scholarly journals Domino statistics of the two-periodic Aztec diamond

2016 ◽  
Vol 294 ◽  
pp. 37-149 ◽  
Author(s):  
Sunil Chhita ◽  
Kurt Johansson
Keyword(s):  
Aztec Diamond

scholarly journals Asymptotic domino statistics in the Aztec diamond

2015 ◽  
Vol 25 (3) ◽  
pp. 1232-1278 ◽  
Author(s):  
Sunil Chhita ◽  
Kurt Johansson ◽  
Benjamin Young
Keyword(s):  
Aztec Diamond

scholarly journals A Note on Domino Shuffling

10.37236/1056 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
É. Janvresse ◽  
T. de la Rue ◽  
Y. Velenik
Keyword(s):  
Large Class ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Sufficient Condition ◽  
Aztec Diamond

We present a variation of James Propp's generalized domino shuffling, which provides an efficient way to obtain perfect matchings of weighted Aztec diamonds. Our modification is specially tailored to deal with cases when some of the weights are zero. This allows us to tile efficiently a large class of planar graphs, by embedding them in a large enough Aztec diamond. We also give a sufficient condition on the size of the latter diamond for the algorithm to succeed.

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 Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

2021 ◽  
Vol 37 ◽  
pp. 295-315
Author(s):  
Derek Young
Keyword(s):  
Lower Bound ◽  
Adjacency Matrix ◽  
Work Group ◽  
Zero Forcing ◽  
Vertex Connectivity ◽  
Aztec Diamond ◽  
Zero Forcing Number ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Graph Parameters

It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph (see [AIM Minimum Rank - Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S. Cioab$\breve{\text{a}}$, D. Cvetkovi$\acute{\text{c}}$, S. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi$\acute{\text{c}}$, H. van der Holst, K. Vander Meulen, and A. Wangsness). Linear Algebra Appl., 428(7):1628--1648, 2008]). In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters that give lower bounds on the maximum nullity of a graph. Inparticular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdiére type parameter and vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.

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 A generalization of Aztec diamond theorem, part II

2016 ◽  
Vol 339 (3) ◽  
pp. 1172-1179 ◽  
Author(s):  
Tri Lai
Keyword(s):  
Aztec Diamond

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.

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