Arctic curves of the $6$V model with partial DWBC and double Aztec rectangles

TL;DR

This paper derives an explicit analytic expression for the arctic curve in the 6V model with partial DWBC at a specific parameter setting, using the tangent method, and confirms results with numerical simulations.

Contribution

It provides the first explicit formula for the arctic curve in the 6V model with partial DWBC at \\Delta=0, linking it to domino tilings of double Aztec rectangles.

Findings

Explicit arctic curve formula derived for the 6V model at \\Delta=0.

Arctic curve matches that of domino tilings of double Aztec rectangles for certain parameters.

Numerical simulations confirm the analytical results.

Abstract

Previous numerical studies have shown that in the disordered and anti-ferroelectric phases the six-vertex ($6$V) model with partial domain wall boundary conditions (DWBC) exhibits an arctic curve whose exact shape is unknown. The model is defined on a $s\times n$ square lattice ($s\leq n$). In this paper, we derive the analytic expression of the arctic curve, for $a=b=1$ and $c=\sqrt{2}$ ($\Delta=0$), while keeping the ratio $s/n \,\in [0,1]$ as a free parameter. The computation relies on the tangent method. We also consider domino tilings of double Aztec rectangles and show via the tangent method that, for particular parameters, the arctic curve is identical to that of the $6$V model with partial DWBC. Our results are confirmed by extensive numerical simulations.