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89
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 98 (13 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.
The Complexity of Counting in Sparse, Regular, and Planar Graphs
 SIAM Journal on Computing
, 1997
"... We show that a number of graphtheoretic counting problems remain NPhard, indeed #Pcomplete, in very restricted classes of graphs. In particular, it is shown that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to ..."
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Cited by 91 (0 self)
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We show that a number of graphtheoretic counting problems remain NPhard, indeed #Pcomplete, in very restricted classes of graphs. In particular, it is shown that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. To achieve these results, a new interpolationbased reduction technique which preserves properties such as constant degree is introduced. In addition, the problem of approximately counting minimum cardinality vertex covers is shown to remain NPhard even when restricted to graphs of maximal degree 3. Previously, restrictedcase complexity results for counting problems were elusive; we believe our techniques may help obtain similar results for many other counting problems. 1 Introduction Ever since the introduction of NPcompleteness in the early 1970's, the primary focus of complexity theory has been on decision ...
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 41 (16 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
The Computational Complexity of Some Classical Problems from Statistical Physics
 In Disorder in Physical Systems
, 1990
"... this paper is to attempt to review and classify the di# culty of a range of problems, arising in the statistical mechanics of physical systems and to which I was introduced by J.M. Hammersley in the early sixties. Their common characteristics at the time were that they all seemed hard and there was ..."
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Cited by 25 (0 self)
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this paper is to attempt to review and classify the di# culty of a range of problems, arising in the statistical mechanics of physical systems and to which I was introduced by J.M. Hammersley in the early sixties. Their common characteristics at the time were that they all seemed hard and there was little existing mathematical machinery which was of much use in dealing with them. Twenty years later the situation has not changed dramatically; there do exist some mathematical techniques which appear to be tools in trade for this area, subadditive functions and transfer matrices for example, but they are still relatively few and despite a great deal of e#ort the number of exact answers which are known to the many problems posed is extremely small. Below we shall attempt to explain why this should be so by showing how the problems originally studied are special cases of a wide range of problems which can, in a well defined sense, be regarded as the most intractable enumeration problems that can sensibly be posed
The Planar Dimer Model With Boundary: A Survey.
 CRM Proceedings and Lecture Notes
, 1998
"... this paper we would like to give a short survey of some of these new results. ..."
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Cited by 21 (1 self)
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this paper we would like to give a short survey of some of these new results.
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 20 (8 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
ON DIMER COVERINGS OF RECTANGLES OF FIXED WIDTH
, 1985
"... For fixed k let A n denote the number of dimer coverings of a k × n rectangle. Various properties of the generating function] ~ An xn are obtained, in particular answering questions of Klarner and Pollack and of Hock and McQuistan. An explicit expression for the molecular freedom for dimers on a s ..."
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Cited by 18 (0 self)
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For fixed k let A n denote the number of dimer coverings of a k × n rectangle. Various properties of the generating function] ~ An xn are obtained, in particular answering questions of Klarner and Pollack and of Hock and McQuistan. An explicit expression for the molecular freedom for dimers on a saturated k × n lattice space is also obtained. The results are consequences of the explicit formula for An obtained by Kasteleyn and by Temperley and Fisher.