The Scaling Limit of the Correlation of Holes on the Triangular Lattice with Periodic Boundary Conditions

The Scaling Limit of the Correlation of Holes on the Triangular Lattice with Periodic Boundary Conditions

by Mihai Ciucu
     
 

ISBN-10: 0821843265

ISBN-13: 9780821843260

Pub. Date: 04/10/2009

Publisher: American Mathematical Society

The author defines the correlation of holes on the triangular lattice under periodic boundary conditions and studies its asymptotics as the distances between the holes grow to infinity. He proves that the joint correlation of an arbitrary collection of triangular holes of even side-lengths (in lattice spacing units) satisfies, for large separations between the

Overview

The author defines the correlation of holes on the triangular lattice under periodic boundary conditions and studies its asymptotics as the distances between the holes grow to infinity. He proves that the joint correlation of an arbitrary collection of triangular holes of even side-lengths (in lattice spacing units) 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 right-pointing and left-pointing unit triangles in each hole. The author details this parallel by indicating that, as a consequence of the results, 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) approach, 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. He also gives an equivalent phrasing of the results in terms of covering surfaces of given holonomy. From this perspective, two dimensional electrostatic potential energy arises by averaging over all possible discrete geometries of the covering surfaces.

Product Details

ISBN-13:
9780821843260
Publisher:
American Mathematical Society
Publication date:
04/10/2009
Series:
Memoirs of the American Mathematical Society Series, #199
Pages:
100

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