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

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.

Twenty Vertex Model and Domino Tilings of the Aztec Triangle

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.

Density matrices in integrable face models

Using the properties of the local Boltzmann weights of integrable interaction-round-a-face (IRF or face) models we express local operators in terms of generalized transfer matrices. This allows for the derivation of discrete functional equations for the reduced density matrices in inhomogeneous generalizations of these models. We apply these equations to study the density matrices for IRF models of various solid-on-solid type and quantum chains of non-Abelian \mathbold{su(2)_3}𝐬𝐮(2)3 or Fibonacci anyons. Similar as in the six vertex model we find that reduced density matrices for a sequence of consecutive sites can be ‘factorized’, i.e. expressed in terms of nearest-neighbour correlators with coefficients which are independent of the model parameters. Explicit expressions are provided for correlation functions on up to three neighbouring sites.

A Novel Cell Vertex Model Formulation that Distinguishes the Strength of Contraction Forces and Adhesion at Cell Boundaries

The vertex model is a useful mathematical model to describe the dynamics of epithelial cell sheets. However, existing vertex models do not distinguish contraction forces on the cell boundary from adhesion between cells, employing a single parameter to express both. In this paper, we introduce the rest length of the cell boundary and its dynamics into the existing vertex model, giving a novel formulation of the model that treats separately the contraction force and the strength of adhesion between cells. We apply this vertex model to the phenomenon of compartment boundary in the fruit fly pupa, recapturing the observation that increasing the strength of adhesion between cells straightens the compartment boundary, even though contraction forces at cell boundaries remain unchanged. We also discuss possibilities of the novel vertex models by considering the stretching of a cell sheet by external forces.