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

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 N τ. 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.