T-systems, networks and dimers

Abstract: We study the solutions of the T-system for type A, also known as the octahedron equation, viewed as a 2+1-dimensional discrete evolution equation. These may be expressed entirely in terms of the stepped surface over which the initial data are specified, via a suitably defined flat $GL_n$ connection which embodies the integrability of this infinite rank system. By interpreting the connection as the transfer operator for a directed graph or network with weighted edges, we show that the solution at a given point is expressed as the partition function for dimers on a bipartite graph dual to the "shadow" of the point onto the initial data stepped surface. We extend the result to the case of other geometries such as that of the evaporation of a cube corner crystal, and to a reformulation of the Kenyon-Pemantle discrete hexahedron equation.