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Dominos and the Gaussian free field
 Ann. Probab
"... We define a scaling limit of the height function on the domino tiling model (dimer model) on simplyconnected regions in Z 2 and show that it is the “massless free field”, a Gaussian process with independent coefficients when expanded in the eigenbasis of the Laplacian. Résumé On définit une “limite ..."
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We define a scaling limit of the height function on the domino tiling model (dimer model) on simplyconnected regions in Z 2 and show that it is the “massless free field”, a Gaussian process with independent coefficients when expanded in the eigenbasis of the Laplacian. Résumé On définit une “limite d’échelle ” pour la fonction d’hauteur dans le modèle des dimères dans Z 2. Nous montrons que la limite est un “champ libre gaussien”, un processus stochastique gaussien dont les coefficients, dans la base des fonctions propres du laplacien, sont indépendentes. AMS classification: primary 82B20, secondary 60G15
Pfaffian graphs, tjoins, and crossing numbers
"... Abstract. We prove a technical theorem about the numbers of crossings in Tjoins in different drawings of a fixed graph. As a corollary we characterize Pfaffian graphs in terms of their drawings in the plane and give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+ ..."
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Abstract. We prove a technical theorem about the numbers of crossings in Tjoins in different drawings of a fixed graph. As a corollary we characterize Pfaffian graphs in terms of their drawings in the plane and give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+1 and K2j+1,2k+1. This gives a new proof of the HananiTutte theorem. 1.
Drawing Pfaffian graphs
 Proc. 12th Int. Symposium on Graph Drawing
, 2005
"... Abstract. We prove that a graph is Pfaffian if and only if it can be drawn in the plane (possibly with crossings) so that every perfect matching intersects itself an even number of times. 1. ..."
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Abstract. We prove that a graph is Pfaffian if and only if it can be drawn in the plane (possibly with crossings) so that every perfect matching intersects itself an even number of times. 1.
Simpler Projective Plane Embedding
, 2000
"... A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n ..."
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A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...
Computation of Sparse Circulant Permanents via Determinants
, 2002
"... We consider the problem of computing the permanent of circulant matrices. ..."
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We consider the problem of computing the permanent of circulant matrices.
Critical resonance in the . . .
, 2004
"... We study the phase transition in the honeycomb dimer model (equivalently, monotone nonintersecting lattice path model). At the critical point the system has a strong longrange dependence; in particular, periodic boundary conditions give rise to a “resonance” phenomenon, where the partition functio ..."
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We study the phase transition in the honeycomb dimer model (equivalently, monotone nonintersecting lattice path model). At the critical point the system has a strong longrange dependence; in particular, periodic boundary conditions give rise to a “resonance” phenomenon, where the partition function and other properties of the system depend sensitively on the shape of the domain.
On the Dimer Problem and the Ising Problem in Finite 3dimensional Lattices
, 2002
"... We present a new expression for the partition function of the dimer arrangements and the Ising partition function of the 3dimensional cubic lattice. We use the Pfaffian method. The partition functions are expressed by means of expectations of determinants and Pfaffians of matrices associated wit ..."
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We present a new expression for the partition function of the dimer arrangements and the Ising partition function of the 3dimensional cubic lattice. We use the Pfaffian method. The partition functions are expressed by means of expectations of determinants and Pfaffians of matrices associated with the cubic lattice.
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.