ИСТИНА

I will continue the talk of Boris Bychkov and will focus on the applications of the embedding defined by response electrical matrices into the nonnegative Lagrangian Grassmannian IG(n − 1,2n − 2). As an important example, I will show that the Kenyon–Wilson theorem about polynomial formulas of special partition functions of trees could be obtained as a consequence of our embedding. Also I will talk about the relation of our technique with the embedding in the positive part of orthogonal Grassmannian of Ising models. In particular, using the Gorbunov–Talalaev vertex-model technique for electrical networks, I will demonstrate the construction of the full «electrical» analogue of Galashin and Pylyavskyy dimers models for Ising models. Using this construction I will show that the star-triangle transformation for Ising models and electrical networks could be obtained by the superurban renewal transformation of dimer models. Finally, I will demonstrate how the star-triangle transformation for these models related with the Sergeev–Korepanov–Kashaev solution of the Yang–Baxter and Zamolodchikov equations. The talk is based on a joint ongoing work with B. Bychkov, V. Gorbounov and D. Talalaev.