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13
Generalized DominoShuffling
"... This article presents efficient algorithms that work in this context to solve three problems: finding the sum of the weights of the matchings of a weighted Aztec diamond graph A n ; computing the probability that a randomlychosen matching of A n will include a particular edge (where the probabil ..."
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This article presents efficient algorithms that work in this context to solve three problems: finding the sum of the weights of the matchings of a weighted Aztec diamond graph A n ; computing the probability that a randomlychosen matching of A n will include a particular edge (where the probability of a matching is proportional to its weight); and generating a matching of A n at random. The first of these algorithms is equivalent to a special case of Mihai Ciucu's cellular complementation algorithm [1] and can be used to solve many of the same problems. The second of the three algorithms is a generalization of notyetpublished work of Alexandru Ionescu, and can be employed to prove an identity governing a threevariable generating function whose coefficients are all the edgeinclusion probabilities; this formula has been used [2] as the basis for asymptotic formulas for these probabilities, but a proof o
Constrained Codes as Networks of Relations
"... Abstract — We address the wellknown problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a twodimensional ..."
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Abstract — We address the wellknown problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a twodimensional constrained coding system is still an elusive research challenge. The only known exception in the twodimensional case is an exact (however, not rigorous) solution to the (1, ∞)RLL system on the hexagonal lattice. Furthermore, only exponentialtime algorithms are known for the related problem of counting the exact number of constrained twodimensional information arrays. We present the first known rigorous technique that yields an exact capacity of a twodimensional constrained coding system. In addition, we devise an efficient (polynomial time) algorithm for counting the exact number of constrained arrays of any given size. Our approach is a composition of a number of ideas and techniques: describing the capacity problem as a solution to a counting problem in networks of relations, graphtheoretic tools originally developed in the field of statistical mechanics, techniques for efficiently simulating quantum circuits, as well as ideas from the theory related to the spectral distribution of Toeplitz matrices. Using our technique we derive a closed form solution to the capacity related to the PathCover constraint in a twodimensional triangular array (the resulting calculated capacity is 0.72399217...). PathCover is a generalization of the well known onedimensional (0, 1)RLL constraint for which the capacity is known to be 0.69424... Index Terms — capacity of constrained systems, capacity of twodimensional constrained systems, holographic reductions, networks of relations, FKT method, spectral distribution of Toeplitz matrices I.
The Laplacian and ¯ ∂ operators on critical planar graphs Invent
 Math
"... On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and ¯ ∂ operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (norma ..."
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On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and ¯ ∂ operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of ¯ ∂ and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete analytic functions, which, via convolutions gives a general process for constructing discrete analytic functions and discrete harmonic functions on critical planar graphs. 1
Dimers on graphs in nonorientable surfaces
"... Abstract. The main result of this paper is a Pfaffian formula for the partition function of the dimer model on a graph Γ embedded in a closed, possibly nonorientable surface Σ. This formula is suitable for computational purposes, and it is obtained using purely geometrical methods. The key step in t ..."
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Abstract. The main result of this paper is a Pfaffian formula for the partition function of the dimer model on a graph Γ embedded in a closed, possibly nonorientable surface Σ. This formula is suitable for computational purposes, and it is obtained using purely geometrical methods. The key step in the proof consists of a correspondence between some orientations on Γ and the set of pin − structures on Σ. This generalizes (and simplifies) the results of a previous paper [2]. 1.
Conformal invariance of isoradial dimer models
, 2005
"... the case of triangular quadritilings ..."
Partition function of periodic isoradial dimer models
, 2006
"... Isoradial dimer models were introduced in [12] they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of [12], namely that for periodic isoradial dimer models, the growth rate ..."
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Isoradial dimer models were introduced in [12] they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of [12], namely that for periodic isoradial dimer models, the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only. This is a surprising feature of periodic isoradial dimer models, which does not hold in the general periodic dimer case [14]. 1
DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II
, 2007
"... In a previous paper [3], we showed how certain orientations of the edges of a graph Γ embedded in a closed oriented surface Σ can be understood as discrete spin structures on Σ. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dime ..."
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In a previous paper [3], we showed how certain orientations of the edges of a graph Γ embedded in a closed oriented surface Σ can be understood as discrete spin structures on Σ. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on Γ. In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function. These operations allow to reformulate the dimer model as
Discrete Dirac Operators, Critical Embeddings and IharaSelberg Functions
"... The aim of the paper is to formulate a discrete analogue of the claim made by AlvarezGaume et al., ([1]), realizing the partition function of the free fermion on a closed Riemann surface of genus g as a linear combination of 22g Pfaffians of Dirac operators. Let G = (V,E) be a finite graph embedded ..."
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The aim of the paper is to formulate a discrete analogue of the claim made by AlvarezGaume et al., ([1]), realizing the partition function of the free fermion on a closed Riemann surface of genus g as a linear combination of 22g Pfaffians of Dirac operators. Let G = (V,E) be a finite graph embedded in a closed Riemann surface X of genus g, xe the collection of independent variables associated with each edge e of G (collected in one vector variable x) and Σ the set of all 22g spinstructures on X. We introduce 22g rotations rots and (2E  × 2E) matrices ∆(s)(x), s ∈ Σ, of the transitions between the oriented edges of G determined by rotations rots. We show that the generating function for the even subsets of edges of G, i.e., the Ising partition function, is a linear combination of the square roots of 22g IharaSelberg functions I(∆(s)(x)) also called Feynman functions. By a result of Foata– Zeilberger holds I(∆(s)(x)) = det(I − ∆′(s)(x)), where ∆′(s)(x) is obtained from ∆(s)(x) by replacing some entries by 0. Thus each Feynman function is computable in polynomial time. We suggest that in the case of critical embedding of a bipartite graph G, the Feynman functions provide suitable discrete analogues for the Pfaffians of discrete Dirac operators.