scholarly journals Twenty Vertex Model and Domino Tilings of the Aztec Triangle

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Philippe Di Francesco
Keyword(s):  
Domain Wall ◽  
Boundary Conditions ◽  
Vertex Model ◽  
Domino Tilings ◽  
Tiling Problem ◽  
Domino Tiling ◽  
The Common ◽  
Type Boundary

We show that the number of configurations of the 20 Vertex model on certain domains with domain wall type boundary conditions is equal to the number of domino tilings of Aztec-like triangles, proving a conjecture of the author and Guitter. The result is based on the integrability of the 20 Vertex model and uses a connection to the U-turn boundary 6 Vertex model to re-express the number of 20 Vertex configurations as a simple determinant, which is then related to a Lindström-Gessel-Viennot determinant for the domino tiling problem. The common number of configurations is conjectured to be $2^{n(n-1)/2}\prod_{j=0}^{n-1}\frac{(4j+2)!}{(n+2j+1)!}=1, 4, 60, 3328, 678912...$ The enumeration result is extended to include refinements of both numbers.  

scholarly journals Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings

10.37236/8809 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Philippe Di Francesco ◽  
Emmanuel Guitter
Keyword(s):  
Domain Wall ◽  
Triangular Lattice ◽  
Roots Of Unity ◽  
Vertex Model ◽  
Domino Tilings ◽  
Enumeration Problems ◽  
Shaped Hole ◽  
Type Boundary ◽  
Type 3 ◽  
Ice Model

We consider the triangular lattice ice model (20-Vertex model) with four types of domain-wall type boundary conditions. In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole. The proof of this statement makes extensive use of integrability and of a connection to the 6-Vertex model. The type 3 configurations are conjectured to be in same number as domino tilings of a particular triangle. The four enumeration problems are reformulated in terms of four types of Alternating Phase Matrices with entries $0$ and sixth roots of unity, subject to suitable alternation conditions. Our result is a generalization of the ASM-DPP correspondence. Several refined versions of the above correspondences are also discussed. 

scholarly journals EXACT SOLUTION OF THE SIX-VERTEX MODEL WITH DOMAIN WALL BOUNDARY CONDITIONS: CRITICAL LINE BETWEEN DISORDERED AND ANTIFERROELECTRIC PHASES

2012 ◽  
Vol 01 (04) ◽  
pp. 1250012 ◽  
Author(s):  
PAVEL BLEHER ◽  
THOMAS BOTHNER
Keyword(s):  
Domain Wall ◽  
Boundary Conditions ◽  
Critical Line ◽  
Hilbert Problem ◽  
Steepest Descent Method ◽  
Vertex Model ◽  
Wall Boundary ◽  
Large N ◽  
Wall Boundary Conditions ◽  
Domain Wall Boundary Conditions

In the present paper we obtain the large N asymptotics of the partition function ZN of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and antiferroelectric phases. Using the weights a = 1 - x, b = 1 + x, c = 2, |x| < 1, we prove that, as N → ∞, ZN = CFN2N1/12(1 + O(N-1)), where F is given by an explicit expression in x and the x-dependency in C is determined. This result reproduces and improves the one given in the physics literature by Bogoliubov, Kitaev and Zvonarev [Boundary polarization in the six-vertex model, Phys. Rev. E65 (2002) 026126]. Furthermore, we prove that the free energy exhibits an infinite-order phase transition between the disordered and antiferroelectric phases. Our proofs are based on the large N asymptotics for the underlying orthogonal polynomials which involve a non-analytical weight function, the Deift–Zhou non-linear steepest descent method to the corresponding Riemann–Hilbert problem, and the Toda equation for the tau-function.

scholarly journals oundary One-Point Function of the Rational Six-Vertex Model with Partial Domain Wall Boundary Conditions: Explicit Formulas and Scaling Properties

2021 ◽  
Author(s):  
Mikhail D. Minin ◽  
◽  
Andrei G. Pronko ◽  
Keyword(s):  
Domain Wall ◽  
Boundary Conditions ◽  
Saddle Point Method ◽  
Vertex Model ◽  
Point Function ◽  
Wall Boundary ◽  
Partial Domain ◽  
Wall Boundary Conditions ◽  
Domain Wall Boundary Conditions ◽  
The One

We consider the six-vertex model with the rational weights on an s by N square lattice with partial domain wall boundary conditions. We study the one-point function at the boundary where the free boundary conditions are imposed. For a finite lattice, it can be computed by the quantum inverse scattering method in terms of determinants. In the large N limit, the result boils down to an explicit terminating series in the parameter of the weights. Using the saddle-point method for an equivalent integral representation, we show that as s next tends to infinity, the one-point function demonstrates a step-wise behavior; at the vicinity of the step it scales as the error function. We also show that the asymptotic expansion of the one-point function can be computed from a second-order ordinary differential equation.

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