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

TL;DR

The paper establishes a precise interpolation between off-critical local height probabilities on a plane and critical partition functions on a cylinder by extending regime-III RSOS models to a 4N-quadrant spiral geometry. Through Baxter’s corner transfer matrix method and a sequence of factorization, conjugate-modulus transformations, and configuration-sum analyses, the authors show that the essential one-dimensional configuration sums depend on the product Nτ, and that in the limits N→1 and N→∞ with τ→τ0 or Nτ→τ0 respectively, these objects reproduce the plane LH probabilities and cylinder partition functions. This culminates in demonstrating that, after conformal transformations, the off-critical LH probability on a plane at τ0 away from criticality equals a critical cylinder partition function with aspect-ratio τ0, echoing Saleur–Bauer’s results. The work thus highlights a deep geometry–physics correspondence, with LH probabilities interpreted as affine and Virasoro characters and the spiral construction providing a unified interpolating framework between two fundamental representations in 2D critical phenomena.

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.

Figures (11)

• Figure 2.1: A configuration in the restricted solid-on-solid model $\mathcal{L}_{\, 4, 5}$, where the state-variables take values in $\{\textswab{1}, \textswab{2}, \textswab{3}, \textswab{4}\}$.
• Figure 2.2: The weight $w \left\lgroup h_{\, 1}, h_{\, 2}, h_{\, 3}, h_{\, 4} \, \vert \, u \right\rgroup$ associated to a face on the lattice with state-variables $\left\lgroup h_{\, 1}, h_{\, 2}, h_{\, 3}, h_{\, 4} \right\rgroup$ at the corners, and a spectral parameter $u$.
• Figure 2.3: The Dynkin diagram $A_{\, 4}$ that corresponds to the restricted solid-on-solid model $\mathcal{L}_{\, 4, 5}$, where the state-variables take values in $\left\lgroup \textswab{1}, \textswab{2}, \textswab{3}, \textswab{4}\right\rgroup$.
• Figure 2.4: The Dynkin diagram $D_{\, 5}$ that corresponds to the restricted solid-on-solid model $\mathcal{L}^{\, D}_{\, 5, 6}$, where the state-variables take values in $\left\lgroup \textswab{1}, \textswab{2}, \textswab{3}, \textswab{4}, {\setul{-0.9em}{}\ul{\textswab{4}}} \right\rgroup$.
• Figure 2.5: The Dynkin diagram of the affine Lie algebra $\widehat{A}_{\, 4}$ that corresponds to the cyclic solid-on-solid model $\mathcal{L}^{\, cyc}_{\, 4, 5}$, where the state-variables take values in $\left\lgroup \textswab{1}, \textswab{2}, \textswab{3}, \textswab{4} \right\rgroup$, and the height variables $\textswab{1}$ and $\textswab{4}$ are regarded as nearest-neighbours.