  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 w in P is proportional to
<psi_1^c(w_1)psi_1^c(w_2)...psi_1^c(w_{2N-1})psi_1^c(w_{2N})Psi_s(w)>_P. The
w_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 at arbitrary points in the rectangle, in the
hexagon, and for the case s=N, in the 2N-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 find very good agreement.
