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 ν-measures 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 sous-ensembles 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 sous-ensembles cylindriques. Comme application on calcule, pour µ et ν, les correlations d’évenements distants, ainsi que la ν-variance de la fonction “hauteur ” sur X ′. 1.

We show that a number of graph-theoretic counting problems remain NP-hard, indeed #P-complete, 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 NP-hard 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 NP-completeness in the early 1970's, the primary focus of complexity theory has been on decision ...

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

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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 #P-complete without the moduli. Going beyond symmetric signatures, we define d-admissibility and d-realizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1

this paper we would like to give a short survey of some of these new results.

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 theory--the Holant Theorem--in 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.

A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of sign-nonsingular 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

: The following problem, which stems from the "flux phase" problem in condensed matter physics, is analyzed and extended here: One is given a planar graph (or lattice) with prescribed vertices, edges and a weight jt xy j on each edge (x; y). The flux phase problem (which we partially solve) is to find the real phase function on the edges, `(x; y), so that the matrix T := fjt xy jexp[i`(x; y)]g minimizes the sum of the negative eigenvalues of \GammaT . One extension of this problem which is also partially solved is the analogous question for the Falicov-Kimball model. There one replaces the matrix \GammaT by \GammaT + V , where V is a diagonal matrix representing a potential. Another extension of this problem, which we solve completely for planar, bipartite graphs, is to maximize jdet T j. Our analysis of this determinant problem is closely connected with Kasteleyn's 1961 theorem (for arbitrary planar graphs) and, indeed, yields an alternate, and we believe more transparent proof of it...