Graphical Approach to the Nonintersecting String Model: Star-triangle Equation, Inversion Relation, and Exact Solution

A general q-state nonintersecting string model for q Z 2 is studied, and solved, using a graphical approach. This model, of which special cases were first introduced by Stroganov and Schultz, has recently drawn further attention because of its relation to Potts models. In its simplest version, a uniform, separable nonintersecting string model on a quadratic lattice, it corresponds to a q*-state Potts model with Potts-spins on every other lattice face. We first formulate the related star-triangle equation which we solve in terms of a line-variable parametrization. Such a parametrization of a solution of the star-triangle equation leads to a simple and direct graphical derivation of an inversion relation for the partition function. Our graphical analysis also shows that the inversion relation holds if certain boundary effects can be neglected, as we shall give a special finite lattice from which the inversion relation can be read off immediately. This relation is next solved under appropriate analyticity assumptions, again using a simple and direct approach, and the result is applied to obtain exact solutions for specific string models.