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.

In this paper we consider the density, at a point z = x + iy, of critical percolation clusters that touch the left (PL(z)), right (PR(z)), or both (PLR(z)) sides of a rectangular system, with open boundary conditions on the top and bottom sides. While each of these quantities is nonuniversal and indeed vanishes in the continuum limit, the ratio C(z) = PLR(z)/ p PL(z)PR(z)Π h , where Π h is the probability of left-right crossing given by Cardy, is a universal function of z. With wired (fixed) boundary conditions on the left-and right-hand sides, high-precision numerical simulations and theoretical arguments show that C(z) goes to a constant C0 = 2 7/2 3 −3/4 π 5/2 Γ(1/3) −9/2 = 1.0299268 . . . for points far from the ends, and varies by no more than a few percent for all z values. Thus PLR(z) factorizes over the entire rectangle to very good approximation. In addition, the numerical observation that C(z) depends upon x but not upon y leads to an explicit expression for C(z) via conformal field theory for a long rectangle (semi-infinite strip). We also derive explict expressions for PL(z), PR(z), and PLR(z) in this geometry, first by assuming y-independence and then by a full analysis that obtains these quantities exactly with no assumption on the y behavior. In this geometry we obtain, in addition, the corresponding quantities in the case of open boundary conditions, which allows us to calculate C(z) in the open system. We give some theoretical results for an arbitrary rectangle as well. Our results also enable calculation of the finite-size corrections to the factorization near an isolated anchor point, for the case of clusters anchored at points. Finally, we present numerical results for a rectangle with periodic b.c. in the horizontal direction, and find C(z) approaches a constant value C1 ≈ 1.022.

Using conformal field theory, we derive several new crossing formulae at the two-dimensional percolation point. High-precision simulation confirms these results. Integrating them gives a unified derivation of Cardy's formula for the horizontal crossing probability h (r), Watts' formula for the horizontalvertical crossing probability hv (r) and Cardy's formula for the expected number of clusters crossing horizontally N h (r). The main step in our approach implies the identification of the derivative of one primary operator with another. We present operator identities that support this idea and suggest the presence of additional symmetry in c = 0 conformal field theories.

In this article, we use our results from [36][37][38][39] to generalize known formulas for crossing probabilities. Prior crossing results date back to J. Cardy's 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 [22]. Here, we predict a new formula for crossing probabilities of a continuum limit loop-gas model on a planar lattice inside a 2N -sided polygon. In this model, boundary loops exit and then re-enter the polygon through its vertices, with exactly one loop passing once through each vertex, and these loops join the vertices pairwise in some specified connectivity through the polygon's exterior. The boundary loops also connect the vertices through the interior, which we regard as a crossing event. For particular values of the loop fugacity, this formula specializes to FK cluster (resp. spin cluster) crossing probabilities of a critical Q-state random cluster (resp. Potts) model on a lattice inside the polygon in the continuum limit. This includes critical percolation as the Q = 1 random cluster model. These latter crossing probabilities are conditioned on a particular side-alternating free/fixed (resp. fluctuating/fixed) boundary condition on the polygon's perimeter, related to how the boundary loops join the polygon's vertices pairwise through the polygon's exterior in the associated loop-gas model. For Q ∈ {2, 3, 4}, we compare our predictions of these random cluster (resp. Potts) model crossing probabilities in a rectangle (N = 2) and in a hexagon (N = 3) with high-precision computer simulation measurements. We find that the measurements agree with our predictions very well for Q ∈ {2, 3} and reasonably well if Q = 4. arXiv:1608.00170v1 [cond-mat.stat-mech] 30 Jul 2016 2 C. Random cluster model crossing events with free/fixed boundary conditions 16 D. Potts model crossing events with fluctuating/fixed boundary conditions 20 IV. Convergence to loop-gas models 21 A. Definition of loop-gas models 21 B. Critical random cluster models in relation to loop-gas models 24 C. Critical Potts model in relation to loop-gas models 25 D. Crossing probability for loop-gas models 28 V. The continuum limit and conformal field theory 29 A. Smeared partition functions and conformal field theory 29 B. Crossing weights and crossing probability 30 VI. Formula for rectangle crossing probability 35 VII. Simulation results for rectangle and hexagon crossing probability 39 A. Simulation results for rectangle crossing probability 39 B. Simulation results for hexagon crossing probability 41 VIII. Summary 53 IX. Acknowledgements 54 A. Transformation to the polygon via CFT corner operators 54 B. Color scheme partition functions for the random cluster model 56

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.