A Variational Principle for Domino Tilings
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From Tiling Problems to Random Matrices
From tiling problems to random matrices Tom Claeys Brussels Summer School of Mathematics September 4, 2018 Outline Tiling problems 1. Some fun to start with 2. Large random tilings Non-intersecting random walks and Brownian bridges 1. From hexagon tilings to nonintersecting paths 2. From nonintersecting paths to Brownian bridges Random matrices 1. From nonintersecting Brownian bridges to random matrix eigenvalues 2. Asymptotic properties of random matrix eigenvalues Tiling problems Given a twodimensional domain and a collection of (shapes of) tiles, we try to cover the domain with the tiles. Tiling problems Rules ✓ Allowed: translations of tiles ✓ Forbidden: rotations, intersections, cutting tiles, crossing the border of the domain Questions ✓ Solvability? Can the domain be covered with tiles? ✓ What is the number of possible tilings? ✓ Do different tilings share certain properties? Tiling problems A possible tiling Tiling problems A possible tiling Tiling problems A possible tiling Tiling problems A possible tiling Tiling problems A possible tiling Tiling problems A possible tiling Tilings of a 2 × n rectangle We speak of a domino tiling if the tiles or 1 × 2 and 2 × 1 rectangles. First training example: tiling of a rectangle of height 2 Tilings of a 2 × n rectangle Number of tilings 1 2 3 5 8 13 Number of tilings of a 2 × n rectangle The Fibonacci sequence! Checkerboard tilings Second training example: tiling a square of size 8 × 8 Checkerboard tilings One tiling of the checkerboard Checkerboard tilings And another one ... Number of tilings of a checkerboard? ✓ 12 988 816 Tilings of the mutilated checkerboard Third training example: two boxes removed Number of tilings of the mutilated checkerboard? Tilings of the mutilated checkerboard Third training example: two boxes removed Number of tilings of the mutilated checkerboard? ✓ None ! Tilings of the mutilated checkerboard Third training example: two boxes removed Theorem (GOMORY 1973) If we remove a white and a black box from the checkerboard, there exists always a tiling. -
Asymptotics of Multivariate Sequences, Part III: Quadratic Points
Asymptotics of multivariate sequences, part III: quadratic points Yuliy Baryshnikov 1 Robin Pemantle 2,3 ABSTRACT: We consider a number of combinatorial problems in which rational generating func- tions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deformations as well as Fourier-Laplace transforms of generalized functions. We apply the results of the theory to specific combinatorial problems, such as Aztec diamond tilings, cube groves, and multi-set permutations. Keywords: generalized function, Fourier transform, Fourier-Laplace, lacuna, multivariate generating function, hyperbolic polynomial, amoeba, Aztec diamond, quantum random walk, random tiling, cube grove. Subject classification: Primary: 05A16 ; Secondary: 83B20, 35L99. 1Bell Laboratories, Lucent Technologies, 700 Mountain Avenue, Murray Hill, NJ 07974-0636, [email protected] labs.com 2Research supported in part by National Science Foundation grant # DMS 0603821 3University of Pennsylvania, Department of Mathematics, 209 S. 33rd Street, Philadelphia, PA 19104 USA, pe- [email protected] Contents 1 Introduction 1 1.1 Background and motivation . 1 1.2 Methods and organization . 4 1.3 Comparison with other techniques . 7 2 Notation and preliminaries 8 2.1 The Log map and amoebas . 9 2.2 Dual cones, tangent cones and normal cones . 10 2.3 Hyperbolicity for homogeneous polynomials . 11 2.4 Hyperbolicity and semi-continuity for log-Laurent polynomials on the amoeba boundary 14 2.5 Critical points . 20 2.6 Quadratic forms and their duals . -
Tiling Problems: from Dominoes, Checkerboards, and Mazes to Discrete Geometry
TILING PROBLEMS: FROM DOMINOES, CHECKERBOARDS, AND MAZES TO DISCRETE GEOMETRY BERKELEY MATH CIRCLE 1. Looking for a number Consider an 8 × 8 checkerboard (like the one used to play chess) and consider 32 dominoes that each may cover two adjacent squares (horizontally or vertically). Question 1 (***). What is the number N of ways in which you can cover the checkerboard with the dominoes? This question is difficult, but we will answer it at the end of the lecture, once we will have familiarized with tilings. But before we go on Question 2. Compute the number of domino tilings of a n × n checkerboard for n small. Try n = 1; 2; 3; 4; 5. Can you do n = 6...? Question 3. Can you give lower and upper bounds on the number N? Guess an estimate of the order of N. 2. Other settings We can tile other regions, replacing the n × n checkerboard by a n × m rectangle, or another polyomino (connected set of squares). We say that a region is tileable by dominos if we can cover entirely the region with dominoes, without overlap. Question 4. Are all polyominoes tileable by dominoes? Question 5. Can you find necessary conditions that polyominoes have to satisfy in order to be tileable by dominos? What if we change the tiles? Try the following questions. Question 6. Can you tile a 8×8 checkerboard from which a square has been removed with triminos 1 × 3 (horizontal or vertical)? Question 7 (*). If you answered yes to the previous question, can you tell what are the only possible squares that can be removed so that the corresponding board is tileable? Date: February 14th 2012; Instructor: Adrien Kassel (MSRI); for any questions, email me. -
Treb All De Fide Gra U
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UPCommons. Portal del coneixement obert de la UPC Grau en Matematiques` T´ıtol:Tilings and the Aztec Diamond Theorem Autor: David Pardo Simon´ Director: Anna de Mier Departament: Mathematics Any academic:` 2015-2016 TREBALL DE FI DE GRAU Facultat de Matemàtiques i Estadística David Pardo 2 Tilings and the Aztec Diamond Theorem A dissertation submitted to the Polytechnic University of Catalonia in accordance with the requirements of the Bachelor's degree in Mathematics in the School of Mathematics and Statistics. David Pardo Sim´on Supervised by Dr. Anna de Mier School of Mathematics and Statistics June 28, 2016 Abstract Tilings over the plane R2 are analysed in this work, making a special focus on the Aztec Diamond Theorem. A review of the most relevant results about monohedral tilings is made to continue later by introducing domino tilings over subsets of R2. Based on previous work made by other mathematicians, a proof of the Aztec Dia- mond Theorem is presented in full detail by completing the description of a bijection that was not made explicit in the original work. MSC2010: 05B45, 52C20, 05A19. iii Contents 1 Tilings and basic notions1 1.1 Monohedral tilings............................3 1.2 The case of the heptiamonds.......................8 1.2.1 Domino Tilings.......................... 13 2 The Aztec Diamond Theorem 15 2.1 Schr¨odernumbers and Hankel matrices................. 16 2.2 Bijection between tilings and paths................... 19 2.3 Hankel matrices and n-tuples of Schr¨oderpaths............ 27 v Chapter 1 Tilings and basic notions The history of tilings and patterns goes back thousands of years in time. -
Aztec Diamond Theorem by Combing Lattice Paths
A BIJECTION PROVING THE AZTEC DIAMOND THEOREM BY COMBING LATTICE PATHS FRÉDÉRIC BOSIO AND MARC VAN LEEUWEN Abstract. We give a bijective proof of the Aztec diamond theorem, stating that there are 2n(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 × 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 2n(n+1)/2 in number; it transforms them into non-intersecting families. 1. Introduction The term “Aztec diamond”, introduced by Elkies, Kuperberg, Larsen and Propp [EKLP92], refers to a diamond-shaped set of squares in the plane, ob- tained by taking a triangular array of squares aligned against two perpendicular axes, and completing it with its mirror images in those two axes; the order of the diamond is the number of squares along each of the sides of the triangular array. Their main result concerns counting the number of domino tilings (i.e., partitions into subsets of two adjacent squares) of the Aztec diamond. n+1 Theorem 1 (Aztec diamond theorem). There are exactly 2( 2 ) domino tilings of the Aztec diamond of order n. -
[Math.CO] 25 Jan 2005 Eirshlra H A/Akct Mathematics City IAS/Park As the Institute
Tilings∗ Federico Ardila † Richard P. Stanley‡ 1 Introduction. 4 3 6 Consider the following puzzle. The goal is to 5 2 cover the region 1 7 For that reason, even though this is an amusing puzzle, it is not very intriguing mathematically. This is, in any case, an example of a tiling problem. A tiling problem asks us to cover a using the following seven tiles. given region using a given set of tiles, com- pletely and without any overlap. Such a cov- ering is called a tiling. Of course, we will fo- 1 2 3 4 cus our attention on specific regions and tiles which give rise to interesting mathematical problems. 6 7 5 Given a region and a set of tiles, there are many different questions we can ask. Some The region must be covered entirely with- of the questions that we will address are the out any overlap. It is allowed to shift and following: arXiv:math/0501170v3 [math.CO] 25 Jan 2005 rotate the seven pieces in any way, but each Is there a tiling? piece must be used exactly once. • One could start by observing that some How many tilings are there? • of the pieces fit nicely in certain parts of the About how many tilings are there? region. However, the solution can really only • be found through trial and error. Is a tiling easy to find? • ∗ This paper is based on the second author’s Clay Is it easy to prove that a tiling does not Public Lecture at the IAS/Park City Mathematics • Institute in July, 2004. -
Perfect Sampling Algorithm for Schur Processes Dan Betea, Cédric Boutillier, Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel, Mirjana Vuletić
Perfect sampling algorithm for Schur processes Dan Betea, Cédric Boutillier, Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel, Mirjana Vuletić To cite this version: Dan Betea, Cédric Boutillier, Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel, et al.. Perfect sampling algorithm for Schur processes. Markov Processes And Related Fields, Polymat Publishing Company, 2018, 24 (3), pp.381-418. hal-01023784 HAL Id: hal-01023784 https://hal.archives-ouvertes.fr/hal-01023784 Submitted on 5 Sep 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Perfect sampling algorithms for Schur processes D. Betea∗ C. Boutillier∗ J. Bouttiery G. Chapuyz S. Corteelz M. Vuleti´cx August 31, 2018 Abstract We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing con- ditions. This class contains several fundamental combinatorial objects as special cases, such as plane partitions, tilings of Aztec diamonds, pyramid partitions and more generally steep domino tilings of the plane. Our algorithm, which is of polynomial complexity, is both exact (i.e. the output follows exactly the target probability law, which is either Boltzmann or uniform in our case), and entropy optimal (i.e. -
Aztec Diamonds and Digraphs, and Hankel Determinants of Schröder
Aztec Diamonds and Digraphs, and Hankel Determinants of Schr¨oder Numbers Richard A. Brualdi Stephen Kirkland Department of Mathematics Department of Mathematics University of Wisconsin University of Regina Madison, WI 53706 Regina, Sask., Canada S4S 082 [email protected] [email protected] June 2, 2003 Abstract The Aztec diamond of order n is a certain configuration of 2n(n + 1) unit squares. We give a new proof of the fact that the number Πn of tilings of the Aztec diamond of order n with dominoes equals 2n(n+1)/2. We determine a sign-nonsingular matrix of order n(n + 1) whose determinant gives Πn. We reduce the calculation of this determinant to that of a Hankel matrix of order n whose entries are large Schr¨oder numbers. To calculate that determinant we make use of the J-fraction expansion of the generating function of the Schr¨oder numbers. 1 Introduction Let n be a positive integer. The Aztec diamond of order n is the union ADn of all the unit squares with integral vertices (x, y) satisfying x + y n+1. The Aztec diamond of order | | | | ≤ 1 consists of the 4 unit squares which have the origin (0, 0) as one of their vertices. The Aztec diamonds of orders 2 and 4 are shown in Figure 1. Aztec diamonds are invariant under rotation by 90 degrees, and by reflections in the horizontal and vertical axes. The part of the Aztec diamond of order n that lies in the positive quadrant consists of a staircase pattern of n, n 1,..., 1 unit squares. -
Domino Tiling, Gene Recognition, and Mice Lior
Domino Tiling, Gene Recognition, and Mice by Lior Samuel Pachter B.S. in Mathematics California Institute of Technology (1994) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1999 © Lior Pachter, MCMXCIX. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part, and to grant others the right to do so. A uth or ........................................... Department of Mathematics May 4, 1999 Certified by ....... ... .. .................................... Bonnie A. Berger Associate Professor of Applied Mathematics Thesis Supervisor Accepted by ..... ....................................... Michael Sipser Chairman, Applied Mathematics Committee Accepted by . ..................... INSTITUTE Richard Melrose Chairman, Department Committee on Graduate Students LIBRARIES 2 Domino Tiling, Gene Recognition, and Mice by Lior Samuel Pachter Submitted to the Department of Mathematics on May 17, 1999, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract The first part of this thesis outlines the details of a computational program to identify genes and their coding regions in human DNA. Our main result is a new algorithm for identifying genes based on comparisons between orthologous human and mouse genes. Using our new technique we are able to improve on the current best gene recognition results. Testing on a collection of 117 genes for which we have human and mouse orthologs, we find that we predict 84% of the coding exons in genes correctly on both ends. Our nucleotide sensitivity and specificity is 95% and 98% respectively. -
Modular Design of Hexagonal Phased Arrays Through Diamond Tiles
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TAP.2019.2963561 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2019 1 Modular Design of Hexagonal Phased Arrays through Diamond Tiles Paolo Rocca, Senior Member, IEEE, Nicola Anselmi, Member, IEEE, Alessandro Polo, Member, IEEE, and Andrea Massa, Fellow, IEEE Abstract—The modular design of planar phased array anten- are considered an enabling technology in these fields thanks nas with hexagonal apertures is addressed by means of innovative to their capability of guaranteeing the necessary quality-of- diamond-shaped tiling techniques. Both tiling configuration and service and a suitable level of safety and reliability. On sub-array coefficients are optimized to fit user-defined power- mask constraints on the radiation pattern. Towards this end, the other hand, even though the continuous development of suitable surface-tiling mathematical theorems are customized to electronics (e.g., fast analog-to-digital converters and massive the problem at hand to guarantee optimal performance in case systems-on-chip) and material science (e.g., artificial materials of low/medium-size arrays, while the computationally-hard tiling and meta-materials) as well as the introduction of innovative of large arrays is yielded thanks to an effective integer-coded manufacturing processes (e.g., 3D printing and flexible elec- GA-based exploration of the arising high-cardinality solution spaces. By considering ideal as well as real array models, a set tronics), phased arrays are still far from being commercial- of representative benchmark problems is dealt with to assess the off-the-shelf (COTS) devices. -
A Simple Proof of the Aztec Diamond Theorem
A Simple Proof of the Aztec Diamond Theorem Sen-Peng Eu1,∗ and Tung-Shan Fu2,† 1Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan [email protected] 2Mathematics Faculty, National Pingtung Institute of Commerce, Pingtung 900, Taiwan [email protected] Abstract 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 in terms of Hankel determinants of the large and small Schr¨oder numbers. MSC2000: 05A15 Keywords: Aztec diamond, Hankel matrix, Schr¨oder numbers, lattice paths 1 Introduction The Aztec diamond of order n, denoted by Az(n), is defined as the union of all the unit squares with integral corners (x, y) satisfying |x| + |y| ≤ n +1. A domino is simply a 1-by-2 or 2-by-1 rectangles with integral corners. A domino tiling of a region R is a set of non-overlapping dominos the union of which is R. Figure 1 shows the Aztec diamond of order 3 and a domino tiling. The Aztec diamond theorem, which is first proved by Elkies arXiv:math/0412041v1 [math.CO] 2 Dec 2004 et al. in [4], indicates that the number an of domino tilings of the Aztec diamond of order n is 2n(n+1)/2. They gave four proofs by relating the tilings to alternating sign matrices, monotone triangles, representations of general linear groups, and domino shuffling. Other approaches to this theorem appeared in [2, 3, 6]. Ciucu [3] derived the recurrence relation n an = 2 an−1 by means of perfect matchings of celluar graphs. -
Airy Point Process at the Liquid-Gas Boundary Vincent Beffara, Sunil Chhita, Kurt Johansson
Airy point process at the liquid-gas boundary Vincent Beffara, Sunil Chhita, Kurt Johansson To cite this version: Vincent Beffara, Sunil Chhita, Kurt Johansson. Airy point process at the liquid-gas boundary. Annals of Probability, Institute of Mathematical Statistics, 2018, 46 (5), pp.2973 - 3013. 10.1214/17-aop1244. hal-01632943v2 HAL Id: hal-01632943 https://hal.archives-ouvertes.fr/hal-01632943v2 Submitted on 19 Sep 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Annals of Probability 2018, Vol. 46, No. 5, 2973–3013 https://doi.org/10.1214/17-AOP1244 © Institute of Mathematical Statistics, 2018 AIRY POINT PROCESS AT THE LIQUID-GAS BOUNDARY BY VINCENT BEFFARA1,SUNIL CHHITA1 AND KURT JOHANSSON2 Université Grenoble Alpes, Durham University and Royal Institute of Technology KTH Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling.