# Dimer Models and Conformal Structures.

@article{Astala2020DimerMA, title={Dimer Models and Conformal Structures.}, author={Kari Astala and Erik Duse and Istv'an Prause and Xiao Zhong}, journal={arXiv: Analysis of PDEs}, year={2020} }

Dimer models have been the focus of intense research efforts over the last years. This paper grew out of an effort to develop new methods to study minimizers or the asymptotic height functions of general dimer models and the geometry of their frozen boundaries. We prove a complete classification of the regularity of minimizers and frozen boundaries for all dimer models for a natural class of polygonal (simply or multiply connected) domains much studied in numerical simulations and elsewhere…

## Figures from this paper

## 11 Citations

Local geometry of the rough-smooth interface in the two-periodic Aztec diamond

- MathematicsThe Annals of Applied Probability
- 2022

Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially;…

The domino shuffling algorithm and Anisotropic KPZ stochastic growth

- Mathematics
- 2019

The domino-shuffling algorithm can be seen as a stochastic process describing the irreversible growth of a $(2+1)$-dimensional discrete interface. Its stationary speed of growth $v_{\mathtt w}(\rho)$…

Homogenization of iterated singular integrals with applications to random quasiconformal maps.

- Mathematics
- 2020

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let
$(F_j)_{j…

The genus-zero five-vertex model

- Mathematics
- 2021

We study the free energy and limit shape problem for the five-vertex model with periodic “genus zero” weights. We derive the exact phase diagram, free energy and surface tension for this model. We…

Improved Hölder regularity for strongly elliptic PDEs

- MathematicsJournal de Mathématiques Pures et Appliquées
- 2020

Gradient variational problems in $\mathbb{R}^2$

- Mathematics
- 2020

We prove a new integrability principle for gradient variational problems in $\mathbb{R}^2$, showing that solutions are explicitly parameterized by $\kappa$-harmonic functions, that is, functions…

Conformal Structure of Autonomous Leray-Lions Equations in the Plane and Linearisation by Hodograph Transform

- Mathematics
- 2022

We give suﬃcient conditions for when an autonomous elliptic Leray-Lions equation in the plane has a conformal structure. This allows the Leray-Lions equation to be linearised in a special form…

Limit Shape of Perfect Matchings on Rail-Yard Graphs

- MathematicsInternational Mathematics Research Notices
- 2022

We obtain limit shape of perfect matchings on a large class of rail-yard graphs with right boundary condition given by the empty partition and left boundary condition given by either by a staircase…

Dimer-dimer correlations at the rough-smooth boundary

- Computer Science
- 2021

It is found that dimer-dimer correlations of the two-periodic Aztec diamond initially decay exponentially when the magnetic coordinates are very close to the bounded complementary component of the associated amoebae, they then transition to a power law decay once far enough apart.

GOE fluctuations for the maximum of the top path in alternating sign matrices

- Mathematics
- 2021

The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter ∆. When ∆ = 0, the so-called free-fermion point, the model is in natural…

## References

SHOWING 1-10 OF 55 REFERENCES

Height Fluctuations in the Honeycomb Dimer Model

- Mathematics
- 2004

We study a model of random surfaces arising in the dimer model on the honeycomb lattice. For a fixed “wire frame” boundary condition, as the lattice spacing ϵ → 0, Cohn, Kenyon and Propp [3] showed…

Limit shapes and the complex Burgers equation

- Mathematics
- 2005

In this paper we study surfaces in R3 that arise as limit shapes in random surface models related to planar dimers. These limit shapes are surface tension minimizers, that is, they minimize a…

Dimers and amoebae

- Mathematics
- 2003

We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive…

Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices

- MathematicsAnnales de l'Institut Fourier
- 2021

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…

Universal edge fluctuations of discrete interlaced particle systems

- Mathematics
- 2017

We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently…

Domino tilings of the Aztec diamond with doubly periodic weightings

- MathematicsThe Annals of Probability
- 2019

In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular a family of models which, for any $ k \in \mathbb{N} $, includes models with $ k $ smooth…

A variational principle for domino tilings

- Mathematics
- 2000

1.1. Description of results. A domino is a 1 x 2 (or 2 x 1) rectangle, and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps. In…

The Shape of a Typical Boxed Plane Partition

- Mathematics
- 1998

Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all…