Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice

Jensen, Iwan

doi:10.1088/0305-4470/32/28/304

Cited by 171 publications

(227 citation statements)

References 42 publications

(53 reference statements)

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“…θ and β coincide with the DP values up to the fourth and third digit, respectively. The evaluation of z is less (4) 0.276(2) 1.58(2) DP [43] 0.159464(6) 0.276486 (6) Table I. accurate, since it relies on a data collapse of finite size estimates (10).…”

Section: Short-range Interactionsmentioning

confidence: 99%

Chaotic synchronizations of spatially extended systems as nonequilibrium phase transitions

Cencini

Tessone

Torcini

2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Two replicas of spatially extended chaotic systems synchronize to a common spatio-temporal chaotic state when coupled above a critical strength. As a prototype of each single spatio-temporal chaotic system a lattice of maps interacting via power-law coupling is considered. The synchronization transition is studied as a non-equilibrium phase transition, and its critical properties are analyzed at varying the spatial interaction range as well as the nonlinearity of the dynamical units composing each system. In particular, continuous and discontinuous local maps are considered. In both cases the transitions are of the second order with critical indexes varying with the exponent characterizing the interaction range. For discontinuous maps it is numerically shown that the transition belongs to the anomalous directed percolation (ADP) family of universality classes, previously identified for Lévy-flight spreading of epidemic processes. For continuous maps, the critical exponents are different from those characterizing ADP, but apart from the nearest-neighbor case, the identification of the corresponding universality classes remains an open problem. Finally, to test the influence of deterministic correlations for the studied synchronization transitions, the chaotic dynamical evolutions are substituted by suitable stochastic models. In this framework and for the discontinuous case, it is possible to derive an effective Langevin description that corresponds to that proposed for ADP.

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Section: Short-range Interactionsmentioning

confidence: 99%

Chaotic synchronizations of spatially extended systems as nonequilibrium phase transitions

Cencini

Tessone

Torcini

2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The threshold value p c as well as the values of DP critical exponents have recently been estimated very accurately using low-density series expansions [34,35,36], so the thermal critical exponents of collapsing directed animals are also known with high precision. Note, however, that the IDA model is more general than DP as it remains well defined for repulsive interactions (w < 1), whereas the equivalence with DP is only valid for w > 2.…”

Section: Collapse Transition and Percolation Thresholdmentioning

confidence: 99%

On directed interacting animals and directed percolation

Knežević

Vannimenus

2002

J. Phys. A: Math. Gen.

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Abstract.We study the phase diagram of fully directed lattice animals with nearest-neighbour interactions on the square lattice. This model comprises several interesting ensembles (directed site and bond trees, bond animals, strongly embeddable animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n = 17. The crossover region in the vicinity of the collapse transition is analyzed in detail and the crossover exponent φ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that φ is equal to the Fisher exponent σ governing the size distribution of large directed percolation clusters.

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“…14b we show the effective exponents α (c) G (L) (local declivities of the previous log-log plot), which suggests an asymptotic exponent α (c) G ∼ 1.6. On the other hand, the best known estimates of DP exponents give ν /ν ⊥ ≈ 1.581 [27]. Although the applicability of this ballistic-like model to real deposition processes is very limited, the robustness of the DP class suggests that systems with deposition of poisoning species but different aggregation mechanisms may also present transitions in that class.…”

Section: Growth With Poisoning Speciesmentioning

confidence: 95%

“…The CP and various other models have the same critical exponents of DP, thus belonging to the DP universality class. The best estimates of exponent β T in d = 1 and d = 2 are β T = 0.276486 ± 0.000008 [27] and β T = 0.584 ± 0.004 [28], respectively. Other critical exponents describe the spatial and temporal correlations in these dynamical processes, whose characteristic lengths are illustrated in Fig.…”

Section: B Phase Transitions In Systems With Absorbing Statesmentioning

confidence: 99%

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Depinning transitions in interface growth models

Reis

2003

Braz. J. Phys.

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Pinning-depinning transitions are roughening transitions separating a growing phase and pinned (or blocked) one, and are frequently connected to transitions into absorbing states. In this review, we discuss lattice growth models exhibiting this type of dynamic transition. Driven growth in media with impurities, the competition between deposition and desorption and deposition of poisoning species are some of the physical mechanisms responsible for the transitions, leading to different types of stochastic growth rules. The growth models are classified according to the those mechanisms and possible applications are shown, which include suggestions of experimental realizations of directed percolation transitions. I IntroductionThe study of surface and interface growth processes attracts much interest from the technological point of view mainly because it helps to understand the growth mechanisms of nanostructures and, consequently, their physical properties [1]. From the theoretical point of view, they motivated significant advances in non-equilibrium Statistical Mechanics and related fields [2,3]. Theoretical modeling is usually based on stochastic differential equations or on discrete atomistic models. In various models, changing one parameter leads to a roughening transition between a rough and a smooth phase. Two well known examples are the transition in the Kardar-Parisi-Zhang equation in dimensions d ≥ 3, separating regimes with linear (smooth) and nonlinear growth, and temperature-induced roughening transitions in equilibrium conditions [2]. On the other hand, in the so-called depinning transitions, the interface propagates only in the rough phase, being pinned in the smooth phase. Several mechanisms may be responsible for this type of transition, such as the presence of impurities in the growth media, competition between deposition and evaporation or formation of poisoning species at the growing surface. One of the interesting features of depinning transitions is that they frequently can be mapped onto transitions into absorbing states, whose most prominent example is directed percolation (DP). Different connections of depinning transitions to DP are found in discrete and continuous growth models, as well as connections with other classes of transitions to absorbing states [4] and, eventually, with statistical equilibrium transitions. Besides the fundamental interest of these systems, the large variety of interface growth phenomena observed in nature suggests them as strong candidates to experimental realizations of models of dynamic transitions, such as DP (see e. g. Ref.[5]).The aim of the present work is to review the basic features of lattice growth models exhibiting depinning transitions. Before introducing these systems, we will briefly summarize the theory and phenomenology concerning interface growth and phase transitions into absorbing states (Sec. II). Then we will discuss the problem of growing interfaces in disordered media with quenched disorder, in which transitions in the DP class or in the cl...

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Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice

Cited by 171 publications

References 42 publications

Chaotic synchronizations of spatially extended systems as nonequilibrium phase transitions

Chaotic synchronizations of spatially extended systems as nonequilibrium phase transitions

On directed interacting animals and directed percolation

Depinning transitions in interface growth models

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