The asymmetric five vertex model on a rectangle

2026/06/20 by Jan de Gier, Richard Kenyon, Michael Wheeler +1 · 1 voice
Physics and Astronomy · Mathematics · #math-ph #math.CO #math.PR

paper · pdf

Abstract

We derive a determinantal expression for the inhomogeneous asymmetric five vertex model in a rectangular geometry with arbitrary boundary conditions at the bottom and top. Standard non-intersecting lattice path, or free fermion, approaches are not applicable and the determinantal form thus is not immediate.

Citations

• The five-vertex model as a discrete log-gas
• Limit shapes from harmonicity: dominos and the five vertex model
• Determinant formulas for the five-vertex model
• Vertex models for Canonical Grothendieck polynomials and their duals
• Double Grothendieck polynomials and colored lattice models
• Gradient variational problems in ℝ²
• Integrability approach to Feher-Nemethi-Rimanyi-Guo-Sun type identities for factorial Grothendieck polynomials
• Grothendieck polynomials and the Boson-Fermion correspondence
• Colored five-vertex models and Demazure atoms
• Coloured stochastic vertex models and their spectral theory
• Littlewood-Richardson coefficients for Grothendieck polynomials from\n integrability
• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• On a family of symmetric rational functions
• Stochastic six-vertex model
• K-theoretic boson-fermion correspondence and melting crystals
• Vertex models, TASEP and Grothendieck polynomials
• K-theoretic analogues of factorial Schur P- and Q-functions
• The Arctic Curve of the Domain-Wall Six-Vertex Model
• The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime
• Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase
• Exact solution of the six-vertex model with domain wall boundary conditions. Antiferroelectric phase
• Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase
• Limit shapes and the complex burgers equation
• Boxed Plane Partitions as an Exactly Solvable Boson Model
• Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram
• A variational principle for domino tilings
• Symmetry classes of alternating-sign matrices under one roof
• Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions
• The shape of a typical boxed plane partition
• Interacting dimers on the honeycomb lattice: An exact solution of the five-vertex model
• Interacting Domain Walls and the Five-Vertex Model
• Quantum Inverse Scattering Method and Correlation Functions
• Calculation of norms of Bethe wave functions

Discussions

• The "six-vertex" terminology comes from statistical physics, where $R$ -matrices represent vertex Boltzmann weights on a two-dimensional square lattice with two-state bonds (arrows or spins $1, 2$ ). [stackexchange, 6 points]