2016/01/08 by Tomack Gilmore, Gilmore, Tomack
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CO #math.MP
paper · pdf · doi:10.48550/arxiv.1601.01965
26 pages, 13 figures. In this version: proof of assertion made in remark 10 of previous version given in the form of Theorem 8 and Corollary 9; typos corrected
arxiv created 2016/02/02 · arxiv updated 2016/02/04
Consider a family of collinear, equilateral triangular holes of any even side length lying within a sea of unit rhombi. The results presented below show that as the distance between the holes grows large, the interaction between them may be approximated, up to a multiplicative constant, by taking the exponential of the negative of the electrostatic energy of the system obtained by viewing the holes as a set of point charges, each with a signed magnitude given by a certain statistic. Furthermore it is shown that the interaction between a family of left pointing collinear triangular holes and a free boundary may be approximated (again up to some multiplicative constant) by taking the exponential of the negative of the electrostatic energy of the system obtained by considering the holes as a set of point charges and the boundary a straight equipotential conductor. These two differing systems of point charges can be related via the method of image charges, a well-known physical law that also surfaces in the following mathematical analysis of enumeration formulas that count tilings of certain regions of the plane by unit rhombi.