T-Systems, Networks and Dimers

Abstract

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.

Arctic curves of the octahedron equation

“…This establishes the connection between a general set of solutions of the octahedron equation with given initial data, and the partition functions of statistical lattice models of dimers, whose local Boltzmann weights are defined in terms of these data. This was recently extended to more general initial conditions, giving rise to dimer models on specific graphs [13]. Note also that a large class of dimer models on periodic graphs was recently shown to have both integrable and cluster algebra structures as well [22].…”

“…One way to think about the T -system is to consider i, j as labeling points on the square lattice and k as a discrete time (as the vertical axis of a cubic lattice). In this sense the T -system (2.1) describes the evolution in time of a given initial data (see [13]). In this paper we will work with a flat initial data, meaning that the value of the T variables is specified on the (i, j, 0) and (i, j, 1) planes, namely we fix:…”

“…The solution T i, j,k , for i+ j+k = 1 mod 2, i, j ∈ Z, k 0, of the A ∞ T -system (2.1) with initial data (2.2) was identified as the partition function for domino tilings of the Aztec diamond, or dually to the dimer coverings of the Aztec diamond graph A i, j,k [13,40]. The graph A i, j,k has vertices at points of the lattice Z 2 .…”

Beyond Aztec Castles: Toric Cascades in the dP 3 Quiver

“…For mutation-infinite cluster algebras, providing a combinatorial formula for every single cluster variable of that algebra is quite a challenge. However, for some specific mutation-infinite cluster algebras, there has been significant work which provides combinatorial formulas for cluster variables lying in certain subsets [BPW09,DiF,G11,S07].…”

Plane partitions in the work of Richard Stanley and his school

“…It is interesting to note that this -at the time, isolated, and maybe somewhat obscure -object has now become part of a fascinating theory, namely the theory of discrete integral systems (cf. e.g [23][29].…”