The arctic curve of the domain-wall six-vertex model

Abstract

The problem of the form of the `arctic' curve of the six-vertex model with domain wall boundary conditions in its disordered regime is addressed. It is well-known that in the scaling limit the model exhibits phase-separation, with regions of order and disorder sharply separated by a smooth curve, called the arctic curve. To find this curve, we study a multiple integral representation for the emptiness formation probability, a correlation function devised to detect spatial transition from order to disorder. We conjecture that the arctic curve, for arbitrary choice of the vertex weights, can be characterized by the condition of condensation of almost all roots of the corresponding saddle-point equations at the same, known, value. In explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle-point equations. The arctic curve is obtained in parametric form and appears to be a non-algebraic curve in general; it turns into an algebraic one in the so-called root-of-unity cases. The arctic curve is also discussed in application to the limit shape of $q$-enumerated (with $0<q\le 4$) large alternating sign matrices. In particular, as $q\to 0$ the limit shape tends to a nontrivial limiting curve, given by a relatively simple equation.

39 pages, 2 figures; minor corrections

References in corpus (6)

• Domino tilings and the six-vertex model at its free fermion point
• Emptiness formation probability in the domain-wall six-vertex model
• The Influence of Boundary Conditions in the Six-Vertex Model
• The role of orthogonal polynomials in the six-vertex model and its combinatorial applications
• Step fluctuations for a faceted crystal
• q-Distributions on boxed plane partitions

Cited by in corpus (46)

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• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain
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• The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime
• The limit shape of large alternating sign matrices
• Free fermions at the edge of interacting systems
• The Frustration of being Odd: How Boundary Conditions can destroy Local Order
• Functional relations for the six vertex model with domain wall boundary conditions
• Thermodynamics of the six-vertex model in an L-shaped domain
• Multiply-refined enumeration of alternating sign matrices
• Inhomogeneous Gaussian Free Field inside the interacting arctic curve
• Arctic curve of the free-fermion six-vertex model in an L-shaped domain
• Extreme boundary conditions and random tilings
• Algebraic arctic curves in the domain-wall six-vertex model
• The density profile of the six vertex model with domain wall boundary conditions
• On the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions
• Phase separation in the six-vertex model with a variety of boundary conditions
• A numerical study of the F-model with domain-wall boundaries
• Emptiness formation probability and Painlevé V equation in the XY spin chain
• The frustration-free fully packed loop model
• Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions
• Arctic curves of the twenty-vertex model with domain wall boundaries
• Thermodynamic limit of the six-vertex model with reflecting end
• The Arctic curve for Aztec rectangles with defects via the Tangent Method
• Boundary correlation functions of the six and nineteen vertex models with domain wall boundary conditions
• Generalized emptiness formation probability in the six-vertex model
• Exact time evolution formulae in the XXZ spin chain with domain wall initial state
• Emptiness formation probability of the six-vertex model and the sixth Painlevé equation
• Domain wall fluctuations of the six-vertex model at the ice point
• Arctic curves of the four-vertex model
• Arctic curves of the Reflecting Boundary Six Vertex and of the Twenty Vertex models
• Integral formulas and antisymmetrization relations for the six-vertex model
• Concavity analysis of the tangent method
• The partition function of the four-vertex model in inhomogeneous external field and trace statistics
• Non-probabilistic fermionic limit shapes
• Factorization in the multirefined tangent method
• Double tangent method for two-periodic Aztec diamonds
• Boundary correlations for the six-vertex model with reflecting end boundary condition
• Arctic curve of the free-fermion six-vertex model with reflecting end boundary condition
• Multiple Meixner-Pollaczek polynomials and the six-vertex model
• Arctic curves of the $6$V model with partial DWBC and double Aztec rectangles
• Evaluation of integrals for the emptiness formation probability in the square-ice model
• Coupling Functions for Domino tilings of Aztec diamonds
• Emptiness Instanton in Quantum Polytropic Gas
• Scaling limit of domino tilings on a pentagonal domain