Entropy and Boundary Conditions in Random Lozenge Tilings
Abstract
The tilings of lozenges in 2 dimensions and of rhomboedra in 3 dimensions are studied when they are constrained by fixed boundary conditions. We establish a link between those conditions and free or periodic boundary ones: the entropy is written as a functional integral which is treated via a saddle-point method. Then we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary. This inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones. This method uses a representation of tilings by membranes embedded in a higher-dimensional hypercubic lattice. It is illustrated in the case of 60 degree lozenge tilings.
Keywords
Cite
@article{arxiv.cond-mat/9804062,
title = {Entropy and Boundary Conditions in Random Lozenge Tilings},
author = {N. Destainville},
journal= {arXiv preprint arXiv:cond-mat/9804062},
year = {2007}
}
Comments
20 pages, LaTeX, 6 figures, submitted to J. Phys. A: Math. Gen