Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

We demonstrate that the invaded cluster algorithm, recently introduced by Machta et al, is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits. On two quasiperiodic graphs which were not investigated in this respect before, the twelvefold symmetric square-triangle tiling and the tenfold symmetric T\"ubingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from the Onsager value. But also notable exceptions to the criterion are found which consist not only of the exactly solvable cases, in agreement with a recent exact result, but also of the self-dual ones and maybe more.Comment: 15 pages, 5 figures; v2: Fig. 5b replaced, minor change

Single-crossover dynamics: finite versus infinite populations

Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we summarise and adapt a deterministic approach, as valid for infinite populations, which assumes continuous time and single crossover events. The corresponding nonlinear system of differential equations permits a closed solution, both in terms of the type frequencies and via linkage disequilibria of all orders. To include stochastic effects, we then consider the corresponding finite-population model, the Moran model with single crossovers, and examine it both analytically and by means of simulations. Particular emphasis is on the connection with the deterministic solution. If there is only recombination and every pair of recombined offspring replaces their pair of parents (i.e., there is no resampling), then the {\em expected} type frequencies in the finite population, of arbitrary size, equal the type frequencies in the infinite population. If resampling is included, the stochastic process converges, in the infinite-population limit, to the deterministic dynamics, which turns out to be a good approximation already for populations of moderate size.Comment: 21 pages, 4 figure

Symmetries and reversing symmetries of toral automorphisms

Toral automorphisms, represented by unimodular integer matrices, are investigated with respect to their symmetries and reversing symmetries. We characterize the symmetry groups of GL(n,Z) matrices with simple spectrum through their connection with unit groups in orders of algebraic number fields. For the question of reversibility, we derive necessary conditions in terms of the characteristic polynomial and the polynomial invariants. We also briefly discuss extensions to (reversing) symmetries within affine transformations, to PGL(n,Z) matrices, and to the more general setting of integer matrices beyond the unimodular ones.Comment: 34 page

Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.Comment: 16 pages; revised and slightly expanded versio

Classification of one-dimensional quasilattices into mutual local-derivability classes

One-dimensional quasilattices are classified into mutual local-derivability (MLD) classes on the basis of geometrical and number-theoretical considerations. Most quasilattices are ternary, and there exist an infinite number of MLD classes. Every MLD class has a finite number of quasilattices with inflation symmetries. We can choose one of them as the representative of the MLD class, and other members are given as decorations of the representative. Several MLD classes of particular importance are listed. The symmetry-preserving decorations rules are investigated extensively.Comment: 42 pages, latex, 5 eps figures, Published in JPS

Discrete Tomography of Planar Model Sets

Discrete tomography is a well-established method to investigate finite point sets, in particular finite subsets of periodic systems. Here, we start to develop an efficient approach for the treatment of finite subsets of mathematical quasicrystals. To this end, the class of cyclotomic model sets is introduced, and the corresponding consistency, reconstruction and uniqueness problems of the discrete tomography of these sets are discussed.Comment: 27 pages, 9 figure

An ultrametric state space with a dense discrete overlap distribution: Paperfolding sequences

We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support is the full interval [-1; +1]. The space of paperfolding sequences has an ultrametric structure. Our example provides an illustration of some properties which were suggested to occur for pure states in spin glass models

Resonant excitations of the 't Hooft-Polyakov monopole

The spherically symmetric magnetic monopole in an SU(2) gauge theory coupled to a massless Higgs field is shown to possess an infinite number of resonances or quasinormal modes. These modes are eigenfunctions of the isospin 1 perturbation equations with complex eigenvalues, $E_n=\omega_n-i\gamma_n$, satisfying the outgoing radiation condition. For $n\to\infty$, their frequencies $\omega_n$ approach the mass of the vector boson, $M_W$, while their lifetimes $1/\gamma_n$ tend to infinity. The response of the monopole to an arbitrary initial perturbation is largely determined by these resonant modes, whose collective effect leads to the formation of a long living breather-like excitation characterized by pulsations with a frequency approaching $M_W$ and with an amplitude decaying at late times as $t^{-5/6}$.Comment: 4 page

Schwinger Boson Formulation and Solution of the Crow-Kimura and Eigen Models of Quasispecies Theory

We express the Crow-Kimura and Eigen models of quasispecies theory in a functional integral representation. We formulate the spin coherent state functional integrals using the Schwinger Boson method. In this formulation, we are able to deduce the long-time behavior of these models for arbitrary replication and degradation functions. We discuss the phase transitions that occur in these models as a function of mutation rate. We derive for these models the leading order corrections to the infinite genome length limit.Comment: 37 pages; 4 figures; to appear in J. Stat. Phy

Counting Berg partitions

We call a Markov partition of a two dimensional hyperbolic toral automorphism a Berg partition if it contains just two rectangles. We describe all Berg partitions for a given hyperbolic toral automorphism. In particular there are exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same connectivity matrix (k, l, m, n)