Critical points of two-dimensional bootstrap percolation-like cellular automata

Cellular automata in two dimensions that generalize the bootstrap percolation dynamics are considered, focusing on the thresholdp _c of the initial density for convergence to total occupancy to occur; these models are classified according top _c being 0, 1, or strictly between these extreme values. Explicit upper and lower bounds are provided in the third case.     

Hierarchy of critical exponents of currents on -simplex fractals

Using the symmetry of ( d +1)-simplex fractals with decimation number b =2, the current distribution has been determined. Then using the renormalization group technique, based on the independent Schur's invariant polynomials of current distributions, the multifractal spectrum of even moments of current distributions has been evaluated analytically up to order six for an arbitrary value of d. Also the scaling exponents of order 8 and order 10 have been calculated numerically up to d =30. Received: 19 November 1997 / Revised: 21 January 1998 / Accepted: 9 February 1998     

Homology of Fortuin–Kasteleyn Clusters of Potts Models on the Torus

Topological properties of Fortuin–Kasteleyn clusters are studied on the torus. Namely, the probability that their topology yields a given subgroup of the first homology group of the torus is computed for Q=1, 2, 3 and 4. The expressions generalize those obtained by Pinson for percolation (Q=1). Numerical results are also presented for three tori of different moduli. They agree with the theoretical predictions for Q=1, 2 and 3. For Q=4 agreement is not ruled out but logarithmic corrections are probably present and they make it harder to decide.     

Random-Cluster Representation for the Blume–Capel Model

We present in this paper a way to perform the mapping of the spin-1 Blume–Capel model into a random-cluster model, and analyze thermodynamic properties of the former model in terms of geometric properties of clusters generated in the random-cluster representation. It is shown that there are two different relevant types of cluster, and that one of them is the exact analogue of the type of cluster generated in the Ising model. We use this result to derive expressions for thermodynamical properties on the second-order transition line which are equivalent to the ones found in the Ising model. The other type of cluster is responsible for the first-order transitions, and we may see the tricritical point as a point where both types of cluster compete on the same footing.     

Phase transitions in diluted magnets: Critical behavior,percolation, and random fields

Transition metal halides provide realizations of Ising,XY, and Heisenberg antiferromagnets in one, two, and three dimensions. The interactions, which are of short range, are generally well understood. By dilution with nonmagnetic species such as Zn⁺⁺ or Mg⁺⁺ one is able to prepare site-random alloys which correspond to random systems of particular interest in statistical mechanics. By mixing two magnetic ions such as Fe⁺⁺ and Co⁺⁺ one can produce magnetic crystals with competing interactions-either in the form of competing anisotropies or competing ferromagnetic and antiferromagnetic interactions. In this paper the results of a series of neutron scattering experiments on these systems carried out at Brookhaven over the past several years are briefly reviewed. First the critical behavior in Rb₂Mn_0.5Ni_0.5F₄ and Fe_cZn_1–cF₂ which correspond to two-dimensional and three-dimensional random Ising systems, respectively, are discussed. Percolation phenomena have been studied in Rb₂Mn_cMg_l–cF₄, Rb₂Co_cMg_l–cF₄, KMn_cZ_l-cF₃, and Mn_cZn_l–cF₂ which correspond to two-and three-dimensional Heisenberg and Ising models, respectively. In these casesc is chosen to be in the neighborhood of the nearest-neighbor percolation concentration. Application of a uniform field to the above systems generates a random staggered magnetic field; this has facilitated a systematic study of the random field problem. As we shall discuss in detail, a variety of novel, unexpected phenomena have been observed.     

Bootstrapping the change-point of a hazard rate

This paper concerns the asymptotic validity of the bootstrap method in a non-regular model. Specifically, it is shown that the parametric bootstrap of the change-point parameter in the change-point hazard rate model works.     

The existence of phase V in the Mandelbrot percolation process

Dekking and Meester defined six phases for a subclass of random Cantor sets consisting of those generated by Bernoulli random substitutions. They proved that the random Sierpinski carpet passed through all these phases asp tended from 0 to 1, but the were not able to prove the existencne of phase V in the Mandelbrot percolation process. In this paper, we accomplish the proof by improving their methods.Research supported by the Chinese Natural Science Foundation.     

三维自助式类渗流细胞自动机的临界值

本文研究三维自助式类渗流细胞自动机的临界值．得出临界值pc＝0的一些充分条件和必要条件．当pc＞0时，利用定向点渗流的临界值，给出了pc上、下界的显式估计．     

Propagation and trapping of excitations on percolation clusters

A study is presented of migration of optical or magnetic excitations on percolation clusters which terminates upon reaching a trapping site. The theory is based on the extension of results from the theory of random walks to systems without translational invariance, together with the use of scaling concepts. For the case of an excitation which resides on one type of atom in a randomly mixed crystal near the percolation threshold, new power laws for the time and concentration dependences of the mean number of sites visited at timet of the kinetics of arrival at traps are obtained. Some of these results are also tested for the first time by numerical simulations.