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The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions
 Mem. Amer. Math. Soc
"... Abstract. We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of latticetriangular holes of even sides satisfies, ..."
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Abstract. We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of latticetriangular holes of even sides satisfies, for large separations between the holes, a Coulomb law and a superposition principle that perfectly parallel the laws of two dimensional electrostatics, with physical charges corresponding to holes, and their magnitude to the difference between the number of rightpointing and leftpointing unit triangles in each hole. We detail this parallel by indicating that, as a consequence of our result, the relative probabilities of finding a fixed collection of holes at given mutual distances (when sampling uniformly at random over all unit rhombus tilings of the complement of the holes) approaches, for large separations between the holes, the relative probabilities of finding the corresponding two dimensional physical system of charges at given mutual distances. Physical temperature corresponds to a parameter refining the background triangular lattice. We give an equivalent phrasing of our result in terms of covering surfaces of given holonomy. From this perspective, two dimensional electrostatics arises by averaging over all possible discrete geometries of the covering surfaces.
The interaction of a gap with a free boundary in a two dimensional dimer system
 Comm. Math. Phys
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The interaction of collinear gaps of arbitrary charge in a two dimensional dimer system
 Comm. Math. Phys
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A TRIANGULAR GAP OF SIZE TWO In A Sea Of Dimers On A 60° Angle
"... We consider a triangular gap of side two in a 60 ◦ angle on the triangular lattice whose sides are zigzag lines. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of th ..."
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Cited by 3 (3 self)
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We consider a triangular gap of side two in a 60 ◦ angle on the triangular lattice whose sides are zigzag lines. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its five images in the sides of the angle. This provides a new aspect of the parallel between the correlation of gaps in dimer packings and electrostatics developed by the first author in previous work.
A DUAL OF MACMAHON’S THEOREM ON PLANE PARTITIONS
"... A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In this paper we present a counterpart of this formula, corresponding to the exterior of a concave hexagon ob ..."
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A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In this paper we present a counterpart of this formula, corresponding to the exterior of a concave hexagon obtained by turning 120 ◦ after drawing each side (MacMahon’s hexagon is obtained by turning 60◦ after each step).
A TRIANGULAR GAP OF SIZE TWO IN A SEA OF DIMERS IN A 90 ◦ ANGLE WITH MIXED BOUNDARY CONDITIONS, AND A HEAT FLOW CONJECTURE FOR THE GENERAL CASE
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Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics
, 2014
"... In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model via the socalled height mapping and the nature of boundary condition ..."
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In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model via the socalled height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predict particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms.
Dimers and analytic torsion I
, 2013
"... In the dimer model, a configuration consists of a perfect matching of a fixed graph. If the underlying graph is planar and bipartite, such a configuration is associated to a height function. For appropriate “critical ” (weighted) graphs, this height function is known to converge in the fine mesh lim ..."
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In the dimer model, a configuration consists of a perfect matching of a fixed graph. If the underlying graph is planar and bipartite, such a configuration is associated to a height function. For appropriate “critical ” (weighted) graphs, this height function is known to converge in the fine mesh limit to a Gaussian free field, following in particular Kenyon’s work. In the present article, we study the asymptotics of smoothed and local field observables from the point of view of families of CauchyRiemann operators and their determinants. This allows in particular to obtain a functional invariance principle for the field; characterise completely the limiting field on toroidal graphs as a compactified free field; analyse electric correlators; and settle the FisherStephenson conjecture on monomer correlators. The analysis is based on comparing the variation of determinants of families of (continuous) CR operators with that of their discrete (finite dimensional) approximants. This relies in turn on estimating precisely inverting kernels, in particular near singularities. In order to treat correlators of “singular”