1201.6405 (Steven M. Flores et al.)

Cluster pinch-point densities in polygons    [PDF]

Steven M. Flores, Peter Kleban, Robert M. Ziff

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 P harboring such a model
in its interior and with 2N sides exhibiting free/fixed side-alternating
boundary conditions, "boundary" clusters anchor to the fixed sides of P. At the
critical point and in the continuum limit, the density (i.e., frequency of
occurrence) of pinch-point events between s distinct boundary clusters at a
bulk point z in P is proportional to
_P. The
x_i are the vertices of P, psi_1^c is a conformal field theory (CFT) corner
one-leg operator, and Psi_s is a CFT bulk 2s-leg operator. In this article, 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 in the rectangle and in the hexagon, and for the
case s=N, an arbitrary 2N-gon. 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
of a rectangle. We compare our results with high-precision simulations of
critical percolation and Ising FK clusters in a rectangle of aspect ratio 2 and
in a regular hexagon and find very good agreement.

View original: http://arxiv.org/abs/1201.6405