Percolation techniques in disordered spin flip dynamics: Relation to the unique invariant measure

Gielis, Guy; Maes, Christian

doi:10.1007/bf02102431

Cited by 11 publications

(15 citation statements)

References 14 publications

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“…On the site DP line, the DK PCA displays an inactive-active phase transition at the critical point p * 1 = p * 2 = 0.705 489(4) [16], corresponding to ε * = 0.294 511(4), within the rigorous bounds mentioned before (the numbers between parentheses indicate the uncertainty in the last digit or digits of the data). Curiously, the relationship between Stavskaya's PCA and the DK PCA seems to have gone unnoticed in previous investigations [6][7][8][9][10][11], although a coupling scheme with an "independent oriented percolation" process equivalent with site DP was used in [8]. It is worth mentioning that Stavskaya's model, together with another PCA introduced by the same epoch, Vasil'ev's model [6,17]which corresponds to the p 2 = 0 line in the DK PCA or, equivalently, to a probabilistic version of CA rule 18 in Wolfram's classification scheme [18]-predates the DK PCA and related models by almost two decades, but did not receive much attention, not even when CA and PCA reentered the mainstream scientific agenda in the 1980s.…”

Section: Stavskaya's Modelmentioning

confidence: 99%

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Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton

Mendonça

2011

Phys. Rev. E

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Stavskaya's model is a one-dimensional probabilistic cellular automaton (PCA) introduced in the end of the 1960s as an example of a model displaying a nonequilibrium phase transition. Although its absorbing state phase transition is well understood nowadays, the model never received a full numerical treatment to investigate its critical behavior. In this Brief Report we characterize the critical behavior of Stavskaya's PCA by means of Monte Carlo simulations and finite-size scaling analysis. The critical exponents of the model are calculated and indicate that its phase transition belongs to the directed percolation universality class of critical behavior, as would be expected on the basis of the directed percolation conjecture. We also explicitly establish the relationship of the model with the Domany-Kinzel PCA on its directed site percolation line, a connection that seems to have gone unnoticed in the literature so far.

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Section: Stavskaya's Modelmentioning

confidence: 99%

“…However, while many rigorous results exist for this model [1][2][3][5][6][7][8][9][10][11], it has never received a full numerical treatment to estimate its critical point and critical exponents. In this Brief Report we proceed to such an investigation of Stavskaya's model by Monte Carlo simulations and finite-size scaling techniques.…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton

Mendonça

2011

Phys. Rev. E

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“…The relaxation for typical realizations of the disorder in a general class of (quenched) random spin systems has been studied by Gielis and Maes (1996). Their result for the biased random voter model reads in our notations as follows (see also Klein 1994).…”

Section: In (215) C Dα Is An Increasing Function Defined On [0 ∞) mentioning

confidence: 99%

“…A powerful tool in the analysis of these systems is the so-called logarithmic Sobolev inequality. Gielis and Maes (1996) tackle the problem of estimating the asymptotic relaxation in a more general class of random spin systems with quenched disorder. Their method consists of a coupling of the spin system to the contact process and the use of percolation techniques on the graphical representation of the contact process.…”

Section: Introductionmentioning

confidence: 99%

Pixel Circuit for Organic Light-Emitting Diode-on-Silicon Microdisplays Using the Source Follower Structure

Kim¹,

Kwak²,

Lim³

et al. 2010

Jpn. J. Appl. Phys.

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We study the relaxation properties of the voter model with i.i.d. random bias. We prove under mild conditions that the disorder-averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent d/(d + α), where 0 < α 2 depends on the transition rates of the non-biased voter model. Under an additional assumption, we show that the above upper bound is optimal. The main ingredient of our proof is a result of Donsker and Varadhan. § Postdoctoraal onderzoeker FWO. Bursaal IWT. 0305-4470/99/447653+12$30.00

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“…In our analyzed case all the interactions have absolute value equal to 1, so we obtain T A 4d 2 . We will not give the proof of this point that can be found in [14] and we concentrate our attention on the proof of the second point.…”

Section: Auto-correlation Time For the Edwards-anderson Spin-glass Momentioning

confidence: 99%

Glauber dynamics of spin glasses at low and high temperature

Santis¹

2002

Annales De l'Institut Henri Poincare (B) Probability and Statistics

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We consider an increasing sequence of finite boxes L ⊂ Z 2 and a reversible stochastic frustrated Ising model having invariant measures satisfying free boundary conditions. We show that the spectral gap associated with the Edwards-Anderson model has a different asymptotic behavior in low and in high temperature. In low temperature, associated with the spectral gap, there is a qualitatively slower relaxation to equilibrium than there is in high temperature. Some geometrical lemmas are employed in the paper to show that some regions are almost independent from their exterior. We use for this aim a Peierls' argument.  2002 Éditions scientifiques et médicales Elsevier SAS AMS classification: 82C44; 82C26; 82C22 RÉSUMÉ.-Nous considérons, dans cet article, une suite strictement croissante de boites finies L ⊂ Z 2 et un modèle d'Ising stochastique réversible non ferromagnétique ayant des mesures invariantes avec conditions de frontière libre. Nous montrons que le frou spectral associé au modèle d'Edwards-Anderson a des comportements asymptotiques différents à haute et basse température. A basse température, la relaxation vers l'équilibre est qualitativement plus lente qu'à haute température. Nous employons dans cet article des lemmes géomêtriques pour montrer que certaines régions sont presque indépendantes de leur exterieur. À cette fin, nous utilisons un argument de type Peierls.  2002 Éditions scientifiques et médicales Elsevier SAS

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Percolation techniques in disordered spin flip dynamics: Relation to the unique invariant measure

Cited by 11 publications

References 14 publications

Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton

Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton

Pixel Circuit for Organic Light-Emitting Diode-on-Silicon Microdisplays Using the Source Follower Structure

Glauber dynamics of spin glasses at low and high temperature

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