Off-critical local height probabilities on a plane and critical partition functions on a cylinder

Abstract

We compute off-critical local height probabilities in regime-III restricted solid-on-solid models in a 4N-quadrant spiral geometry, with periodic boundary conditions in the angular direction, and fixed boundary conditions in the radial direction, as a function of N, the winding number of the spiral, and τ, the departure from criticality of the model, and observe that the result depends only on the product Nτ. In the limit N→1, τ→τ0, such that τ0 is finite, we recover the off-critical local height probability on a plane, τ0-away from criticality. In the limit N→∞, τ→0, such that Nτ=τ0 is finite, and following a conformal transformation, we obtain a critical partition function on a cylinder of aspect-ratio τ0. We conclude that the off-critical local height probability on a plane, τ0-away from criticality, is equal to a critical partition function on a cylinder of aspect-ratio τ0, in agreement with a result of Saleur and Bauer.