Predictions of bond percolation thresholds for the kagom\'e and Archimedean (3,122)(3,12^2) lattices

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2) bond problems, and the other gives estimates of $p_c$ for the homogeneous kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.Comment: 4 pages, 5 figure

Chromatic Polynomials for Families of Strip Graphs and their Asymptotic Limits

We calculate the chromatic polynomials $P((G_s)_m,q)$ and, from these, the asymptotic limiting functions $W(\{G_s\},q)=\lim_{n \to \infty}P(G_s,q)^{1/n}$ for families of $n$-vertex graphs $(G_s)_m$ comprised of $m$ repeated subgraphs $H$ adjoined to an initial graph $I$. These calculations of $W(\{G_s\},q)$ for infinitely long strips of varying widths yield important insights into properties of $W(\Lambda,q)$ for two-dimensional lattices $\Lambda$. In turn, these results connect with statistical mechanics, since $W(\Lambda,q)$ is the ground state degeneracy of the $q$-state Potts model on the lattice $\Lambda$. For our calculations, we develop and use a generating function method, which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant $W(\{G_s\},q)$ function in the limit $n \to \infty$. From this, we obtain the exact continuous locus of points ${\cal B}$ where $W(\{G_s\},q)$ is nonanalytic in the complex $q$ plane. This locus is shown to consist of arcs which do not separate the $q$ plane into disconnected regions. Zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities ${\cal B}$. We find that as the width of the infinitely long strips is increased, the arcs comprising ${\cal B}$ elongate and move toward each other, which enables one to understand the origin of closed regions that result for the (infinite) 2D lattice.Comment: 48 pages, Latex, 12 encapsulated postscript figures, to appear in Physica

Taylor-Socolar hexagonal tilings as model sets

The Taylor-Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor-Socolar tilings into an algebraic setting which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice $Q$ in the plane. If $X_Q$ denotes the set of all Taylor-Socolar tilings with centers on $Q$ then $X_Q$ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of $Q$. The $Q$-adic completion $\bar{Q}$ of $Q$ is a natural factor of $X_Q$ and the natural mapping $X_Q \longrightarrow \bar{Q}$ is bijective except at a dense set of points of measure 0 in $\bar{Q}$. We show that $X_Q$ consists of three LI classes under translation. Two of these LI classes are very small, namely countable $Q$-orbits in $X_Q$. The other is a minimal dynamical system which maps surjectively to $\bar{Q}$ and which is variously $2:1$, $6:1$, and $12:1$ at the singular points. We further develop the formula of Socolar and Taylor (2011) that determines the parity of the tiles of a tiling in terms of the co-ordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull $X_Q$; more precisely the two hulls are mutually locally derivable.Comment: 45 pages, 33 figure

There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies on the induced conformal structure and proceeds by invoking the residue theorem. Similar methods can be used to prove a theorem of Dress on infinite triangulations of the plane with exactly two irregular vertices. The non-existence results for torus decompositions provide infinite families of graphs which cannot be embedded in the torus.Comment: 14 pages, 11 figures, only minor changes from first version, to appear in Geometriae Dedicat

Critical surfaces for general inhomogeneous bond percolation problems

We present a method of general applicability for finding exact or accurate approximations to bond percolation thresholds for a wide class of lattices. To every lattice we sytematically associate a polynomial, the root of which in $[0,1]$ is the conjectured critical point. The method makes the correct prediction for every exactly solved problem, and comparison with numerical results shows that it is very close, but not exact, for many others. We focus primarily on the Archimedean lattices, in which all vertices are equivalent, but this restriction is not crucial. Some results we find are kagome: $p_c=0.524430...$, $(3,12^2): p_c=0.740423...$, $(3^3,4^2): p_c=0.419615...$, $(3,4,6,4):p_c=0.524821...$, $(4,8^2):p_c=0.676835...$, $(3^2,4,3,4)$: $p_c=0.414120...$ . The results are generally within $10^{-5}$ of numerical estimates. For the inhomogeneous checkerboard and bowtie lattices, errors in the formulas (if they are not exact) are less than $10^{-6}$.Comment: Submitted to J. Stat. Mec

How model sets can be determined by their two-point and three-point correlations

We show that real model sets with real internal spaces are determined, up to translation and changes of density zero by their two- and three-point correlations. We also show that there exist pairs of real (even one dimensional) aperiodic model sets with internal spaces that are products of real spaces and finite cyclic groups whose two- and three-point correlations are identical but which are not related by either translation or inversion of their windows. All these examples are pure point diffractive. Placed in the context of ergodic uniformly discrete point processes, the result is that real point processes of model sets based on real internal windows are determined by their second and third moments.Comment: 19 page

Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces

Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with p-adic topologies or even with mixed Euclidean/p-adic topologies. We show that a number of well known tilings precisely fit this form, including the chair tiling and the Robinson square tilings. Thus the scope of the cut and project formalism is considerably larger than is usually supposed. Applying the powerful consequences of model sets we derive the diffractive nature of these tilings.Comment: 11 pages, 2 figures; dedicated to Peter Kramer on the occasion of his 65th birthda

Ground State Entropy of Potts Antiferromagnets: Bounds, Series, and Monte Carlo Measurements

We report several results concerning $W(\Lambda,q)=\exp(S_0/k_B)$, the exponent of the ground state entropy of the Potts antiferromagnet on a lattice $\Lambda$. First, we improve our previous rigorous lower bound on $W(hc,q)$ for the honeycomb (hc) lattice and find that it is extremely accurate; it agrees to the first eleven terms with the large-$q$ series for $W(hc,q)$. Second, we investigate the heteropolygonal Archimedean $4 \cdot 8^2$ lattice, derive a rigorous lower bound, on $W(4 \cdot 8^2,q)$, and calculate the large-$q$ series for this function to $O(y^{12})$ where $y=1/(q-1)$. Remarkably, these agree exactly to all thirteen terms calculated. We also report Monte Carlo measurements, and find that these are very close to our lower bound and series. Third, we study the effect of non-nearest-neighbor couplings, focusing on the square lattice with next-nearest-neighbor bonds.Comment: 13 pages, Latex, to appear in Phys. Rev.

Determination of the bond percolation threshold for the Kagome lattice

The hull-gradient method is used to determine the critical threshold for bond percolation on the two-dimensional Kagome lattice (and its dual, the dice lattice). For this system, the hull walk is represented as a self-avoiding trail, or mirror-model trajectory, on the (3,4,6,4)-Archimedean tiling lattice. The result pc = 0.524 405 3(3) (one standard deviation of error) is not consistent with the previously conjectured values.Comment: 10 pages, TeX, Style file iopppt.tex, to be published in J. Phys. A. in August, 199