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 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.

scholarly journals Cluster algebras of unpunctured surfaces and snake graphs

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Gregg Musiker ◽  
Ralf Schiffler
Keyword(s):  
Perfect Matchings ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Polynomial Expansion ◽  
International Audience ◽  
Polynomial Expansions

International audience We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma}$ . Nous étudions des algèbres amassées avec coefficients principaux associées aux surfaces. Nous présentons une formule directe pour les développements de Laurent des variables amassées dans ces algèbres en terme de couplages parfaits d'un certain graphe $G_{T,\gamma}$ que l'on construit a partir de la surface en recollant des pièces élémentaires que l'on appelle carreaux. Nous donnons aussi une seconde formule pour ces développements en termes de sous-graphes de $G_{T,\gamma}$ .

scholarly journals Three notions of tropical rank for symmetric matrices

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Dustin Cartwright ◽  
Melody Chan
Keyword(s):  
Convex Hull ◽  
Symmetric Matrices ◽  
Star Tree ◽  
Close Study ◽  
International Audience ◽  
Donnent Lieu ◽  
Tropical Varieties

International audience We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull. Nous introduisons et étudions trois notions différentes de rang tropical pour des matrices symétriques et des matrices de dissimilarité, en utilisant des décompositions minimales en matrices symétriques de rang 1, en matrices d'arbres étoiles, et en matrices d'arbres. Nos résultats donnent lieu à une étude détaillée des ensembles des sécantes tropicales de certaines jolies variétés tropicales, y compris la grassmannienne tropicale. En particulier, nous déterminons la dimension de chaque ensemble des sécantes, l'enveloppe convexe de la variété, ainsi que, dans la plupart des cas, le plus petit ensemble des sécantes qui est égal à l'enveloppe convexe.

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 The Zeta Mahler measure of $(z^n − 1)/(z − 1)$

2019 ◽  
Author(s):  
Arunabha Biswas ◽  
M Ram Murty
Keyword(s):  
Unit Circle ◽  
Generating Function ◽  
Laurent Polynomial ◽  
Mahler Measure ◽  
International Audience

International audience We consider the k-higher Mahler measure $m_k (P) $ of a Laurent polynomial $P$ as the integral of ${\log}^k |P | $ over the complex unit circle and zeta Mahler measure as the generating function of the sequence ${m_k (P)}$. In this paper we derive a few properties of the zeta Mahler measure of the polynomial $P_n (z) := (z^N − 1)/(z − 1) $ and propose a conjecture.

Compounds of Skew-Symmetric Matrices

1964 ◽  
Vol 16 ◽  
pp. 473-478 ◽  
Author(s):  
Marvin Marcus ◽  
Adil Yaqub
Keyword(s):  
Symmetric Matrices ◽  
The Real ◽  
Real Solutions ◽  
Interesting Paper ◽  
The Matrix ◽  
Image Position ◽  
Square Matrix

In a recent interesting paper (3) H. Schwerdtfeger answered a question of W. R. Utz (4) on the structure of the real solutions A of A* = B, where A is skew-symmetric. (Utz and Schwerdtfeger call A* the "adjugate" of A ; A* is the n-square matrix whose (i, j) entry is (—1)i+j times the determinant of the (n — 1)-square matrix obtained by deleting row i and column j of A. The word "adjugate," however, is more usually applied to the matrix (AT)*, where AT denotes the transposed matrix of A ; cf. (1, 2).)The object of the present paper is to find all real n-square skew-symmetric solutions A to the equation

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.

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