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 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 Kasteleyn Cokernels

10.37236/1645 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Greg Kuperberg
Keyword(s):  
Normal Form ◽  
Smith Normal Form ◽  
Principal Ideal Domain ◽  
Aztec Diamond ◽  
Plane Partitions ◽  
Symmetry Classes ◽  
Matrix Operations ◽  
The Matrix ◽  
Lozenge Tilings ◽  
Planar Regions

We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in enumerating matchings of planar graphs, up to matrix operations on their rows and columns. If such a matrix is defined over a principal ideal domain, this is equivalent to considering its Smith normal form or its cokernel. Many variations of the enumeration methods result in equivalent matrices. In particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus matrices. We apply these ideas to plane partitions and related planar of tilings. We list a number of conjectures, supported by experiments in Maple, about the forms of matrices associated to enumerations of plane partitions and other lozenge tilings of planar regions and their symmetry classes. We focus on the case where the enumerations are round or $q$-round, and we conjecture that cokernels remain round or $q$-round for related "impossible enumerations" in which there are no tilings. Our conjectures provide a new view of the topic of enumerating symmetry classes of plane partitions and their generalizations. In particular we conjecture that a $q$-specialization of a Jacobi-Trudi matrix has a Smith normal form. If so it could be an interesting structure associated to the corresponding irreducible representation of SL$(n,C)$. Finally we find, with proof, the normal form of the matrix that appears in the enumeration of domino tilings of an Aztec diamond.

Euclidean path integrals and quantum mechanics (QM)

2021 ◽  
pp. 18-41
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Correlation Functions ◽  
Path Integrals ◽  
Continuous Phase ◽  
Functional Integral ◽  
Smooth Transition ◽  
Path Integral Representation ◽  
The Matrix ◽  
Functional Measure

Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.

scholarly journals W-INFINITY WARD IDENTITIES AND CORRELATION FUNCTIONS IN THE c=1 MATRIX MODEL

1992 ◽  
Vol 07 (11) ◽  
pp. 937-953 ◽  
Author(s):  
SUMIT R. DAS ◽  
AVINASH DHAR ◽  
GAUTAM MANDAL ◽  
SPENTA R. WADIA
Keyword(s):  
Matrix Model ◽  
Correlation Functions ◽  
Scaling Limit ◽  
Three Dimensional ◽  
Dimensional Theory ◽  
Fermionic Field ◽  
Ward Identities ◽  
The Matrix ◽  
Dependent Potential ◽  
General Time

We explore consequences of W-infinity symmetry in the fermionic field theory of the c=1 matrix model. We derive exact Ward identities relating correlation functions of the bilocal operator. These identities can be expressed as equations satisfied by the effective action of a three-dimensional theory and contain non-perturbative information about the model. We use these identities to calculate the two-point function of the bilocal operator in the double scaling limit. We extract the operator whose two-point correlator has a single pole at an (imaginary) integer value of the energy. We then rewrite the W-infinity charges in terms of operators in the matrix model and use this to derive constraints satisfied by the partition function of the matrix model with a general time dependent potential.

scholarly journals QCD Factorization and PDFs from Lattice QCD Calculation

2015 ◽  
Vol 37 ◽  
pp. 1560041 ◽  
Author(s):  
Yan-Qing Ma ◽  
Jian-Wei Qiu
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Lattice Qcd ◽  
Correlation Functions ◽  
Parton Distribution ◽  
Distribution Functions ◽  
Matrix Elements ◽  
Parton Distribution Functions ◽  
Qcd Factorization ◽  
The Matrix

In this talk, we review a QCD factorization based approach to extract parton distribution and correlation functions from lattice QCD calculation of single hadron matrix elements of quark-gluon operators. We argue that although the lattice QCD calculations are done in the Euclidean space, the nonperturbative collinear behavior of the matrix elements are the same as that in the Minkowski space, and could be systematically factorized into parton distribution functions with infrared safe matching coefficients. The matching coefficients can be calculated perturbatively by applying the factorization formalism on to asymptotic partonic states.

scholarly journals A Formal Approach to the Description and Manipulation of Structured Mathematical Objects

2005 ◽  
Vol Volume 3, Special Issue... ◽  
Author(s):  
Bernard Fotsing Talla ◽  
Georges-Edouard Kouamou
Keyword(s):  
Square Root ◽  
Two Dimensional ◽  
Attribute Grammars ◽  
Formal Approach ◽  
Mathematical Symbols ◽  
Nous Montrons ◽  
The Matrix ◽  
Mathematical Formulas ◽  
International Audience ◽  
Structured Objects

International audience We present in this paper a formal approach of description, posting and handling of the mathematical structured objects; based on the formalism of attribute grammars. We are interested particularly in the problem of two-dimensional and bidirectional posting of certain expressions and mathematical formulas. Indeed, in more of the two-dimensional character that presents certain mathematical symbols like the square root or the matrix, we also note the problem of posting rightto-left of an Arab text in a context planned for a posting left-to-right of an Indo-European text, or a bidirectional posting mixing the two modes. After a study of some solutions suggested in the literature, we show how the method of attribute grammars adapts easily to these types of problem. Nous présentons dans ce papier une approche formelle de description, d'affichage et de manipulation des objets structurés mathématiques ; basée sur le formalisme des grammaires attribuées. Nous nous intéressons particulièrement au problème d'affichage bidimensionnel et bidirectionnel de certaines expressions et formules mathématiques. En effet, en plus du caractère bidimensionnel que présentent certains symboles comme la racine carrée ou la matrice, on note le problème d'affichage de droite à gauche d'un texte arabe dans un contexte prévu pour un affichage de gauche à droite d'un texte indo-européen, ou encore un affichage bidirectionnel mélangeant les deux modes. Après une étude de quelques méthodes proposées dans la littérature, nous montrons comment la méthode des grammaires attribuées s'adapte facilement à ces types de problèmes.

scholarly journals Is the binomial coefficient $2n \choose n$ square free?

1995 ◽  
Vol Volume 18 ◽  
Author(s):  
G Velammal
Keyword(s):  
Binomial Coefficient ◽  
International Audience

International audience In this paper, we prove the Erd\"os conjecture that the binomial coefficient ${2n \choose n}$ is never square free, for all $n>4$.

scholarly journals Matrix-Ball Construction of affine Robinson-Schensted correspondence

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Michael Chmutov ◽  
Pavlo Pylyavskyy ◽  
Elena Yudovina
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Type A ◽  
Group Symmetry ◽  
Dominant Weight ◽  
Dominance Condition ◽  
The Matrix ◽  
International Audience ◽  
Affine Type

International audience In his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson- Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combi- natorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between extended affine symmetric group and triples (P, Q, ρ) where P and Q are tabloids and ρ is a dominant weight. The weights ρ get a natural interpretation in terms of the Affine Matrix-Ball Construction. Finally, we prove that fibers of the inverse map possess a Weyl group symmetry, explaining the dominance condition on weights.

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