On the Anisotropic Walk on the Supercritical Percolation Cluster

We investigate in this work the asymptotic behavior of an anisotropic random walk on the supercritical cluster for bond percolation on ^d, d2. In particular we show that for small anisotropy the walk behaves in a ballistic fashion, whereas for strong anisotropy the walk is sub-diffusive. For arbitrary anisotropy, we also prove the directional transience of the walk and construct a renewal structure.     

A Note on Anisotropic Percolation

We consider an anisotropic independent bond percolation model on , i.e. we suppose that the vertical edges of are open with probability p and closed with probability 1–p, while the horizontal edges of are open with probability p and closed with probability 1– p, with 0 < p, < 1. Let , with x₁ < x₂, and . It is natural to ask how the two point connectivity function P_p,({0 x}) behaves, and whether anisotropy in percolation probabilities implies the strict inequality P_p,({0 x})> P_p,({0 x}). In this note we give affirmative answer at least for some regions of the parameters involved.Mathematics Subject Classifications (2000). 82B20, 82B41, 82B43.     

On Large Isolated Regions in Supercritical Percolation

We consider supercritical vertex percolation in ^d with any non-degenerate uniform oriented pattern of connection. In particular, our results apply to the more special unoriented case. We estimate the probability that a large region is isolated from . In particular, pancakes with a radius r and constant thickness, parallel to a constant linear subspace L, are isolated with probability, whose logarithm grows asymptotically as r ^dim(L) if percolation is possible across L and as r ^dim(L)–1 otherwise. Also we estimate probabilities of large deviations in invariant measures of some cellular automata.     

Percolation in a Voronoi Competition-Growth Model

We study a model in which two entities (e.g., plant species, political ideas,...) compete for space on a plane, starting from randomly distributed seeds and growing deterministically at possibly different rates. An entity which forms an infinite cluster is considered to dominate over the other (which then cannot percolate). We analyze the occurrence of such a form of domination in situations in which one entity starts from a much larger density of seeds than the other one, but the latter one grows at a much faster rate than the former one. The model studied here generalizes the problem of Voronoi percolation.     

Metastability Thresholds for Anisotropic Bootstrap Percolation in Three Dimensions

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability thresholds for a fairly general class of models. In our proofs, we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the ‘easiest growth direction’ in the model. In contrast to anisotropic bootstrap percolation in two dimensions, in three dimensions the order of the metastability threshold for anisotropic bootstrap percolation can be equal to that of isotropic bootstrap percolation.     

Finite-Size Effects for Anisotropic Bootstrap Percolation: Logarithmic Corrections

In this note we analyse an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.     

Phase Transitions in a Highly Anisotropic Antiferromagnetic Heisenberg Toda Chain

The behavior of the antiferromagnetic spin 112 Heisenberg Toda chain is investigated in a space of interactions which include exchange anisotropy, bond alternation and Toda-like spinlattice coupling by means of coherent-state for spin. We analyse magnetically driven lattice instabilities and find that multiphase structure only occurs under the condition, of bond alternation. Three phase diagrams and tricritical points are obtained and some relevant physical properties are discussed.     

On the Localized Phase of a Copolymer in an Emulsion: Supercritical Percolation Regime

In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, A and B, each occurring with density . The emulsion is a random mixture of liquids of two types, A and B, organised in large square blocks occurring with density p and 1 − p, respectively, where p ϵ (0, 1). The copolymer in the emulsion has an energy that is minus α times the number of AA-matches minus β times the number of BB-matches, where without loss of generality the interaction parameters can be taken from the cone . To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In [7], a variational expression was derived for the quenched free energy per monomer in the limit as the length n of the copolymer tends to infinity and the blocks in the emulsion have size L _n such that L _n → ∞ and L _n /n → 0. Under this restriction, the free energy is self-averaging with respect to both types of randomness. It was found that in the supercritical percolation regime p ≥ p _c , with p _c the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of p. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the A-blocks to a phase where it is partially localized near the AB-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase. In the subcritical percolation regime p < p _c , the phase diagram is much more complex. This regime will be treated in a forthcoming paper.     

Anomalous Diffusion in a Model with a Variable Percolation Radius

Anomalous diffusion models for random 1-D cluster and comb structures of length L = 100 with finite fingers and different boundary conditions are considered. The effect of electric field on anomalous diffusion is discussed. The cases with different percolation radii are compared. The comb-structure model with periodic boundary conditions is shown to be useful in studying various types of anomalous diffusion. A new diffusion type, where the average rate is higher than the typical rate, is predicted. Physical causes for this diffusion are revealed.     

A Note on Truncated Long-Range Percolation with Heavy Tails on Oriented Graphs

We consider oriented long-range percolation on a graph with vertex set \({\mathbb {Z}}^d \times {\mathbb {Z}}_+\) and directed edges of the form \(\langle (x,t), (x+y,t+1)\rangle \), for x, y in \({\mathbb {Z}}^d\) and \(t \in {\mathbb {Z}}_+\). Any edge of this form is open with probability \(p_y\), independently for all edges. Under the assumption that the values \(p_y\) do not vanish at infinity, we show that there is percolation even if all edges of length more than k are deleted, for k large enough. We also state the analogous result for a long-range contact process on \({\mathbb {Z}}^d\).     

The Central Limit Theorem for Supercritical Oriented Percolation in Two Dimensions

We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the standard normal law. This resolves a longstanding open problem pointed out to in several instances in the literature. The result applies also to the continuous-time analog of the process, viz. the basic one-dimensional contact process. We also derive general random-indices central limit theorems for associated random variables as byproducts of our proof.     

Analysis of a Drop-Push Model for Percolation and Coagulation

In this article, we study a type of a one dimensional percolation and coagulation model whose basic features include a sequential dropping of particles on a substrate followed by their transport via a pushing mechanism. Particles are dropped onto a one dimensional lattice and carry out a random walk until they encounter an empty site where they become stuck. In such a model, calculating the probability of coalescence of two arbitrary clusters of particles, we embed a certain coalescence process, called the additive Marcus-Lushnikov process, which converges to a particular solution of the Smoluchowski equation. Throughout, we study the asymptotic behavior of the arrangement of empty sites and of the total displacement of all particles as well as the partial displacement of some particles, when the number of sites and of the particles tend to infinite.     

The Percolation Transition in the Zero-Temperature Domany Model

We analyze a deterministic cellular automaton σ ^?=(σ ⁿ :n≥0) corresponding to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice $\mathbb{N}$ . The state space $\mathcal{S}_\mathbb{H} = \left\{ { - 1, + 1} \right\}^\mathbb{H}$ consists of assignments of ?1 or +1 to each site of $\mathbb{H}$ and the initial state $\sigma ^0 = \left\{ {\sigma _{^x }^0 } \right\}_{x \in \mathbb{H}}$ is chosen randomly with P(σ ⁰ _x=+1)=p∈[0,1]. The sites of $\mathbb{H}$ are partitioned in two sets $\mathcal{A}$ and $\mathcal{B}$ so that all the neighbors of a site x in $\mathcal{A}$ belong to $\mathcal{B}$ and vice versa, and the discrete time dynamics is such that the σ ^? _x 's with ${x \in \mathcal{A}}$ (respectively, $\mathcal{B}$ ) are updated simultaneously at odd (resp., even) times, making σ ^? _x agree with the majority of its three neighbors. In ref. 1 it was proved that there is a percolation transition at p=1/2 in the percolation models defined by σ ⁿ , for all times n∈[1,∞]. In this paper, we study the nature of that transition and prove that the critical exponents β, ν, and η of the dependent percolation models defined by σ ⁿ , n∈[1,∞], have the same values as for standard two-dimensional independent site percolation (on the triangular lattice).     

An Anisotropic Homogeneous Cosmological Model in a Modified Brans-Dicke Cosmology

Adding the cosmological term, which is assumed to be variable in Brans-Dicke theory we have discussed about a spatially homogeneous and anisotropic cosmological model corresponding to Bianchi type-I solution. The physical and geometrical properties of this model has been discussed. Finally this model has been transformed to the original form (1961) of Brans-Dicke theory (including a variable cosmological term).     

Thermal Entanglement in a Two-Qutrit Spin-1 Anisotropic Heisenberg Model

The negativity (N) as a measure of thermal entanglement (TE) is studied for a two-qutrit spin-1 anisotropic Heisenberg XXZ chain with Dzyaloshinskii-Moriya (DM)...