scholarly journals A Generalization of Aztec Diamond Theorem, Part I

10.37236/3691 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Square Lattice ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
New Family ◽  
Schröder Paths

We consider a new family of 4-vertex regions with zigzag boundary on the square lattice with diagonals drawn in. By proving that the number of tilings of the new regions is given by a power 2, we generalize both Aztec diamond theorem and Douglas' theorem. The proof extends an idea of Eu and Fu for Aztec diamonds, by using a  bijection between domino tilings and non-intersecting Schröder paths, then applying Lindström-Gessel-Viennot methodology.

scholarly journals A Bijection Proving the Aztec Diamond Theorem by Combing Lattice Paths

10.37236/2809 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Frédéric Bosio ◽  
Marc A. A. Van Leeuwen
Keyword(s):  
Lattice Paths ◽  
Square Grid ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
Bijective Proof ◽  
Schröder Paths ◽  
Intersecting Families ◽  
Grid Points

We give a bijective proof of the Aztec diamond theorem, stating that there are $2^{n(n+1)/2}$ domino tilings of the Aztec diamond of order $n$. The proof in fact establishes a similar result for non-intersecting families of $n+1$ Schröder paths, with horizontal, diagonal or vertical steps, linking the grid points of two adjacent sides of an $n\times n$ square grid; these families are well known to be in bijection with tilings of the Aztec diamond. Our bijection is produced by an invertible "combing'' algorithm, operating on families of paths without non-intersection condition, but instead with the requirement that any vertical steps come at the end of a path, and which are clearly $2^{n(n+1)/2}$ in number; it transforms them into non-intersecting families.

Spanning Trees of the Generalised Union Jack Lattice

2016 ◽  
Vol 71 (4) ◽  
pp. 331-335
Author(s):  
Lingyun Chen ◽  
Weigen Yan
Keyword(s):  
Boundary Condition ◽  
Spanning Trees ◽  
Square Lattice ◽  
Aztec Diamond

AbstractThe Union Jack lattice UJL(n, m) with toroidal boundary condition can be obtained from an n×m square lattice with toroidal boundary condition by inserting a new vertex vf to each face f and adding four edges (vf, ui(f)), where u1(f), u2(f), u3(f), and u4(f) are four vertices on the boundary of f. The Union Jack lattice has been studied extensively by statistical physicists. In this article, we consider the problem of enumeration of spanning trees of the so-called generalised Union Jack lattice UDn, which is obtained from the Aztec diamond $AD_n^t$ of order n with toroidal boundary condition by inserting a new vertex vf to each face f and adding four edges (vf, ui(f)), where u1(f), u2(f), u3(f) and u4(f) are four vertices on the boundary of f.

scholarly journals Symmetric matrices, Catalan paths, and correlations

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Emmanuel Tsukerman ◽  
Lauren Williams ◽  
Bernd Sturmfels
Keyword(s):  
Laurent Polynomial ◽  
Symmetric Matrices ◽  
Correlation Matrices ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
International Audience ◽  
Square Matrix

International audience Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.

scholarly journals Weakly directed self-avoiding walks

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Axel Bacher ◽  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Square Lattice ◽  
Growth Constant ◽  
Complex Singularity ◽  
Singularity Structure ◽  
New Family ◽  
End Distance ◽  
International Audience ◽  
Directed Walks ◽  
Self Avoiding Walks

International audience We define a new family of self-avoiding walks (SAW) on the square lattice, called $\textit{weakly directed walks}$. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model. Nous définissons une nouvelle famille de chemins auto-évitants (CAE) sur le réseau carré, appelés $\textit{chemins faiblement dirigés}$. Ces chemins ont une caractérisation simple en termes des ponts irréductibles qui les composent. Nous déterminons leur série génératrice. Cette série a une structure de singularité complexe et n'est en particulier pas D-finie. La constante de croissance est environ 2,54, ce qui est supérieur à toutes les familles naturelles de SAW étudiées jusqu'à présent, mais inférieur aux CAE généraux (dont la constante est environ 2,64). Nous prouvons également que la distance moyenne entre les extrémités du chemin croît linéairement. Enfin, nous étudions une variante diagonale du modèle.

scholarly journals Domino tilings of the expanded Aztec diamond

2018 ◽  
Vol 341 (4) ◽  
pp. 1185-1191 ◽  
Author(s):  
Seungsang Oh
Keyword(s):  
Domino Tilings ◽  
Aztec Diamond

scholarly journals Domino Shuffling on Novak Half-Hexagons and Aztec Half-Diamonds

10.37236/668 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Eric Nordenstam ◽  
Benjamin Young
Keyword(s):  
The Arctic ◽  
Arctic Circle ◽  
Aztec Diamond ◽  
Circle Theorem ◽  
New Family

We explore the connections between the well-studied Aztec Diamond graphs and a new family of graphs called the Half-Hexagons, discovered by Jonathan Novak. In particular, both families of graphs have very simple domino shuffling algorithms, which turn out to be intimately related. This connection allows us to prove an "arctic parabola" theorem for the Half-Hexagons as a corollary of the Arctic Circle theorem for the Aztec Diamond.

scholarly journals A Simple Proof of the Aztec Diamond Theorem

10.37236/1915 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu
Keyword(s):  
Simple Proof ◽  
Lattice Paths ◽  
Hankel Determinants ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
Schröder Numbers

Based on a bijection between domino tilings of an Aztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given by means of Hankel determinants of the large and small Schröder numbers.

scholarly journals Correlations for the Novak process

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Eric Nordenstam ◽  
Benjamin Young
Keyword(s):  
Correlation Functions ◽  
Binomial Coefficient ◽  
Domino Tilings ◽  
Aztec Diamond ◽  
The Matrix ◽  
International Audience ◽  
Lozenge Tilings

International audience We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process. This model was introduced by Nordenstam and Young (2011) and has many intriguing similarities with a more well-studied model, domino tilings of the Aztec diamond. The most difficult step in the present paper is to compute the inverse of the matrix whose (i,j)-entry is the binomial coefficient $C(A, B_j-i)$ for indeterminate variables $A$ and $B_1, \dots , B_n.$ Nous étudions des pavages aléatoires d'une region dans le plan par des losanges qui s'appelle le demi-hexagone de Novak et nous calculons les corrélations de ce processus. Ce modèle a été introduit par Nordenstam et Young (2011) et a plusieurs similarités des pavages aléatoires d'un diamant aztèque par des dominos. La partie la plus difficile de cet article est le calcul de l'inverse d'une matrice ou l’élément (i,j) est le coefficient binomial $C(B_j-i, A)$ pour des variables $A$ et $B_1, \dots , B_n$ indéterminés.

Microstructure and crystal symmetry of Pb2Sr2(Y/Ca)Cu3O9−δ

1990 ◽  
Vol 48 (4) ◽  
pp. 64-65
Author(s):  
Y. P. Lin ◽  
J. S. Xue ◽  
J. E. Greedan
Keyword(s):  
Electron Diffraction ◽  
High Temperature Superconductors ◽  
Carbon Film ◽  
Basic Structure ◽  
Crystal Symmetry ◽  
X Ray Diffraction ◽  
X Ray ◽  
Space Groups ◽  
New Family ◽  
Convergent Beam

A new family of high temperature superconductors based on Pb2Sr2YCu3O9−δ has recently been reported. One method of improving Tc has been to replace Y partially with Ca. Although the basic structure of this type of superconductors is known, the detailed structure is still unclear, and various space groups has been proposed. In our work, crystals of Pb2Sr2YCu3O9−δ with dimensions up to 1 × 1 × 0.25.mm and with Tc of 84 K have been grown and their superconducting properties described. The defects and crystal symmetry have been investigated using electron microscopy performed on crushed crystals supported on a holey carbon film.Electron diffraction confirmed x-ray diffraction results which showed that the crystals are primitive orthorhombic with a=0.5383, b=0.5423 and c=1.5765 nm. Convergent Beam Electron Diffraction (CBED) patterns for the and axes are shown in Figs. 1 and 2 respectively.

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