Principal minors and rhombus tilings

The algebraic relations between the principal minors of an $n\times n$ matrix are somewhat mysterious, see e.g. [lin-sturmfels]. We show, however, that by adding in certain \emph{almost} principal minors, the relations are generated by a single relation, the so-called hexahedron relation, which is a composition of six cluster mutations. We give in particular a Laurent-polynomial parameterization of the space of $n\times n$ matrices, whose parameters consist of certain principal and almost principal minors. The parameters naturally live on vertices and faces of the tiles in a rhombus tiling of a convex $2n$-gon. A matrix is associated to an equivalence class of tilings, all related to each other by Yang-Baxter-like transformations. By specializing the initial data we can similarly parametrize the space of Hermitian symmetric matrices over $\mathbb R, \mathbb C$ or $\mathbb H$ the quaternions. Moreover by further specialization we can parametrize the space of \emph{positive definite} matrices over these rings

Discrete line complexes and integrable evolution of minors

Based on the classical Pl\"ucker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in $CP^3$. Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the existence of these integrable line complexes is shown to be guaranteed by Desargues' classical theorem of projective geometry. A remarkable characterisation in terms of correlations of $CP^3$ is also recorded.Comment: 29 pages, 11 figures; updated reference

Arctic curves of the octahedron equation

We study the octahedron relation (also known as the $A_{\infty}$ $T$-system), obeyed in particular by the partition function for dimer coverings of the Aztec Diamond graph. For a suitable class of doubly periodic initial conditions, we find exact solutions with a particularly simple factorized form. For these, we show that the density function that measures the average dimer occupation of a face of the Aztec graph, obeys a system of linear recursion relations with periodic coefficients. This allows us to explore the thermodynamic limit of the corresponding dimer models and to derive exact "arctic" curves separating the various phases of the system.Comment: 39 pages, 21 figures; typos fixed, four references and an appendix adde

A generalisation of the honeycomb dimer model to higher dimensions

This paper studies a generalisation of the honeycomb dimer model to higher dimensions. The generalisation was introduced by Linde, Moore, and Nordahl. Each sample of the model is both a tiling and a height function. First, we derive a surprising identity for the covariance structure of the model. Second, we prove that the surface tension associated with the model is strictly convex, in any dimension. This greatly streamlines the original proof for strict convexity by Sheffield. It implies a large deviations result with a unique minimiser for the rate function, and consequently a variational principle with a unique limit shape. Third, we demonstrate that the model is a perfect matching model on a hypergraph with a generalised Kasteleyn theory: the partition function is given by the Cayley hyperdeterminant of the appropriate hypermatrix. The formula so obtained is very challenging: the author does not expect a closed-form solution for the surface tension. The first two results rely on the development of the boundary swap, which is a versatile technique for understanding the model; it is inspired by the double dimer model, works in any dimension, and may be of independent interest.The author was supported by the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge and the UK Engineering and Physical Sciences Research Council grant EP/L016516/1