of a two-dimensional system, which are anchored in various ways to an edge. These quantities are calculated by use of conformal field theory and computer simulations. We find that they are given by simple functions of the potentials of 2D electrostatic dipoles and that a kind of superposition cum factorization applies. Our results broaden this connection, already known from previous studies, and we present evidence that it is more generally valid. An exact result similar to the Kirkwood superposition approximation emerges.

Twist operator correlation functions in O(n) loop models

Using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the O(n) model at the critical point. We focus on correlation functions containing twist operators, combining these with anchored loops, boundaries with SLE processes and with double SLE processes. We focus further upon n = 0, representing self-avoiding loops, which corresponds to a logarithmic conformal field theory (LCFT) with c = 0. In this limit the twist operator plays the role of a 0-weight indicator operator, which we verify by comparison with known examples. Using the additional conditions imposed by the twist operator null states, we derive a new explicit result for the probabilities that an SLE8/3 winds in various ways about two points in the upper half-plane, e.g. that the SLE passes to the left of both points. The collection of c = 0 logarithmic CFT operators that we use deriving the winding probabilities is novel, highlighting a potential incompatibility caused by the presence of two distinct logarithmic partners to the stress tensor within the theory. We argue that both partners do appear in the theory, one in the bulk and one on the boundary and that the incompatibility is resolved by restrictive bulk–boundary fusion rules.

Exact factorization of correlation functions in two-dimensional critical percolation.

By use of conformal field theory, we discover several exact factorizations of higher-order density correlation functions in critical two-dimensional percolation. Our formulas are valid in the upper half-plane, or any conformally equivalent region. We find excellent agreement of our results with high-precision computer simulations. There are indications that our formulas hold more generally.

Factorization of correlations in two-dimensional percolation on the plane and torus

Recently, Delfino and Viti have examined the factorization of the threepoint density correlation function P3 at the percolation point in terms of the two-point density correlation functions P2. According to conformal invariance, this factorization is exact on the infinite plane, such that the ratio R(z1 ,z 2 ,z 3) = P3(z1 ,z 2 ,z 3)/[P2(z1 ,z 2)P2(z1 ,z 3)P2(z2 ,z 3)] 1/2 is not only universal but also a constant, independent of zi and in fact an operator product expansion coefficient. Delfino and Viti analytically calculated its value (1.022013 ... ) for percolation, in agreement with the numerical value 1.022 found previously in a study of R on the conformally equivalent cylinder. In this paper we confirm the factorization on the plane numerically using periodic lattices (tori) of very large size, which locally approximate a plane. We also investigate the general behavior of R on the torus, and find a minimum value of R ≈ 1.0132 when the three points are maximally separated. In addition, we present a simplified expression for R on the plane as a function of the SLE parameter κ.

Logarithmic operator intervals in the boundary theory of critical percolation

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 ?1, 2 ? ?, identified with the growing hull of the SLE6 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 ?; the stress-tensor and its logarithmic partner; ?? and its logarithmic partner; and the pre-logarithmic operator ?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 CG, 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.

The density of critical percolation clusters touching the boundaries of strips and squares

We consider the density of two-dimensional critical percolation clusters, constrained to touch one or both boundaries, in infinite strips, half-infinite strips and squares, as well as several related quantities for the infinite strip. Our theoretical results follow from conformal field theory and are compared with high-precision numerical simulation. For example, we show that the density of clusters touching both boundaries of an infinite strip of unit width (i.e. crossing clusters) is proportional to (sinπy)−5/48{[cos(πy/2)]1/3+[sin(πy/2)]1/3−1}. We also determine numerically contours for the density of clusters crossing squares and long rectangles with open boundaries on the sides and compare with theory for the density along an edge.

Percolation crossing probabilities in hexagons: a numerical study

In a recent article, one of the authors used

A formula for crossing probabilities of critical systems inside polygons

c=0

General solution of an exact correlation function factorization in conformal field theory

logarithmic conformal field theory to predict crossing-probability formulas for percolation clusters inside a hexagon with free boundary conditions. In this article, we verify these predictions with high-precision computer simulations. Our simulations generate percolation-cluster perimeters with hull walks on a triangular lattice inside a hexagon. Each sample comprises two hull walks, and the order in which these walks strike the bottom and upper left/right sides of the hexagon determines the crossing configuration of the percolation sample. We compare our numerical results with the predicted crossing probabilities, finding excellent agreement.

Factorization of percolation density correlation functions for clusters touching the sides of a rectangle

In this article, we generalize known formulas for crossing probabilities. Prior crossing results date back to J. Cardys prediction of a formula for the probability that a percolation cluster in two dimensions connects the left and right sides of a rectangle at the percolation critical point in the continuum limit. Here, we predict a new formula for crossing probabilities of a continuum limit loop-gas model on a planar lattice inside a