Restricted partition functions of the two-dimensional Ising model on a half-infinite cylinder

We study the Ising model on a half-infinite cylinder at the critical temperature. On the boundary circle, we fix four intervals of constant signs. Let $θ1$⁠, $θ2$⁠, $θ3$⁠, and $θ4$ be the positions where the flips occur, labeled counterclockwise in that order. From each $θ$ starts a contour between clusters of opposite signs. The contour leaving $θ1$ may end only at $θ2$ or $θ4$⁠. Using an argument based on conformal field theory, we give the probability distribution that the contour leaving $θ1$ ends at $θ2$⁠. The behavior of this function when $θ2−θ1→0$ is described by a power law with an exponent $(53)$ that belongs to the Kac table but that corresponds to a nonunitarizable highest-weight representation. We check that this prediction agrees with a Monte Carlo simulation.