Coarsening and persistence in the voter model

Ben‐Naim, E.; Frachebourg, L.; Krapivsky, P. L.

doi:10.1103/physreve.53.3078

Cited by 168 publications

(200 citation statements)

References 24 publications

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“…The probability of this is exponentially small in n. If the domains at q = ∞ were not correlated, the constants A(q) and B(q) would be given by (28,29). The existence of correlations between the lengths of domains (22) alters the coefficients of the decay, but not the exponential decay itself.…”

Section: Large Xmentioning

confidence: 99%

Distribution of domain sizes in the zero temperature Glauber dynamics of the one-dimensional Potts model

1996

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For the zero temperature Glauber dynamics of the q-state Potts model, we calculate the exact distribution of domain sizes by mapping the problem on an exactly soluble one-species coagulation model (A+A → A). In the long time limit, this distribution is universal and from its (complicated) exact expression, we extract its behavior in various regimes. Our results are tested in a simulation and compared to the predictions of a simple approximation proposed recently. Considering the dynamics of domain walls as a reaction diffusion model A + A → A with probability (q − 2)/(q − 1) and A + A → ∅ with probability 1/(q − 1), we calculate the pair correlation function in the long time regime.

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Section: Large Xmentioning

confidence: 99%

Distribution of domain sizes in the zero temperature Glauber dynamics of the one-dimensional Potts model

1996

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“…the cyclic invasion processes can maintain a selforganizing domain structure. Although this phenomenon is investigated extensively in different areas, such as the chemical reactions on crystal surfaces [1,2], biological (Lotka-Volterra) systems [3][4][5][6], Rock-Scissors-Paper (RSP) games in evolutionary game theories [7], cyclically dominated voter models [8][9][10], the mechanism sustaining the polydomain patterns is not well understood. At the same time, this type of spatial self-organizations is believed to play crucial role in the biological evolution [11] and it can provide protection for the participiants against some external invadors [12,13].…”

Section: Introductionmentioning

confidence: 99%

Three-state cyclic voter model extended with Potts energy

Szabó

Szolnoki

2002

Phys. Rev. E

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The cyclically dominated voter model on a square is extended by taking into consideration the variation of Potts energy during the nearest neighbor invasions. We have investigated the effect of surface tension on the self-organizing patterns maintained by the cyclic invasions. A geometrical analysis is also developed to study the three-color patterns. These investigations indicate clearly that in the "voter model" limit the pattern evolution is governed by the loop creation due to the overhanging during the interfacial roughening. Conversely, in the presence of surface tension the evolution is governed by spiral formation whose geometrical parameters depend on the strength of cyclic dominance.

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“…Lett. 77, 1420 (1996)].The 'persistence exponent', θ, which characterizes the decay of the probability that a stochastic variable exceeds a threshold value (typically its mean value) throughout a time interval, has attracted a great deal of recent interest [1][2][3][4][5][6][7][8][9][10][11]. One of the most surprising properties of this exponent is that its value is highly non-trivial even in simple systems.…”

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confidence: 99%

Non-Markovian persistence and nonequilibrium critical dynamics

1997

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The persistence exponent θ for the global order parameter, M (t), of a system quenched from the disordered phase to its critical point describes the probability, p(t) ∼ t −θ , that M (t) does not change sign in the time interval t following the quench. We calculate θ to O(ǫ 2 ) for model A of critical dynamics (and to order ǫ for model C) and show that at this order M (t) is a non-Markov process. Consequently, θ is a new exponent. The calculation is performed by expanding around a Markov process, using a simplified version of the perturbation theory recently introduced by Sire [Phys. Rev. Lett. 77, 1420 (1996)].The 'persistence exponent', θ, which characterizes the decay of the probability that a stochastic variable exceeds a threshold value (typically its mean value) throughout a time interval, has attracted a great deal of recent interest [1][2][3][4][5][6][7][8][9][10][11]. One of the most surprising properties of this exponent is that its value is highly non-trivial even in simple systems. For example, θ is irrational for the q > 2 Potts model in one dimension [6] (where the fraction of spins that have not changed their state in the time t after a quench to T = 0 decays as t −θ ) and is apparently not a simple fraction for the diffusion equation [9,10] (where the fraction of space where the diffusion field has always exceeded its mean decays as t −θ ).A recent study of non-equilibrium model A critical dynamics, where a system coarsens at its critical point starting from a disordered initial condition, looked at the probability P (t 1 , t 2 ) that the global magnetization does not change sign during the interval t 1 < t < t 2 [11]. The persistence exponent for this system is defined by P (t 1 , t 2 ) ∼ (t 1 /t 2 ) θ in the limit t 2 /t 1 → ∞. Explicit results were obtained for the 1D Ising model, the n → ∞ limit of the O(n) model, and to order ǫ = 4 − d near dimension d = 4. For these systems it was found that the value of θ was related to the dynamic critical exponent z, the static critical exponent η, and 'nonequilibrium' exponent λ (which describes the decay of the autocorrelation with the initial condition, φ(x, t)φ(x, 0) ∼ t −λ/z ) by the scaling relation θz = λ − d + 1 − η/2. This relation may be derived from the assumption that the dynamics is Markovian, which is indeed the case for all of the cases considered in that paper.From a consideration of the structure of the diagrams which appear at order ǫ 2 (and higher order), however, it was argued that the dynamics of the global order parameter should not be Markovian to all orders, implying that the exponent θ does not obey exactly that 'Markovian scaling relation ' [11]. This means that θ is a new exponent. Monte-Carlo simulations in 2 dimensions indeed suggest weak violation of the Markov scaling relation [11].In this Rapid Communication we present an explicit calculation of the non-Markovian properties of the global order parameter. The nonequilibrium magnetizationmagnetization correlation function is calculated to order ǫ 2 , and this is then used ...

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Coarsening and persistence in the voter model

Cited by 168 publications

References 24 publications

Distribution of domain sizes in the zero temperature Glauber dynamics of the one-dimensional Potts model

Distribution of domain sizes in the zero temperature Glauber dynamics of the one-dimensional Potts model

Three-state cyclic voter model extended with Potts energy

Non-Markovian persistence and nonequilibrium critical dynamics

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