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The five-vertex model as a discrete log-gas

Filippo Colomo, Michelangelo Mannatzu, Andrei G. Pronko

TL;DR

This work provides a complete resolvent-based analysis of the five-vertex model with scalar-product boundary conditions by reformulating its partition function as a discrete log-gas. It derives the free-energy density in the scaling limit across both square ($\lambda=\mu$) and rectangular ($\lambda\neq\mu$) geometries, reproducing known leading-order results and delivering explicit forms for the log-gas resolvent in all regimes. The study reveals two third-order phase transitions at critical points $x_c$ (and its inverse in the square case), associated with changes between void, band, and saturated regions in the equilibrium measure, and demonstrates a robust framework to connect to limit-shape phenomena and potential boundary correlations via the tangent method. These results lay groundwork for determining Arctic curves and boundary correlation functions in the five-vertex model and related integrable systems.

Abstract

We consider the five-vertex model on a rectangular domain of the square lattice, with the so-called `scalar-product' boundary conditions. We address the evaluation of the free-energy density of the model in the scaling limit, that is when the number of sites is sent to infinity and the mesh of the lattice to zero, while keeping the size of the domain constant. To this aim, we reformulate the partition function of the model in terms of a discrete log-gas, and study its behaviour in the thermodynamic limit. We reproduce previous results, obtained by using a differential equation approach. Moreover, we provide the explicit form of the resolvent in all possible regimes. This work is preliminary to further studies of limit shape phenomena in the model.

The five-vertex model as a discrete log-gas

TL;DR

This work provides a complete resolvent-based analysis of the five-vertex model with scalar-product boundary conditions by reformulating its partition function as a discrete log-gas. It derives the free-energy density in the scaling limit across both square () and rectangular () geometries, reproducing known leading-order results and delivering explicit forms for the log-gas resolvent in all regimes. The study reveals two third-order phase transitions at critical points (and its inverse in the square case), associated with changes between void, band, and saturated regions in the equilibrium measure, and demonstrates a robust framework to connect to limit-shape phenomena and potential boundary correlations via the tangent method. These results lay groundwork for determining Arctic curves and boundary correlation functions in the five-vertex model and related integrable systems.

Abstract

We consider the five-vertex model on a rectangular domain of the square lattice, with the so-called `scalar-product' boundary conditions. We address the evaluation of the free-energy density of the model in the scaling limit, that is when the number of sites is sent to infinity and the mesh of the lattice to zero, while keeping the size of the domain constant. To this aim, we reformulate the partition function of the model in terms of a discrete log-gas, and study its behaviour in the thermodynamic limit. We reproduce previous results, obtained by using a differential equation approach. Moreover, we provide the explicit form of the resolvent in all possible regimes. This work is preliminary to further studies of limit shape phenomena in the model.
Paper Structure (46 sections, 141 equations, 3 figures)

This paper contains 46 sections, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The five-vertex model
  3. The model
  4. Particular cases
  5. Hankel determinant representation
  6. Scaling limit and free-energy density
  7. The function $\Phi(x)$
  8. A discrete log-gas description
  9. The log-gas
  10. Scaling limit
  11. The resolvent
  12. The free-energy density
  13. The case $\lambda=\mu$
  14. Preliminaries
  15. The void-band-void scenario (VBV)
  16. ...and 31 more sections

Figures (3)

  • Figure 1: The six vertex configurations of the six-vertex model, represented in terms of lines, and their Boltzmann weights in the five-vertex model.
  • Figure 2: The $L\times M$ lattice, with scalar-product boundary condition (left), and a possible configuration of the model (right). Here $L=8$, $M=9$, and $N=3$.
  • Figure 3: Transitions betwen scenarios, and regimes, depending whether $x_1,x_2>x_\mathrm{c}$ (left) or $x_1,x_2<x_\mathrm{c}$ (right). Remarkably, despites very different expressions, for given values of $\lambda$ and $\mu$, $x_1$ and $x_2$ appears to be both smaller or both larger than $x_\mathrm{c}$.