Universal Scaling Behavior of Directed Percolation Around the Upper Critical Dimension

Lübeck, S.; Willmann, R. D.

doi:10.1023/b:joss.0000028059.24904.3b

Cited by 17 publications

(34 citation statements)

References 52 publications

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“…The values are partly more precise than previously reported estimates. Similar results for directed site percolation in d = 3, 4, and 5 dimensions were recently reported in [25,32]. Table B1 shows the results for directed bond percolation ranging from 1 to 7 spatial dimensions.…”

Section: Appendix B Dp In High Dimensionssupporting

confidence: 83%

Spreading with immunization in high dimensions

Dammer

Hinrichsen

2004

J. Stat. Mech.: Theor. Exp.

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We investigate a model of epidemic spreading with partial immunization which is controlled by two probabilities, namely, for first infections, p 0 , and reinfections, p. When the two probabilities are equal, the model reduces to directed percolation, while for perfect immunization one obtains the general epidemic process belonging to the universality class of dynamical percolation. We focus on the critical behavior in the vicinity of the directed percolation point, especially in high dimensions d > 2. It is argued that the clusters of immune sites are compact for d ≤ 4. This observation implies that a recently introduced scaling argument, suggesting a stretched exponential decay of the survival probability for p = p c , p 0 ≪ p c in one spatial dimension, where p c denotes the critical threshold for directed percolation, should apply in any dimension d ≤ 3 and maybe for d = 4 as well. Moreover, we show that the phase transition line, connecting the critical points of directed percolation and of dynamical percolation, terminates in the critical point of directed percolation with vanishing slope for d < 4 and with finite slope for d ≥ 4. Furthermore, an exponent is identified for the temporal correlation length for the case of p = p c and p 0 = p c − ǫ, ǫ ≪ 1, which is different from the exponent ν of directed percolation. We also improve numerical estimates of several critical parameters and exponents, especially for dynamical percolation in d = 4, 5.

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Section: Appendix B Dp In High Dimensionssupporting

confidence: 83%

Spreading with immunization in high dimensions

Dammer

Hinrichsen

2004

J. Stat. Mech.: Theor. Exp.

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“…The universal order parameter scaling function Rpbc (0, x, 1) (inset) and the universal fourth order ratio scaling function Qpbc (0, x, 1) as a function of the rescaled field a h h(aLL) d at criticality for d > dc. The analytically obtained scaling functions are in perfect agreement with numerical data of the five-dimensional contact process (CP, implemented on simple cubic lattices of size L = 4, 8, 16, λc = 1.13846 (11)) and of the five-dimensional site-directed percolation process (sDP, implemented via the Domany-Kinzel automaton [24] on lattices of a generalized bcc-like structure [22] of linear size L = 8, 16, 32, pc = 0.0359725(2) [25]). Note that the numerical data already belongs to the asymptotic scaling regime.…”

supporting

confidence: 65%

“…The convincing agreement between the numerical and the field theoretical results indicates that L 0 is sufficiently small for the quantities Eqs. (19)(20)(21)(22). The universal order parameter scaling function Rpbc (0, x, 1) (inset) and the universal fourth order ratio scaling function Qpbc (0, x, 1) as a function of the rescaled field a h h(aLL) d at criticality for d > dc.…”

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Finite-size scaling of directed percolation above the upper critical dimension

Lübeck

Janssen

2005

Phys. Rev. E

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We consider analytically as well as numerically the finite-size scaling behavior in the stationary state near the nonequilibrium phase transition of directed percolation within the mean field regime, i.e., above the upper critical dimension. Analogous to equilibrium, usual finite-size scaling is valid below the upper critical dimension, whereas it fails above. Performing a momentum analysis of associated path integrals we derive modified finite-size scaling forms of the order parameter and its higher moments. The results are confirmed by numerical simulations of corresponding high-dimensional lattice models.

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“…[12]. Monte Carlo simulations have determined the numerical values for the DP critical exponents in dimensions d < 4 to high precision [8,9], and confirmed the logarithmic corrections predicted by the RG [131,132].…”

Section: Renormalization and Dp Critical Exponentssupporting

confidence: 52%

Applications of field-theoretic renormalization group methods to reaction–diffusion problems

Täuber¹,

Howard²,

Vollmayr-Lee³

2005

J. Phys. A: Math. Gen.

278

506

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Abstract. We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction-diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction-diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction kA → ℓA (ℓ < k). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization, and Callan-Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative ǫ = d c − d expansions with respect to the upper critical dimension d c . With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, Lévy flights, persistence problems, and the influence of spatial boundaries. We also analyze field-theoretic approaches to nonequilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: evenoffspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.

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Universal Scaling Behavior of Directed Percolation Around the Upper Critical Dimension

Cited by 17 publications

References 52 publications

Spreading with immunization in high dimensions

Spreading with immunization in high dimensions

Finite-size scaling of directed percolation above the upper critical dimension

Applications of field-theoretic renormalization group methods to reaction–diffusion problems

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