Related papers: Dimer models on astroidal zig-zag graphs

Isoradial dimer models were introduced in \cite{Kenyon3} - they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of…

We prove that the class of cluster integrable systems constructed by Goncharov and Kenyon out of the dimer model on a torus coincides with the one defined by Gekhtman, Shapiro, Tabachnikov, and Vainshtein using Postnikov's perfect networks.…

In this paper we review the asymptotic matching conjectures for $r$-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in $d$-dimensional integer lattice and Bethe lattices. We prove new rigorous upper…

The main result of this paper is a Pfaffian formula for the partition function of the dimer model on a graph G embedded in a closed, possibly non-orientable surface S. This formula is suitable for computational purposes, and it is obtained…

For a family of integer-valued height functions defined over the faces of planar graphs, we establish a relation between the probability of connection by level sets and the spin-spin correlations of the dual $O(2)$ symmetric spin models…

We consider embeddings of 3-regular graphs into 3-dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axis-parallel line) and…

Consider an elliptic parameter $k$; we introduce a family of $Z^u$-Dirac operators $(\mathsf{K}(u))_{u\in\Re(\mathbb{T}(k))}$, relate them to the $Z$-massive Laplacian of [BdTR17b], and extend to the full $Z$-invariant case the results of…

In this paper we develop a general approach to dimer models analogous to Krichever's scheme in the theory of integrable systems. We start with a Riemann surface and the simplest generic meromorphic functions on it and demonstrate how to…

We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of SL$_r$-webs, and is built upon the r-fold dimer model on the network. When…

Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…

Using Kasteleyn's determinant method, we count perfect matchings of rectangular subgraphs of the square grid.

Linde, Moore, and Nordahl introduced a generalisation of the honeycomb dimer model to higher dimensions. The purpose of this article is to describe a number of structural properties of this generalised model. First, it is shown that the…

We consider the inverse problem of finding matrix valued edge or nodal quantities in a graph from measurements made at a few boundary nodes. This is a generalization of the problem of finding resistors in a resistor network from voltage and…

We conjecture that for all regular lattices b(n) is asymptotically of the form in eq.(A1). (-1)^{n+1} b(n) = exp( k(-1) n + k(0) ln(n) + k(1) / n + k(2) / n^(2)...) (A1) We restrict testing this to lattices for which we know the first 20…

Fox coloring provides a combinatorial framework for studying dihedral representations of the knot group. The less well-known concept of Dehn coloring captures the same data. Recent work of Carter-Silver-Williams clarifies the relationship…

We study a generalisation of the double-dimer model that encompasses several models of interest, including the monomer double-dimer model, spatial random permutations, the dimer model, and the spin $O(N)$ model, and which is also related to…

We study asymptotics of perfect matchings on a large class of graphs called the contracting square-hexagon lattice, which is constructed row by row from either a row of a square grid or a row of a hexagonal lattice. We assign the graph…

We consider the monomer-dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights, with the restriction that the only non-zero monomer weights are those on the boundary. We prove a Pfaffian formula…

We analyze height fluctuations in Aztec diamond dimer models with nearly arbitrary periodic edge weights. We show that the centered height function approximates the sum of two independent components: a Gaussian free field on the multiply…

We present a geometrical approach for studying dimers. We introduce a connection for dimer problems on bipartite and non-bipartite graphs. In the bipartite case the connection is flat but has non-trivial ${\bf Z}_2$ holonomy round certain…