ID: math-ph/0311005

Dimers and Amoebae

November 5, 2003

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Dimers on surface graphs and spin structures. I

August 31, 2006

81% Match
David Cimasoni, Nicolai Reshetikhin
Geometric Topology
Mathematical Physics

Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of discrete spin structures.

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Statistical mechanics of random graphs

December 18, 2003

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Zdzislaw Burda, Jerzy Jurkiewicz, Andre Krzywicki
Statistical Mechanics

We discuss various aspects of the statistical formulation of the theory of random graphs, with emphasis on results obtained in a series of our recent publications.

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Asymptotics of height change on toroidal Temperleyan dimer models

July 23, 2014

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Julien Dubédat, Reza Gheissari
Probability

The dimer model is an exactly solvable model of planar statistical mechanics. In its critical phase, various aspects of its scaling limit are known to be described by the Gaussian free field. For periodic graphs, criticality is an algebraic condition on the spectral curve of the model, determined by the edge weights; isoradial graphs provide another class of critical dimer models, in which the edge weights are determined by the local geometry. In the present article, we consi...

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Two dimensional dimers beyond planarity

December 28, 2023

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Lorenzo Taggi, Wei Wu
Probability

We consider a generalisation of the double dimer model which includes several models of interest, such as the monomer double dimer model, the dimer model, the Spin O(N) model, and it is related to the loop O(N) model. We prove that on two-dimension like graphs (such as slabs), both the correlation function and the probability that a loop visits two vertices converge to zero as the distance between such vertices gets large. Our analysis is by introducing a new (complex) spin r...

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The Height fluctuations of an off-critical dimer model on the square grid

February 14, 2011

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Sunil Chhita
Probability

The dimer model on a planar bipartite graph can be viewed as a random surface measure. We study these fluctuations for a dimer model on the square grid with two different classes of weights and provide a condition for their equivalence. In the thermodynamic limit and scaling window, these height fluctuations are shown to be non-Gaussian. They are also rotationally invariant for a certain choice of weights. Finally, we show that the height difference between any two points is ...

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The critical Z-invariant Ising model via dimers: the periodic case

December 19, 2008

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Cédric Boutillier, Tilière Béatrice de
Probability
Mathematical Physics

We study a large class of critical two-dimensional Ising models namely critical Z-invariant Ising models on periodic graphs, example of which are the classical square, triangular and honeycomb lattice at the critical temperature. Fisher introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corr...

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A generalisation of the honeycomb dimer model to higher dimensions

May 30, 2019

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Piet Lammers
Probability

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 samples of the model are in one-to-one correspondence with the perfect matchings of a hypergraph. This leads to a generalised Kasteleyn theory: the partition function of the model equals the Cayley hyperdeterminant of the adjacency hypermatrix...

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Identities between dimer partition functions on different surfaces

August 2, 2016

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David Cimasoni, Anh Minh Pham
Combinatorics
Mathematical Physics

Given a weighted graph $G$ embedded in a non-orientable surface $\Sigma$, one can consider the corresponding weighted graph $\widetilde{G}$ embedded in the so-called orientation cover $\widetilde\Sigma$ of $\Sigma$. We prove identities relating twisted partition functions of the dimer model on these two graphs. When $\Sigma$ is the M\"obius strip or the Klein bottle, then $\widetilde\Sigma$ is the cylinder or the torus, respectively, and under some natural assumptions, these ...

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Free boundary dimers: random walk representation and scaling limit

February 25, 2021

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Nathanael Berestycki, Marcin Lis, Wei Qian
Probability
Mathematical Physics

We study the dimer model on subgraphs of the square lattice in which vertices on a prescribed part of the boundary (the free boundary) are possibly unmatched. Each such unmatched vertex is called a monomer and contributes a fixed multiplicative weight $z>0$ to the total weight of the configuration. A bijection described by Giuliani, Jauslin and Lieb relates this model to a standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this dimer model describes a ...

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Pattern densities in fluid dimer models

March 14, 2006

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Cedric Boutillier
Probability
Mathematical Physics

In this paper, we introduce a family of observables for the dimer model on a bi-periodic bipartite planar graph, called pattern density fields. We study the scaling limit of these objects for liquid and gaseous Gibbs measures of the dimer model, and prove that they converge to a linear combination of a derivative of the Gaussian massless free field and an independent white noise.

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