5 research outputs found
Random Tilings: Concepts and Examples
We introduce a concept for random tilings which, comprising the conventional
one, is also applicable to tiling ensembles without height representation. In
particular, we focus on the random tiling entropy as a function of the tile
densities. In this context, and under rather mild assumptions, we prove a
generalization of the first random tiling hypothesis which connects the maximum
of the entropy with the symmetry of the ensemble. Explicit examples are
obtained through the re-interpretation of several exactly solvable models. This
also leads to a counterexample to the analogue of the second random tiling
hypothesis about the form of the entropy function near its maximum.Comment: 32 pages, 42 eps-figures, Latex2e updated version, minor grammatical
change
The asymptotic determinant of the discrete Laplacian
We compute the asymptotic determinant of the discrete Laplacian on a
simply-connected rectilinear region in R^2. As an application of this result,
we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.Comment: 36 pages, 4 figures, to appear in Acta Mathematic
Tiling a simply connected figure with bars of length 2 or 3
AbstractLet F be a simply connected figure formed from a finite set of cells of the planar square lattice. We first prove that if F has no peak (a peak is a cell of F which has three of its edges in the contour of F), then F can be tiled with rectangular bars formed from 2 or 3 cells. Afterwards, we devise a linear-time algorithm for finding a tiling of F with those bars when such a tiling exists
Tiling with Bars and Satisfaction of Boolean Formulas
AbstractLetFbe a figure formed from a finite set of cells of the planar square lattice. We first prove that the problem of tiling such a figure with bars formed from 2 or 3 cells can be reduced to the logic problem 2-SAT. Afterwards, we deduce a linear-time algorithm of tiling with these bars