Some rigorous results for the vertex model in statistical mechanics

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Abstract
It is shown that the free and periodic boundary conditions are completely equivalent for the ice-rule (six-vertex) models in zero field. With an external direct or staggered field, we establish that in an ice-rule model the free and periodic boundary conditions are equivalent, and also equal to some special boundary conditions, either at sufficiently low temperatures or with sufficiently high fields in the appropriate direction. Regions of constant direct polarization are found. We also establish the existence of the spontaneous staggered polarization in an antiferroelectric using the Peierls argument.