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

TL;DR

This paper connects off-critical local height probabilities in solid-on-solid models to critical partition functions on a cylinder, revealing a deep relationship between off-critical and critical regimes through geometric and conformal transformations.

Contribution

It demonstrates that off-critical local height probabilities on a plane are equivalent to critical partition functions on a cylinder, extending the understanding of critical phenomena in statistical models.

Findings

Off-critical local height probabilities depend only on the product N * τ.

In the limit N→1, τ→τ₀, the probability reduces to the plane case.

In the limit N→∞, τ→0 with N*τ finite, the probability corresponds to a critical partition function 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 $\tau$, the departure from criticality of the model, and observe that the result depends only on the product $N \, \tau$. In the limit $N \rightarrow 1$, $\tau \rightarrow \tau_0$, 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 $N \rightarrow \infty$, $\tau \rightarrow 0$, 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.