An equivalence relation of boundary/initial conditions, and the infinite limit properties

TL;DR

This paper introduces n-equivalences of boundary conditions in lattice models, showing they lead to identical free energies and classifying the six-vertex model's free energy based on boundary arrow densities.

Contribution

It defines n-equivalence of boundary conditions and demonstrates their impact on free energy, connecting boundary arrow densities to model properties and known results.

Findings

Models with n-equivalent boundary conditions share the same free energy.

The free energy of the six-vertex model depends on boundary arrow densities.

When the arrow density is 1/2, the free energy matches the periodic boundary case.

Abstract

The 'n-equivalences' of boundary conditions of lattice models are introduced and it is derived that the models with n-equivalent boundary conditions result in the identical free energy. It is shown that the free energy of the six-vertex model is classified through the density of left/down arrows on the boundary. The free energy becomes identical to that obtained by Lieb and Sutherland with the periodic boundary condition, if the density of the arrows is equal to 1/2. The relation to the structure of the transfer matrix and a relation to stochastic processes are noted.