Cluster pinch-point densities in polygons

Abstract

In a statistical cluster or loop model such as percolation, or more generally the Potts models or O( n ) models, a pinch point is a single bulk point where several distinct clusters or loops touch. In a polygon ##IMG## [http://ej.iop.org/images/1751-8121/45/50/505002/jpa439370ieqn1.gif] {$\mathcal {P$ harboring such a model in its interior and with 2 N sides exhibiting free/fixed side-alternating boundary conditions, boundary clusters anchor to the fixed sides of ##IMG## [http://ej.iop.org/images/1751-8121/45/50/505002/jpa439370ieqn2.gif] {$\mathcal {P$ . At the critical point and in the continuum limit, the density (i.e. frequency of occurrence) of pinch-points between s distinct boundary clusters at a bulk point ##IMG## [http://ej.iop.org/images/1751-8121/45/50/505002/jpa439370ieqn3.gif] {$w\in \mathcal {P$ is proportional to ##IMG## [http://ej.iop.org/images/1751-8121/45/50/505002/jpa439370ueq01.gif] {\begin{eqnarray* \big\langle \psi _1^c(w_1)\psi _1^c(w_2)\ldots \psi _1^c(w_{2N-1)\psi _1^c(w_{2N)\Psi _s(w,\bar{w)\big\rangle _\mathcal {P. \end{eqnarray* The w i are the vertices of ##IMG## [http://ej.iop.org/images/1751-8121/45/50/505002/jpa439370ieqn4.gif] {$\mathcal {P$ , ψ c 1 is a conformal field theory (CFT) corner one-leg operator, and Ψ s is a CFT bulk 2 s -leg operator. In this paper, we use the Coulomb gas formalism to construct explicit contour integral formulas for these correlation functions and thereby calculate the density of various pinch-point configurations at arbitrary points in the rectangle, in the hexagon, and for the case s = N , in the 2 N -sided polygon at the system’s critical point. Explicit formulas for these results are given in terms of algebraic functions or integrals of algebraic functions, particularly Lauricella functions. In critical percolation, the result for s = N = 2 gives the density of red bonds between boundary clusters (in the continuum limit) inside a rectangle. We compare our results with high-precision simulations of critical percolation and Ising FK clusters in a rectangle of aspect ratio two and in a regular hexagon, and we find very good agreement.