One-dimensional contact process: Duality and renormalization

Hooyberghs, Jef; Vanderzande, Carlo

doi:10.1103/physreve.63.041109

Cited by 17 publications

(15 citation statements)

References 40 publications

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“…In the latter case, let P (0) ret (t) denote the probability that, starting the process with all but site 0 inactive, the state at time t returns to the initial state. As it is proved in Appendix A by exploiting the duality property of the contact process [30,31], this return probability precisely equals to the persistence probability P 0 (t) of site 0:…”

Section: Persistence By the Sdrg Methodsmentioning

confidence: 96%

“…Appendix A: Relationship to a return probability By the help of the duality of the contact process [30,31], an exact equivalence between the local persistence and a return probability in a slightly modified system can be proved as follows. In the quantum Hamiltonian formalism [30], the configurations of a system with L sites are described by states |η ≡ L i=1 |η i , where η i = 0, 1 correspond to inactive and active state at site i, respectively. The state of the system at time t, |ψ(t) = η p η (t)|η evolves according to the master equation…”

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confidence: 99%

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Scaling of local persistence in the disordered contact process

Juhász,

Kovács

2020

Preprint

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We study the time-dependence of the local persistence probability during a non-stationary time evolution in the disordered contact process in d = 1, 2, and 3 dimensions. We present a method for calculating the persistence with the strong-disorder renormalization group (SDRG) technique, which we then apply in the critical point analytically for d = 1 and numerically for d = 2, 3. According to the results, the average persistence decays at late times as an inverse power of the logarithm of time, with a universal, dimension-dependent generalized exponent. For d = 1, the distribution of sample-dependent local persistences is shown to be characterized by a universal limit distribution of effective persistence exponents. By a phenomenological approach of rare-region effects in the active phase, we obtain a non-universal algebraic decay of the average persistence for d = 1, and enhanced power laws for d > 1. As an exception, for randomly diluted lattices, the algebraic decay holds to be valid for d > 1, which is explained by the contribution of dangling ends. Results on the time-dependence of average persistence are confirmed by Monte Carlo simulations. We also prove the equivalence of the persistence with a return probability, a valuable tool for the argumentations.

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Section: Persistence By the Sdrg Methodsmentioning

confidence: 96%

mentioning

confidence: 99%

Scaling of local persistence in the disordered contact process

Juhász,

Kovács

2020

Preprint

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“…Note that the common definition of the spread for short-range CP through the second moment of the distance from the origin would be divergent in the LRCP for any finite t, hence the average of the logarithmic distance, which is finite, is considered here instead [13]. Due to the self-dual property of the CP [2,34], the average survival probability P (t) equals to the average density ρ(t) in the case when the process had been started from a fully active (12), we obtain for the asymptotic time-dependence of the average survival probability in the critical point…”

Section: B Scaling At Criticalitymentioning

confidence: 99%

Long-range epidemic spreading in a random environment

2015

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Modeling long-range epidemic spreading in a random environment, we consider a quenched disordered, d-dimensional contact process with infection rates decaying with the distance as 1/r d+σ . We study the dynamical behavior of the model at and below the epidemic threshold by a variant of the strong-disorder renormalization group method and by Monte Carlo simulations in one and two spatial dimensions. Starting from a single infected site, the average survival probability is found to decay as P (t) ∼ t −d/z up to multiplicative logarithmic corrections. Below the epidemic threshold, a Griffiths phase emerges, where the dynamical exponent z varies continuously with the control parameter and tends to zc = d + σ as the threshold is approached. At the threshold, the spatial extension of the infected cluster (in surviving trials) is found to grow as R(t) ∼ t 1/zc with a multiplicative logarithmic correction, and the average number of infected sites in surviving trials is found to increase as Ns(t) ∼ (ln t) χ with χ = 2 in one dimension.

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“…ln t (21) for long times with some unknown scaling functionf (δ). This kind of scaling form can be interpreted in the way that P(t) decays for long times as an inverse power of the time but the power δ that characterizes a given environment at a given time is a random variable having a time-independent probability densityf (δ) in the limit t → ∞.…”

Section: Distribution Of the Survival Probabilitymentioning

confidence: 99%

Distribution of dynamical quantities in the contact process, random walks, and quantum spin chains in random environments

Juhász

2014

Phys. Rev. E

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We study the distribution of dynamical quantities in various one-dimensional disordered models, the critical behavior of which is described by an infinite randomness fixed point. In the disordered contact process, the survival probability P(t) is found to show multiscaling in the critical point, meaning that P(t)=t-δ, where the (environment and time-dependent) exponent δ has a universal limit distribution when t→∞. The limit distribution is determined by the strong disorder renormalization group method analytically in the end point of a semi-infinite lattice, where it is found to be exponential, while, in the infinite system, conjectures on its limiting behaviors for small and large δ, which are based on numerical results, are formulated. By the same method, the survival probability in the problem of random walks in random environments is also shown to exhibit multiscaling with an exponential limit distribution. In addition to this, the (imaginary-time) spin-spin autocorrelation function of the random transverse-field Ising chain is found to have a form similar to that of survival probability of the contact process at the level of the renormalization approach. Consequently, a relationship between the corresponding limit distributions in the two problems can be established. Finally, the distribution of the spontaneous magnetization in this model is also discussed.

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One-dimensional contact process: Duality and renormalization

Cited by 17 publications

References 40 publications

Scaling of local persistence in the disordered contact process

Scaling of local persistence in the disordered contact process

Long-range epidemic spreading in a random environment

Distribution of dynamical quantities in the contact process, random walks, and quantum spin chains in random environments

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