# Finite lattice systems, combinatorics and Painleve equations

Lattice integrable systems have led to a remarkable progress in many research areas of physics and mathematics including quantum groups, topology, combinatorics, 2D conformal field theory and condensed matter physics. This project aims to study recently discovered connections between mathematics of 2D integrable systems on finite lattices, the combinatorial 3-coloring problems on square grids and the theory of special functions.

During the last decade there have been substantial progress in our understanding of properties of 2D lattice integrable systems and their relations to purely combinatorial problems like enumeration of alternative sign matrices, plane partitions, representations of Temperley-Lieb algebra, etc. In 2005 Vladimir Mangazeev and Vladimir Bazhanov studied the famous 8-vertex model of statistical mechanics, originaly solved by R.J.Baxter in 1972 in the limit of an infinite lattice. It was found that for a special value of coupling constant (at so called "supersymmetric point") one can explicitly describe the ground state of the model even on a finite lattice using analytic results from the theory of Painleve differential equations.

Recently our colleague from Sweden, Prof Rosengren (2011), studied the problem of enumeration of 3-coloring configurations of the finite square lattice (i.e. no two adjacent squares share the same color) and discovered the same algebraic structures which appeared in our research on the 8-vertex model. A deeper understanding of this relationship is the main subject of this project.

Our current work is on exploring the physics of these and other fundamental mathematical models which are applicable to future experiments.

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