Cluster pinch-point densities in polygons

Details Cluster pinch-point densities in polygons Cluster pinch-point densities in polygons

2012-01-30

arxiv.org/abs/1201.6405v1

Physics

Mathematical Physics

31 pages, 1 appendix, 21 figures

Scientific paper

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.

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