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Arctic curves of the four-vertex model
Mathematical Physics · Submitted 6 Jul 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We determine the analytic expression of the interface by means of the Tangent Method.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the four-vertex model with a special choice of fixed boundary conditions giving rise to limit shape phenomena. | contribution 0.67 | 0.63 | 0.06 | asserted 0.99 |
| s1 | 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. | background 0.56 | 0.31 | 0.05 | asserted 1.00 |
| s2 | 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. | result 0.97 | 0.56 | 0.44 | asserted 1.00 |
| s3 | We determine the analytic expression of the interface by means of the Tangent Method. | method 0.86 | 0.96 | 0.31 | asserted 1.00 |
| s4 | We supplement this heuristic method with an alternative, rigorous derivation of the arctic curve. | method 0.54 | 0.96 | 0.09 | asserted 1.00 |
| s5 | 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. | method 0.97 | 0.92 | 0.13 | asserted 1.00 |
| s6 | As a by-product, we also deduce that the arctic curve's fluctuations are governed by the Tracy-Widom distribution. | result 1.00 | 0.94 | 0.14 | asserted 1.00 |
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The paper
Original 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.
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2 arXiv papers cite this in 6 sentences
Papers citing it · 4 sentences without a confident label
Cites it as background 2 papers
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The behaviours at these two types of corners are referred to as ferroelectric and anti-ferroelectric phases in [ref] .
Limit shapes from harmonicity: dominos and the five vertex model · Four-vertex model versus lozenge tilings · 0.81
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Such systems appear, for example, in the context of random walks on the lattice [ref] , growth processes [ref] , vertex models [ref] or exclusion processes [ref] .
Importance Sampling for counting statistics in one-dimensional systems · Trajectories of the non-crossing particles: Symmetric Simple Exclusion process · 0.88
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