Arctic curves of the four-vertex model

Abstract

We consider the four-vertex model with a special choice of fixed boundary conditions giving rise to limit shape phenomena. More generally, the considered boundary conditions relate vertex models to scalar products of off-shell Bethe states, boxed plane partitions, and fishnet diagrams in quantum field theory. In the scaling limit, the model exhibits the emergence of an arctic curve separating a central disordered region from six frozen `corners' of ferroelectric or anti-ferroelectric type. We determine the analytic expression of the interface by means of the Tangent Method. We supplement this heuristic method with an alternative, rigorous derivation of the arctic curve. This is based on the exact evaluation of suitable correlation functions, devised to detect spatial transition from order to disorder, in terms of the partition function of some discrete log-gas associated to the orthogonalizing measure of the Hahn polynomials. As a by-product, we also deduce that the arctic curve's fluctuations are governed by the Tracy-Widom distribution.

arXiv comment36 pages, 21 figures; v2: minor changes in the abstract and Sect. 2, one reference added, misprints corrected

References in corpus (29)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• Fredholm Determinants, Differential Equations and Matrix Models
• A variational principle for domino tilings
• Large $N$ Phase Transition in Continuum QCD${}_2$
• The shape of a typical boxed plane partition
• Local statistics for random domino tilings of the Aztec diamond
• Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions
• Stochastic six-vertex model
• The arctic curve of the domain-wall six-vertex model
• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• Directed-loop Monte Carlo simulations of vertex models
• Emptiness formation probability in the domain-wall six-vertex model
• Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
• The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime
• The limit shape of large alternating sign matrices
• The Influence of Boundary Conditions in the Six-Vertex Model
• From Alternating Sign Matrices to the Gaussian Unitary Ensemble
• Four-vertex Model and Random Tilings
• Arctic curve of the free-fermion six-vertex model in an L-shaped domain
• A numerical study of the F-model with domain-wall boundaries
• Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions
• Arctic curves of the twenty-vertex model with domain wall boundaries
• The Arctic curve for Aztec rectangles with defects via the Tangent Method
• Minimal bipartite dimers and higher genus Harnack curves
• Concavity analysis of the tangent method
• Determinant formulas for the five-vertex model
• Arctic curves of the Reflecting Boundary Six Vertex and of the Twenty Vertex models
• Arctic curves of the 20V model on a triangle

Cited by in corpus (3)

• Importance Sampling for counting statistics in one-dimensional systems
• Thermodynamics of the five-vertex model with scalar-product boundary conditions
• Emptiness Instanton in Quantum Polytropic Gas