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41
Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transferimpedances
, 1993
"... Let G be a finite graph or an infinite graph on which Z d acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is infinite, methods from [Pem] show that this still makes sense, producing a random essential spanning f ..."
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Cited by 82 (0 self)
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Let G be a finite graph or an infinite graph on which Z d acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is infinite, methods from [Pem] show that this still makes sense, producing a random essential spanning forest of G. A method for calculating local characteristics (i.e. finitedimensional marginals) of T from the transferimpedance matrix is presented. This differs from the classical matrixtree theorem in that only small pieces of the matrix (ndimensional minors) are needed to compute small (ndimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green’s function for G, which is not a local calculation. However, it is shown how the calculation of the Green’s function may be reduced to a finite computation in the case when G is an infinite graph admitting a Z daction with finite quotient. The same computation also gives the entropy of the law of T. These results are applied to the problem of tiling certain lattices by dominos – the socalled dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous [Al2] on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. Included is a generalization of moments to treevalued random variables and criteria for these generalized moments to determine a distribution.
Local statistics of lattice dimers
, 1997
"... Abstract. We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy µ on the space of tilings of the plane with dominos. We construct a measure ..."
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Cited by 64 (11 self)
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Abstract. We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy µ on the space of tilings of the plane with dominos. We construct a measure ν on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the νmeasures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for µ and ν, and compute the rate of convergence to mixing (the correlation between distant events). For the measure ν we compute the variance of the height function. Resumé. Soit µ la mesure d’entropie maximale sur l’espace X des pavages du plan par des dominos. On calcule explicitement la mesure des sousensembles cylindriques de X. De même, on construit une mesure ν d’entropie maximale sur l’espace X ′ des pavages du plan par losanges, et on calcule explicitement la mesure des sousensembles cylindriques. Comme application on calcule, pour µ et ν, les correlations d’évenements distants, ainsi que la νvariance de la fonction “hauteur ” sur X ′. 1.
On the Theory of Pfaffian Orientations. I. Perfect Matchings and Permanents.
 Electronic Journal of Combinatorics
, 1998
"... Kasteleyn stated that the generating function of the perfect matchings of a graph of genus g may be written as a linear combination of 4 g Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute the permanent of a square matrix. ..."
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Cited by 28 (5 self)
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Kasteleyn stated that the generating function of the perfect matchings of a graph of genus g may be written as a linear combination of 4 g Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute the permanent of a square matrix.
Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 22 (11 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
Planar dimers and Harnack curves
, 2003
"... 1.1 Summary of results In this paper we study the connection between dimers and Harnack curves discovered in [12]. To any periodic edgeweighted planar bipartite graph Γ ..."
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Cited by 14 (2 self)
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1.1 Summary of results In this paper we study the connection between dimers and Harnack curves discovered in [12]. To any periodic edgeweighted planar bipartite graph Γ
On the Theory of Matchgate Computations
 Submitted. Also available at Electronic Colloquium on Computational Complexity Report
, 2007
"... Valiant has proposed a new theory of algorithmic computation based on perfect matchings and the Pfaffian. We study the properties of matchgates—the basic building blocks in this new theory. We give a set of algebraic identities which completely characterize these objects in terms of the GrassmannPl ..."
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Cited by 14 (5 self)
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Valiant has proposed a new theory of algorithmic computation based on perfect matchings and the Pfaffian. We study the properties of matchgates—the basic building blocks in this new theory. We give a set of algebraic identities which completely characterize these objects in terms of the GrassmannPlücker identities. In the important case of 4 by 4 matchgate matrices, which was used in Valiant’s classical simulation of a fragment of quantum computations, we further realize a group action on the character matrix of a matchgate, and relate this information to its compound matrix. Then we use Jacobi’s theorem to prove that in this case the invertible matchgate matrices form a multiplicative group. These results are useful in establishing limitations on the ultimate capabilities of Valiant’s theory of matchgate computations and his closely related theory of Holographic Algorithms. 1
Valiant’s Holant Theorem and Matchgate Tensors (Extended Abstract
 In Proceedings of TAMC 2006: Lecture Notes in Computer Science
"... Abstract We propose matchgate tensors as a natural and proper language to develop Valiant's newtheory of Holographic Algorithms. We give a treatment of the central theorem in this theorythe Holant Theoremin terms of matchgate tensors. Some generalizations are presented. 1 Background In a remarka ..."
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Cited by 13 (7 self)
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Abstract We propose matchgate tensors as a natural and proper language to develop Valiant's newtheory of Holographic Algorithms. We give a treatment of the central theorem in this theorythe Holant Theoremin terms of matchgate tensors. Some generalizations are presented. 1 Background In a remarkable paper, Valiant [9] in 2004 has proposed a completely new theory of Holographic Algorithms or Holographic Reductions. In this framework, Valiant has developed a most novel methodology of designing polynomial time (indeed NC2) algorithms, a methodology by which one can design a custom made process capable of carrying out a seemingly exponential computation with exponentially many cancellations so that the computation can actually be done in polynomial time. The simplest analogy is perhaps with Strassen's matrix multiplication algorithm [5]. Here the algorithm computes some extraneous quantities in terms of the submatrices, which do not directly appear in the answer yet only to be canceled later, but the purpose of which is to speedup computation by introducing cancelations. In the several cases such clever algorithms had been found, they tend to work in a linear algebraic setting, in particular the computation of the determinant figures prominently [8, 2, 6]. Valiant's new theory manages to create a process of custom made cancelation which gives polynomial time algorithms for combinatorial problems which do not appear to be linear algebraic. In terms of its broader impact in complexity theory, one can view Valiant's new theory as another algorithmic design paradigm which pushes back the frontier of what is solvable by polynomial time. Admittedly, at this early stage, it is still premature to say what drastic consequence it might have on the landscape of the big questions of complexity theory, such as P vs. NP. But the new theory has already been used by Valiant to devise polynomial time algorithms for a number of problems for which no polynomial time algorithms were known before.
On the Theory of Pfaffian Orientations. II. Tjoins, kCuts, and Duality of Enumeration
, 1998
"... This is a continuation of our paper "A Theory of Pfaffian Orientations I: Perfect Matchings and Permanents". We present a new combinatorial way to compute the generating functions of T joins and kcuts of graphs. As a consequence, we show that the computational problem to find the maximum weight of ..."
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Cited by 11 (2 self)
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This is a continuation of our paper "A Theory of Pfaffian Orientations I: Perfect Matchings and Permanents". We present a new combinatorial way to compute the generating functions of T joins and kcuts of graphs. As a consequence, we show that the computational problem to find the maximum weight of an edgecut is polynomially solvable for the instances (G; w) where G is a graph embedded on an arbitrary fixed orientable surface and the weight function w has only a bounded number of different values. We also survey the related results concerning a duality of the Tutte polynomial, and present an application for the weight enumerator of a binary code. In a continuation of this paper which is in preparation we present an application to the Ising problem of threedimensional crystal structures.
Pólya’s permanent problem
 Electron. J. Combin
, 1996
"... A square real matrix is signnonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of signnonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determ ..."
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Cited by 10 (0 self)
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A square real matrix is signnonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of signnonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms. 1
Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers
 J. Combin. Theory Ser. B
"... 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 2 n(n+1)/2. We determine a signnonsingular matrix of order n(n + 1) whose determinant gives Πn. We redu ..."
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Cited by 8 (0 self)
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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 2 n(n+1)/2. We determine a signnonsingular 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 Schröder numbers. To calculate that determinant we make use of the Jfraction expansion of the generating function of the Schröder numbers. 1