On discrete three-dimensional equations associated with the local Yang-Baxter relation

The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to three dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable three-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov model is related to the star-triangle relation in the Ising model. In this Letter the corresponding discrete equation is derived. In the continuous limit it leads to a differential three-dimensional equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy.Some related operator solutions to the tetrahedron equation are also constructed.St. Petersburg Branch of the Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, RussiaURA 14-36 du CNRS, associée à l'ENS de Lyon, au LAPP d'Annecy et à l Universitè de Savoie, France