The problem of directed percolation

Obukhov, S P

doi:10.1016/0378-4371(80)90105-3

Cited by 137 publications

(101 citation statements)

References 5 publications

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“…In the latter case the fluctuations diverge at the transition point whereas they remain finite (jump) in case of DP [34,35]. Another difference between both universality classes concerns the value of the upper critical dimension, namely D c = 4 for ordinary DP [36,37] and D c = 3 for TDP [22,23].…”

Section: Tricritical Directed Percolationmentioning

confidence: 99%

Tricritical Directed Percolation

Lübeck

2006

J Stat Phys

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We consider a modification of the contact process incorporating higher-order reaction terms. The original contact process exhibits a non-equilibrium phase transition belonging to the universality class of directed percolation. The incorporated higher-order reaction terms lead to a non-trivial phase diagram. In particular, a line of continuous phase transitions is separated by a tricritical point from a line of discontinuous phase transitions. The corresponding tricritical scaling behavior is analyzed in detail, i.e., we determine the critical exponents, various universal scaling functions as well as universal amplitude combinations.

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Section: Tricritical Directed Percolationmentioning

confidence: 99%

Tricritical Directed Percolation

Lübeck

2006

J Stat Phys

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“…Remarkably, the action (76) was first encountered and analyzed in particle physics under the guise of Reggeon field theory [123,124]. It was subsequently noticed that it actually represents a stochastic ('Gribov') process [125,126], and its equivalence to the geometric problem of directed percolation was established [127,128,120]. In directed bond percolation, randomly placed bonds connecting regular lattice sites can only be traversed in a given preferred special direction, which is to be identified with t in the dynamical problem.…”

Section: The Directed Percolation (Dp) Universality Classmentioning

confidence: 99%

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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“…The action (29) with λ = 0 is known as Reggeon field theory (Moshe 1978 [22]); it satisfies a characteristic symmetry, namely invariance under so-called rapidity inversion S(x, t) ↔ − S(x, −t). Remarkably, it has moreover been established that the field theory action (29) describes the scaling properties of critical directed percolation clusters [45,46,47]. The fluctuation-corrected universal power laws governing the vicinity of the phase transition can be extracted by renormalization group methods (reviewed for directed percolation in Ref.…”

Section: Active To Absorbing State Phase Transitionsmentioning

confidence: 99%

Field Theoretic Methods

Täuber

2009

Encyclopedia of Complexity and Systems Science

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1Summary. Many complex systems are characterized by intriguing spatio-temporal structures. Their mathematical description relies on the analysis of appropriate correlation functions. Functional integral techniques provide a unifying formalism that facilitates the computation of such correlation functions and moments, and furthermore allows a systematic development of perturbation expansions and other useful approximative schemes. It is explained how nonlinear stochastic processes may be mapped onto exponential probability distributions, whose weights are determined by continuum field theory actions. Such mappings are madeexplicit for (1) stochastic interacting particle systems whose kinetics is defined through a microscopic master equation; and (2) nonlinear Langevin stochastic differential equations which provide a mesoscopic description wherein a separation of time scales between the relevant degrees of freedom and background statistical noise is assumed. Several well-studied examples are introduced to illustrate the general methodology.

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The problem of directed percolation

Cited by 137 publications

References 5 publications

Tricritical Directed Percolation

Tricritical Directed Percolation

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

Field Theoretic Methods

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