Dimers and Amoebae

We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Despite the lack of periodicity we prove that each infinite tiling admits 'perfect matchings' in which every vertex is touched by one dimer. We introduce an auxiliary 'AB$^*$' tiling obtained from the AB tiling by deleting all 8-fold coordinated vertices. The AB$^*$ tiling is again two-dimensional, infinite, and quasiperiodic. The AB$^*$ tiling has a single connected component, which admit...

We study the number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths, in which the tiles are "vertically decreasing". We use these quantities to compute pairing probabilities in the double-dimer model: Given a planar bipartite graph $G$ with special vertices, called nodes, on the outer face, the double-dimer model is formed by the superposition of a uniformly random dimer configuration (perfect matching) of $G$ together with a random dimer configuratio...

We consider the disordered monomer-dimer model on cylinder graphs $\mathcal{G}_n$, i.e., graphs given by the Cartesian product of the line graph on $n$ vertices, and a deterministic graph. The edges carry i.i.d. random weights, and the vertices also carry i.i.d. random weights, not necessarily from the same distribution. Given the random weights, we define a Gibbs measure on the space of monomer-dimer configurations on $\mathcal{G}_n$. We show that the associated free energy ...

Given a bounded Riemann surface $M$ of finite topological type, we show the existence of a universal and conformally invariant scaling limit for the Temperleyan cycle-rooted spanning forest on any sequence of graphs which approximate $M$ in a reasonable sense (essentially, the invariance principle holds and the walks satisfy a crossing assumption). In combination with the companion paper arxiv:1908.00832, this proves the existence of a universal, conformally invariant scaling...

We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of $\mathbb Z^2$, i.e. subsets of edges such that each vertex is covered exactly once ("close-packing" condition). Dimer configurations are in bijection with discrete height functions, defined on faces $\boldsymbol{\xi}$ of $\mathbb Z^2$. The non-interacting model is "integrable" and solvable via Kasteleyn theory; it is known that all the momen...

We consider the monomer-dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises the one obtained by Giuliani, Jauslin and Lieb for graphs embedded in the disk. Our proof is based on an extension of a bijective method mentioned in their paper, together with the Pfaffian formula for the dimer partition fun...

Amoebae from tropical geometry and the Mahler measure from number theory play important roles in quiver gauge theories and dimer models. Their dependencies on the coefficients of the Newton polynomial closely resemble each other, and they are connected via the Ronkin function. Genetic symbolic regression methods are employed to extract the numerical relationships between the 2d and 3d amoebae components and the Mahler measure. We find that the volume of the bounded complement...

We present a general result which shows that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds true assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations. This includes the c...

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 periodic edge weights with period $1\times n$, and consider the probability measure of perfect matchings in which the probability of each configuration is proportional to the product of edge weights. We show that the partition function of perfe...

We discuss some diverse open problems in the dimer model, motivated by a geometric viewpoint. This is part of a conference proceedings for the OPAC 2022 conference.