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

Jean-Francois de KEMMETER; Bryan DEBIN; Philippe RUELLE

doi:10.1088/1751-8121/ac7c48

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

Jean-Francois de KEMMETER, Bryan DEBIN, Philippe RUELLE

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Abstract

Previous numerical studies have shown that in the disordered and anti-ferroelectric phases the six-vertex (6V) model with partial domain wall boundary conditions (DWBC) exhibits an arctic curve whose exact shape is unknown. The model is defined on a s × n square lattice (s ≤ n). In this paper, we derive the analytic expression of the arctic curve, for a = b = 1 and c=2 (Δ= 0), while keeping the ratio s/n ϵ [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 6V model with partial DWBC. Our results are confirmed by extensive numerical simulations.

Original language	English
Article number	305004
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	55
Issue number	30
DOIs	https://doi.org/10.1088/1751-8121/ac7c48
Publication status	Published - 29 Jul 2022

Keywords

arctic phenomenon
domino tilings
double Aztec rectangle
integrable lattice model
six-vertex model

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10.1088/1751-8121/ac7c48

6V_and_DoubleAztecRectanglesSubmitted manuscript, 7.84 MB

https://iopscience.iop.org/article/10.1088/1751-8121/ac7c48

Cite this

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title = "Arctic curves of the 6V model with partial DWBC and double Aztec rectangles",

abstract = "Previous numerical studies have shown that in the disordered and anti-ferroelectric phases the six-vertex (6V) model with partial domain wall boundary conditions (DWBC) exhibits an arctic curve whose exact shape is unknown. The model is defined on a s × n square lattice (s ≤ n). In this paper, we derive the analytic expression of the arctic curve, for a = b = 1 and c=2 (Δ= 0), while keeping the ratio s/n ϵ [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 6V model with partial DWBC. Our results are confirmed by extensive numerical simulations.",

keywords = "arctic phenomenon, domino tilings, double Aztec rectangle, integrable lattice model, six-vertex model",

author = "{de KEMMETER}, Jean-Francois and Bryan DEBIN and Philippe RUELLE",

note = "Funding Information: BD and JFDK contributed equally to this work. The authors thank Timoteo Carletti and Filippo Colomo for useful discussions. BD acknowledges the financial support of the Fonds Sp{\'e}ciaux de Recherche (FSR) of the Universit{\'e} catholique de Louvain, the Fonds de la Recherche Scientifique (FNRS) and the Fonds Wetenschappelijk Onderzoek—Vlaanderen (FWO) under EOS Project No. 30889451. JFDK is supported by a FNRS Aspirant Fellowship under the Grant FC38477. PR is a Senior Research Associate of FRS-FNRS. Part of the results were obtained using the computational resources provided by the {\textquoteleft}Consortium des Equipements de Calcul Intensif{\textquoteright} (CECI), funded by the Fonds de la Recherche Scientifique de Belgique (FRS-FNRS) under Grant No. 2.5020.11 and by the Walloon Region. Publisher Copyright: {\textcopyright} 2022 IOP Publishing Ltd.",

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PY - 2022/7/29

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N2 - Previous numerical studies have shown that in the disordered and anti-ferroelectric phases the six-vertex (6V) model with partial domain wall boundary conditions (DWBC) exhibits an arctic curve whose exact shape is unknown. The model is defined on a s × n square lattice (s ≤ n). In this paper, we derive the analytic expression of the arctic curve, for a = b = 1 and c=2 (Δ= 0), while keeping the ratio s/n ϵ [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 6V model with partial DWBC. Our results are confirmed by extensive numerical simulations.

AB - Previous numerical studies have shown that in the disordered and anti-ferroelectric phases the six-vertex (6V) model with partial domain wall boundary conditions (DWBC) exhibits an arctic curve whose exact shape is unknown. The model is defined on a s × n square lattice (s ≤ n). In this paper, we derive the analytic expression of the arctic curve, for a = b = 1 and c=2 (Δ= 0), while keeping the ratio s/n ϵ [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 6V model with partial DWBC. Our results are confirmed by extensive numerical simulations.

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

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

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Reaction-diffusion processes on temporal and non-normal networks

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