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 $4 N$-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 $N \rightarrow 1$, $τ\rightarrow τ_0$, such that $τ_0$ is finite, we recover the off-critical local height probability on a plane, $τ_0$-away from criticality. In the limit $N \rightarrow \infty$, $τ\rightarrow 0$, such that $N \, τ= τ_0$ is finite, and following a conformal transformation, we obtain a critical partition function on a cylinder of aspect-ratio $τ_0$. We conclude that the off-critical local height probability on a plane, $τ_0$-away from criticality, is equal to a critical partition function on a cylinder of aspect-ratio $τ_0$, in agreement with a result of Saleur and Bauer.