A multilayer contact process

Arxiv preprint cond-mat/9812243, 1998

We study the Gierer-Meinhardt model of reaction-diffusion on a site-disordered square lattice. Let p be the site occupation probability of the square lattice. For p greater than a critical value p c , the steady state consists of stripe-like patterns with long-range connectivity. For p < p c , the connectivity is lost. The value of p c is found to be much greater than that of the site percolation threshold for the square lattice. In the vicinity of p c , the cluster-related quantities exhibit power-law scaling behaviour. The method of finite-size scaling is used to determine the values of the fractal dimension d f , the ratio, γ ν , of the average cluster size exponent γ and the correlation length exponent ν and also ν itself. The values appear to indicate that the disordered GM model belongs to the universality class of ordinary percolation.

2003

We analyze a deterministic cellular automaton σ · = (σn: n ≥ 0) corresponding to the zero-temperature case of Domany’s stochastic Ising ferromagnet on the hexagonal lattice H. The state space SH = {−1,+1} H consists of assignments of −1 or +1 to each site of H and the initial state σ0 = {σ0 x}x∈H is chosen randomly with P(σ0 x = +1) = p ∈ [0,1]. The sites of H are partitioned in two sets A and B so that all the neighbors of a site x in A belong to B and vice versa, and the discrete time dynamics is such that the σ · x’s with x ∈ A (respectively, B) are updated simultaneously at odd (resp., even) times, making σ · x agree with the majority of its three neighbors. In [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 ...

Physical Review E, 2005

It is argued that some phase-transitions observed in models of non-equilibrium wetting phenomena are related to contact processes with long-range interactions. This is investigated by introducing a model where the activation rate of a site at the edge of an inactive island of length ℓ is 1 + aℓ −σ . Mean-field analysis and numerical simulations indicate that for σ > 1 the transition is continuous and belongs to the universality class of directed percolation, while for 0 < σ < 1, the transition becomes first order. This criterion is then applied to discuss critical properties of various models of non-equilibrium wetting.

J Theor Probability, 2002

We study the contact process in Z d and a family of two-parametric oriented percolation models in Z d × Z +. It is proved that the derivative at the endpoint of the critical curve for percolation exists and its absolute value coincides with the critical rate for the corresponding contact process.

Theoretical and Mathematical Physics, 1985

Percolation models in which the centers of defects are distributed randomly in space in accordance with Poisson's law and the shape of each defect is also random are considered. Methods of obtaining rigorous estimates of the critical densities are described. It is shown that the number of infinite clusters can take only three values: 0, 1, or ~. Models in which the defects have an elongated shape and a random orientation are investigated in detail. In the two-dimensional case, it is shown that the critical volume concentration of the defects is proportional to all, where l and a are, respectively, the major and minor axes of the defect; the mean number of (direct) bonds per defect when percolation occurs is bounded.

Communications in Mathematical Physics, 1987

The equality of two critical points-the percolation threshold p H and the point p τ where the cluster size distribution ceases to decay exponentiallyis proven for all translation invariant independent percolation models on homogeneous d-dimensional lattices (d^ 1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter M(β, h\ which for h = Q reduces to the percolation density P^-at the bond density p = l-e~β in the single parameter case. These are: (1) M^hdM/dh + M 2 + βMdM/dβ, and (2) dM/dβ^\J\MdM/dh. Inequality (1) is intriguing in that its derivation provides yet another hint of a "φ 3 structure" in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents β and δ. One of these resembles an Ising model inequality of Frόhlich and Sokal and yields the mean field bound (5^2, and the other implies the result of Chayes and Chayes that β^ί. An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation /?((5-1)^1 and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.

Journal of Physics A: Mathematical and General, 2002

A criterion for the uniqueness of limiting Gibbs states in classical models with unique ground states is formulated. Various applications of this criterion formulated in the terminology of percolation theory are discussed.

arXiv (Cornell University), 2001

Using a Boltzmann-like equation, we investigate the nonequilibrium dynamics of nonperturbative fluctuations within the context of Ginzburg-Landau models. As an illustration, we examine how a two-phase system initially prepared in a homogeneous, low-temperature phase becomes populated by precursors of the opposite phase as the temperature is increased. We compute the critical value of the order parameter for the onset of percolation, which signals the breakdown of the conventional dilute gas approximation.

Theoretical and Mathematical Physics, 1985

Percolation models in which defect centers are distributed randomly in space in accordance with Poisson's law and the shape of each defect is also random are considered. Coincidence of two critical points is proved. One of these corresponds to the time when the mean number of defects connected to a given defect becomes infinite. The other corresponds to the existence of percolation in an arbitrarily large region of space.

Physical Review E, 2007

We investigate the critical behavior of a model with two coupled critical densities, one of which is diffusive. The model simulates the propagation of an epidemic process in a population, which uses the underlying lattice to leave a track of the recent disease history. We determine the critical density of the population above which the system reaches an active stationary state with a finite density of active particles. We also perform a scaling analysis to determine the order parameter, the correlation length, and critical relaxation exponents. We show that the model does not belong to the usual directed percolation universality class and is compatible with the class of directed percolation with diffusive and conserved fields.