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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 EMERGENCE OF THE ELECTROSTATIC FIELD AS A FEYNMAN SUM IN RANDOM TILINGS WITH HOLES
, 2009
"... We consider random lozenge tilings on the triangular lattice with holes Q1,..., Qn in some fixed position. For each unit triangle not in a hole, consider the average orientation of the lozenge covering it. We show that the scaling limit of this discrete field is the electrostatic field obtained wh ..."
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We consider random lozenge tilings on the triangular lattice with holes Q1,..., Qn in some fixed position. For each unit triangle not in a hole, consider the average orientation of the lozenge covering it. We show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole Qi as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside Qi. This is then restated in terms of random surfaces, yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is, in the scaling limit, a sum of helicoids.
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).
Enumerations of Lozenge Tilings, Lattice Paths, and PERFECT MATCHINGS AND THE WEAK LEFSCHETZ PROPERTY
, 2013
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