Limit shapes for Domain-Wall (colored) vertex models

TL;DR

The paper analyzes limit shapes (arctic curves) for domain-wall boundary colored vertex models, proving an equivalence to a colorblind variant and deriving Arctic curves using a sliding map and the tangent method. For the special case \(t=0\), it provides exact limit shapes, including SE/SW portions and gap/frozen scenarios, with translations and shear relations linking colored and NILP pictures. It extends the framework to \(q\)-weighted paths, obtaining q-deformed arctic curves via a curved tangent method and highlighting the persistence of shear/translation phenomena. Numerical simulations based on Metropolis-Hastings sampling corroborate the analytic predictions and illuminate non-free-fermion regimes, gap effects, and tropical limits. The work advances understanding of arctic phenomena beyond free-fermion models in colored vertex systems and sets the stage for further exploration of non-free-fermion limit shapes.

Abstract

We study partition functions with domain-wall like boundary conditions for path models issued from colored vertex models. These models display an arctic phenomenon, as attested by numerical simulations. We show that the colored vertex model is equivalent to a certain single-color ``colorblind" vertex model. In a special case of the weights for the colorblind touching paths, we derive the arctic curve using a bijective sliding map to non-intersecting paths, for which arctic curves were previously derived using the tangent method. The resulting arctic curves are only piecewise analytic, as in the known non-free fermion cases of Six vertex model with domain-wall boundaries and its relatives. We also prove a shear phenomenon, that some portions of the arctic curve are sheared versions of the analytic continuation of other portions, as already observed in the uniformly weighted Six and Twenty vertex models.

Figures (26)

• Figure 1: Example colored domain-wall boundary condition and a possible configuration for $n=4$. The colors are order blue$<$green$<$orange$<$red. Here $\sigma=(3,1,4,2)$. The coloring on the top boundary is given by $\sigma^{-1}=(2,4,1,3)$. The weight of the configuration is $x_3^4x_4^2t^4$.
• Figure 2: Simulations for $n=200$ with $x_1=\ldots=x_n=1$.
• Figure 3: $n\times n$ square grid with Domain Wall Boundaries (left): paths start on the W edge and end on the N edge. A sample configuration (right).
• Figure 4: Two example of the one-to-many mapping from a colorblind configuration to a collection of colored configurations such that the total weight is preserved. Here $n=3$, $x_1=x_2=x_3=1$, and we order the colors blue$<$green$<$red.
• Figure 5: For $t=0$, the sliding map is a bijection from the touching path configurations (with only vertical multiply occupied edges) to strictly non-intersecting paths.

Theorems & Definitions (29)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6