Inversion relations and symmetry groups for Potts models on the triangular lattice

Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on H3 is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in H2 where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

Phase transitions and thermodynamic properties of triangular strongly-diluted antiferromagnetic Potts model

This paper describes components of a branch-and-cut algorithm for solving integer linear programs having a large symmetry group. It describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group. Pruning and orbit computations are performed by backtracking procedures using a Schreier-Sims table for representing the symmetry group. Applications to hard set covering problems, generation of covering designs and error correcting codes are given.

In this paper, we show that any Coxeter graph which defines a higher rank Coxeter group must have disjoint induced subgraphs each of which defines a hyperbolic or higher rank Coxeter group where the groups in question do not necessarily act cocompactly or cofinitely. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups and to produce structure theorems for hyperbolic Coxeter groups.

We give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct, e.g., new groups having some interesting asphericity properties.

Kantian philosophy of space, time and gravity is significantly affected in three ways by particle physics. First, particle physics deflects Schlick’s General Relativity-based critique of synthetic a priori knowledge. Schlick argued that since geometry was not synthetic a priori, nothing was—a key step toward logical empiricism. Particle physics suggests a Kant-friendlier theory of space-time and gravity presumably approximating General Relativity arbitrarily well, massive spin-2 gravity, while retaining a flat space-time geometry that is indirectly observable at large distances. The theory’s roots include Seeliger and Neumann in the 1890s and Einstein in 1917 as well as 1920s–1930s physics. Such theories have seen renewed scientific attention since 2000 and especially since 2010 due to breakthroughs addressing early 1970s technical difficulties. Second, particle physics casts additional doubt on Friedman’s constitutive a priori role for the principle of equivalence. Massive spin-2 gravity presumably should have nearly the same empirical content as General Relativity while differing radically on foundational issues. Empirical content even in General Relativity resides in partial differential equations, not in an additional principle identifying gravity and inertia. Third, Kant’s apparent claim that Newton’s results could be known a priori is undermined by an alternate gravitational equation. The modified theory has a smaller (Galilean) symmetry group than does Newton’s. What Kant wanted from Newton’s gravity is impossible due its large symmetry group, but is closer to achievable given the alternative theory.

Topology of the space of periodic ground states in the antiferromagnetic Ising and Potts models in selected spatial structures

We present Monte Carlo studies and analysis of the frustrated antiferromagnetic Potts model of a triangular lattice. This Potts model shows a remarkably rich range of structures, and striking similarities to the high pressure phases of hydrogen which are typified by hexagonal close packed layered structures []. There are four known H2 molecular phases, all of which are isostructural to within the resolution of x-ray diffraction. Experimentally, the phase lines have been mapped by spectroscopy, which cannot reveal the structure. Study by density functional theory (DFT) has suggested a large number of candidate structures, based on the hexagonal-close packing of H2 molecules. The Potts model exhibits structures similar to DFT candidate hydrogen phases I, II and III: the range of different Potts model structures suggests that the hydrogen system in the ‘phase II’ region, may exhibit more than a single phase. It also suggests reorientational excitations which may be detectable in spectroscopy.

The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström's theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem's requirements.

We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpi\'nski curve as its Gromov boundary.

The kinetics of domain growth for a q=8 state Potts model on a triangular lattice is studied using the Monte Carlo renormalization-group technique. A block-spin transformation is applied to the evolving configurations of a two-dimensional Potts model with nonconserved order parameter. The growth law for the average size of domains is consistent with curvature-driven growth. A scaling form for the structure factor is obtained and compared to theory. The edge distribution function for the growth kinetics is also studied.

The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.

Group representation theory, bifurcation theory and pattern formation

The authors present some analytical results for the Potts model, using the symmetry group (acting on its parameters) generated by the inverse relation and other symmetries of this model. In particular, they find the critical manifolds and study the relationship between this group and the Lee-Yang singularities in the complex plane. For those cases where the symmetry group is finite, they look at the possible consequences for some colouring problems in graph theory and more specifically for chromatic polynomials.

At special loci in their moduli spaces, Calabi–Yau manifolds are endowed with discrete symmetries. Over the years, such spaces have been intensely studied and have found a variety of important applications. As string compactifications they are phenomenologically favored, and considerably simplify many important calculations. Mathematically, they provided the framework for the first construction of mirror manifolds, and the resulting rational curve counts. Thus, it is of significant interest to investigate such manifolds further. In this paper, we consider several unexplored loci within familiar families of Calabi–Yau hypersurfaces that have large but unexpected discrete symmetry groups. By deriving, correcting, and generalizing a technique similar to that of Candelas, de la Ossa and Rodriguez–Villegas, we find a calculationally tractable means of finding the Picard–Fuchs equations satisfied by the periods of all 3–forms in these families. To provide a modest point of comparison, we then briefly investigate the relation between the size of the symmetry group along these loci and the number of nonzero Yukawa couplings. We include an introductory exposition of the mathematics involved, intended to be accessible to physicists, in order to make the discussion self–contained.