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 $\tau$, the departure from criticality of the model, and observe that the result depends only on the product $N\tau$. In the limit $N\to 1$, $\tau \to {\tau }_0$, such that ${\tau }_0$ is finite, we recover the off-critical local height probability on a plane, ${\tau }_0$-away from criticality. In the limit $N\to \infty$, $\tau \to 0$, such that $N\tau ={\tau }_0$ is finite, and following a conformal transformation, we obtain a critical partition function on a cylinder of aspect-ratio ${\tau }_0$. We conclude that the off-critical local height probability on a plane, ${\tau }_0$-away from criticality, is equal to a critical partition function on a cylinder of aspect-ratio ${\tau }_0$, in agreement with a result of Saleur and Bauer.

28 pages

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