• DOI:10.1007/s00220-022-04459-x
• Corpus ID: 229349060

On the Six-Vertex Model’s Free Energy

@article{DuminilCopin2020OnTS,
  title={On the Six-Vertex Model’s Free Energy},
  author={Hugo Duminil-Copin and Karol Kajetan Kozlowski and Dmitry Krachun and Ioan Manolescu and Tatiana Tikhonovskaia},
  journal={Communications in Mathematical Physics},
  year={2020},
  volume={395},
  pages={1383 - 1430}
}

• H. Duminil-Copin, K. Kozlowski, +2 authors Tatiana Tikhonovskaia
• Published 21 December 2020
• Mathematics
• Communications in Mathematical Physics

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime Δ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta <1$$\end{document}. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when a=b=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=b=1$$\end{document} and c≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ge 1$$\end{document}, and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.