Critical behavior of percolation and Markov fields on branching planes

For an independent percolation model on, whereis a homogeneous tree andis a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probabilityθ(p) is a continuous function ofpat the critical pointpc, and the critical exponents,γ,δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields onare also obtained.

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Critical behavior of percolation and Markov fields on branching planes

Journal of Applied Probability ◽

10.1017/s0021900200044296 ◽

1993 ◽

Vol 30 (03) ◽

pp. 538-547 ◽

Cited By ~ 1

Author(s):

C. Chris Wu

Keyword(s):

Continuous Function ◽

Critical Behavior ◽

Mean Field ◽

Percolation Model ◽

Homogeneous Tree ◽

Dimensional Lattice ◽

One Dimensional ◽

Percolation Probability ◽

Triangle Condition ◽

Markov Fields

For an independent percolation model on , where is a homogeneous tree and is a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probability θ (p) is a continuous function of p at the critical point p c, and the critical exponents , γ, δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields on are also obtained.

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MEAN-FIELD CRITICAL BEHAVIOR AND ERGODICITY BREAK IN A NONEQUILIBRIUM ONE-DIMENSIONAL RSOS GROWTH MODEL

International Journal of Modern Physics C ◽

10.1142/s0129183112500192 ◽

2012 ◽

Vol 23 (03) ◽

pp. 1250019 ◽

Cited By ~ 1

Author(s):

J. RICARDO G. MENDONÇA

Keyword(s):

Phase Transition ◽

Probability Density ◽

Critical Behavior ◽

Random Number Generator ◽

Mean Field ◽

System Size ◽

Ginzburg Landau ◽

One Dimensional ◽

Roughening Transition ◽

Pseudo Random Number Generator

We investigate the nonequilibrium roughening transition of a one-dimensional restricted solid-on-solid model by directly sampling the stationary probability density of a suitable order parameter as the surface adsorption rate varies. The shapes of the probability density histograms suggest a typical Ginzburg–Landau scenario for the phase transition of the model, and estimates of the "magnetic" exponent seem to confirm its mean-field critical behavior. We also found that the flipping times between the metastable phases of the model scale exponentially with the system size, signaling the breaking of ergodicity in the thermodynamic limit. Incidentally, we discovered that a closely related model not considered before also displays a phase transition with the same critical behavior as the original model. Our results support the usefulness of off-critical histogram techniques in the investigation of nonequilibrium phase transitions. We also briefly discuss in the appendix a good and simple pseudo-random number generator used in our simulations.

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Correlation length critical exponent as a function of the percolation radius for one-dimensional chains in bond problems

Journal of Physics Conference Series ◽

10.1088/1742-6596/2094/2/022038 ◽

2021 ◽

Vol 2094 (2) ◽

pp. 022038

Author(s):

T V Yakunina ◽

V N Udodov

Keyword(s):

Critical Exponent ◽

Percolation Threshold ◽

Correlation Length ◽

Adjacency Matrix ◽

Nearest Neighbors ◽

Percolation Model ◽

Dimensional Lattice ◽

One Dimensional ◽

The One

Abstract A one-dimensional lattice percolation model is constructed for the problem of constraints flowing along non-nearest neighbors. In this work, we calculated the critical exponent of the correlation length in the one-dimensional bond problem for a percolation radius of up to 6. In the calculations, we used a method without constructing a covering lattice or an adjacency matrix (to find the percolation threshold). The values of the critical exponent of the correlation length were obtained in the one-dimensional bond problem depending on the size of the system and at different percolation radii. Based on original algorithms that operate on a computer faster than standard ones (associated with the construction of a covering lattice), these results are obtained with corresponding errors.

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POPULATION DYNAMICS IN HETEROGENEOUS ENVIRONMENTS: A DISCRETE MODEL

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127400001262 ◽

2000 ◽

Vol 10 (08) ◽

pp. 1993-2000 ◽

Cited By ~ 1

Author(s):

BASTIEN FERNANDEZ ◽

VALERY TERESHKO

Keyword(s):

Population Dynamics ◽

Periodic Orbit ◽

Discrete Model ◽

Mean Field ◽

Heterogeneous Environments ◽

Dimensional Lattice ◽

One Dimensional ◽

Population Spread ◽

Asymptotic Dynamics ◽

Time Periodic

We study the dynamics of a multidimensional coordinate-dependent mapping governing the time evolution of a population spread over a one-dimensional lattice. The nonlinearity is of mean-field type and the dependence on coordinates, given by the so-called fitness, allows to take into account the spatial heterogeneities of the habitat. A global picture of the dynamics is given in the case without diffusion and in the case with diffusion when the fitness is homogeneous and leads to a periodic orbit. Moreover it is shown that, periodic fitnesses close to homogeneous ones impose their periodicity on the asymptotic dynamics when the latter is time-periodic.

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Overdamped dynamics of long-range systems on a one-dimensional lattice: Dominance of the mean-field mode and phase transition

Physical Review E ◽

10.1103/physreve.86.061130 ◽

2012 ◽

Vol 86 (6) ◽

Cited By ~ 7

Author(s):

Shamik Gupta ◽

Alessandro Campa ◽

Stefano Ruffo

Keyword(s):

Phase Transition ◽

Long Range ◽

Mean Field ◽

Field Mode ◽

Dimensional Lattice ◽

One Dimensional ◽

The Mean

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CRITICAL BEHAVIOR OF A PROBABILISTIC LOCAL AND NONLOCAL SITE-EXCHANGE CELLULAR AUTOMATON

International Journal of Modern Physics C ◽

10.1142/s0129183194000714 ◽

1994 ◽

Vol 05 (03) ◽

pp. 537-545 ◽

Cited By ~ 6

Author(s):

N. BOCCARA ◽

J. NASSER ◽

M. ROGER

Keyword(s):

Cellular Automaton ◽

Critical Behavior ◽

Exchange Process ◽

Mean Field ◽

One Dimensional ◽

Probabilistic Cellular Automaton ◽

Site Exchange ◽

Two Parameters ◽

Automata Network ◽

Dimensional Range

We study the critical behavior of a probabilistic automata network whose local rule consists of two subrules. The first one, applied synchronously, is a probabilistic one-dimensional range-one cellular automaton rule. The second, applied sequentially, exchanges the values of a pair of sites. According to whether the two sites are first-neighbors or not, the exchange is said to be local or nonlocal. The evolution of the system depends upon two parameters, the probability p characterizing the probabilistic cellular automaton, and the degree of mixing m resulting from the exchange process. Depending upon the values of these parameters, the system exhibits a bifurcation similar to a second order phase transition characterized by a nonnegative order parameter, whose role is played by the stationary density of occupied sites. When m is very large, the correlations created by the application of the probabilistic cellular automaton rule are destroyed, and, as expected, the behavior of the system is then correctly predicted by a mean-field-type approximation. According to whether the exchange of the site values is local or nonlocal, the critical behavior is qualitatively different as m varies.

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