ID: math-ph/0311005

Dimers and Amoebae

November 5, 2003

View on ArXiv

Similar papers 2

Dimer Models and Conformal Structures

April 6, 2020

83% Match
Kari Astala, Erik Duse, ... , Zhong Xiao
Analysis of PDEs
Complex Variables
Mathematical Physics

In this work we study the variational problem associated to dimer models, a class of models from integrable probability and statistical mechanics in dimension two which have been the focus of intense research efforts over the last decades. These models give rise to an infinite family of non-differentiable functionals on Lipschitz functions with gradient constraint, determined by solutions of the Dirichlet problem on compact convex polygons for a class of Monge-Amp\`ere equati...

Find SimilarView on arXiv

Dimers on Riemann surfaces I: Temperleyan forests

August 2, 2019

83% Match
Nathanaël Berestycki, Benoit Laslier, Gourab Ray
Probability
Mathematical Physics

This is the first article in a series of two papers in which we study the Temperleyan dimer model on an arbitrary bounded Riemann surface of finite topolgical type. The end goal of both papers is to prove the convergence of height fluctuations to a universal and conformally invariant scaling limit. In this part we show that the dimer model on the Temperleyan superposition of a graph embedded on the surface and its dual is well posed, provided that we remove an appropriate num...

Find SimilarView on arXiv

Dimers on graphs in non-orientable surfaces

April 30, 2008

82% Match
David Cimasoni
Geometric Topology
Mathematical Physics

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 using purely geometrical methods. The key step in the proof consists of a correspondence between some orientations on G and the set of pin^- structures on S. This generalizes (and simplifies) the results of a previous paper [2].

Find SimilarView on arXiv

Dimers on surface graphs and spin structures. II

April 2, 2007

82% Match
David Cimasoni, Nicolai Reshetikhin
Geometric Topology
Mathematical Physics

In a previous paper, we showed how certain orientations of the edges of a graph G embedded in a closed oriented surface S can be understood as discrete spin structures on S. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on G. In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on di...

Find SimilarView on arXiv

Double dimers on planar hyperbolic graphs via circle packings

June 18, 2024

82% Match
Gourab Ray
Probability
Mathematical Physics

In this article we study the double dimer model on hyperbolic Temperleyan graphs via circle packings. We prove that on such graphs, the weak limit of the dimer model exists if and only if the removed black vertex from the boundary of the exhaustion converges to a point on the unit circle in the circle packing representation of the graph. One of our main results is that for such measures, we prove that the double dimer model has no bi-infinite path almost surely. Along the way...

Find SimilarView on arXiv

Height fluctuations in the honeycomb dimer model

May 19, 2004

82% Match
Richard Kenyon
Mathematical Physics
Probability

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 $\epsilon\to0$, Cohn, Kenyon and Propp [CKP] showed the almost sure convergence of a random surface to a non-random limit shape $\Sigma_0$. In [KO], Okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions, in particular constructing a natural conformal structure on them. We show...

Find SimilarView on arXiv

Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces

September 13, 2021

82% Match
Dmitry Chelkak, Benoît Laslier, Marianna Russkikh
Probability
Complex Variables
Mathematical Physics

This is the second paper in the series devoted to the study of the dimer model on t-embeddings of planar bipartite graphs. We introduce the notion of perfect t-embeddings and assume that the graphs of the associated origami maps converge to a Lorentz-minimal surface $\mathrm{S}_\xi$ as $\delta\to 0$. In this setup we prove (under very mild technical assumptions) that the gradients of the height correlation functions converge to those of the Gaussian Free Field defined in the ...

Find SimilarView on arXiv

Weakly non-planar dimers

July 21, 2022

82% Match
Alessandro Roma Tre Giuliani, Bruno Roma Tre Renzi, Fabio TU Wien Toninelli
Probability
Mathematical Physics

We study a model of fully-packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from $\mathbb Z^2$ via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight $\lambda$ of the non-planar edges is small enough, a suitably defined height function scales on large distances to the Gaussian Free Field with a $\lambda$-depend...

Find SimilarView on arXiv

Limit shapes and the complex burgers equation

July 1, 2005

82% Match
Richard Kenyon, Andrei Okounkov
Mathematical Physics
Probability

In this paper we study surfaces in R^3 that arise as limit shapes in a class of random surface models arising from dimer models. The limit shapes are minimizers of a surface tension functional, that is, they minimize, for fixed boundary conditions, the integral of a quantity (the surface tension) depending only on the slope of the surface. The surface tension as a function of the slope has singularities and is not strictly convex, which leads to formation of facets and edges ...

Find SimilarView on arXiv

Random Surfaces

April 3, 2003

81% Match
Scott Sheffield
Probability
Mathematical Physics

We study "random surfaces," which are random real (or integer) valued functions on Z^d. The laws are determined by convex, nearest neighbor, difference potentials that are invariant under translation by a full-rank sublattice L of Z^d; they include many discrete and continuous height models (e.g., domino tilings, square ice, the harmonic crystal, the Ginzburg-Landau grad-phi interface model, the linear solid-on-solid model) as special cases. A gradient phase is an L-ergodic...

Find SimilarView on arXiv