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

Abstract

We consider the six-vertex model with the rational weights on an 𝑠 × 𝑁 square lattice, 𝑠 ≤ 𝑁, 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; in 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.