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Tarpin M, Benitez F, Canet L, Wschebor N. Nonperturbative renormalization group for the diffusive epidemic process. Phys Rev E 2017; 96:022137. [PMID: 28950583 DOI: 10.1103/physreve.96.022137] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2016] [Indexed: 06/07/2023]
Abstract
We consider the Diffusive Epidemic Process (DEP), a two-species reaction-diffusion process originally proposed to model disease spread within a population. This model exhibits a phase transition from an active epidemic to an absorbing state without sick individuals. Field-theoretic analyses suggest that this transition belongs to the universality class of Directed Percolation with a Conserved quantity (DP-C, not to be confused with conserved-directed percolation C-DP, appearing in the study of stochastic sandpiles). However, some exact predictions derived from the symmetries of DP-C seem to be in contradiction with lattice simulations. Here we revisit the field theory of both DP-C and DEP. We discuss in detail the symmetries present in the various formulations of both models. We then investigate the DP-C model using the derivative expansion of the nonperturbative renormalization group formalism. We recover previous results for DP-C near its upper critical dimension d_{c}=4, but show how the corresponding fixed point seems to no longer exist below d≲3. Consequences for the DEP universality class are considered.
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Affiliation(s)
- Malo Tarpin
- LPMMC, Université Grenoble Alpes and CNRS, F-38042 Grenoble, France
| | - Federico Benitez
- Physikalisches Institut, Universität Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland
- ICS, University of Zurich, Winterthurerstr. 190, CH-8057 Zurich, Switzerland
| | - Léonie Canet
- LPMMC, Université Grenoble Alpes and CNRS, F-38042 Grenoble, France
| | - Nicolás Wschebor
- Instituto de Física, Facultad de Ingeniería, Universidad de la República, J.H.y Reissig 565, 11000 Montevideo, Uruguay
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Zachariou N, Expert P, Takayasu M, Christensen K. Generalised Sandpile Dynamics on Artificial and Real-World Directed Networks. PLoS One 2015; 10:e0142685. [PMID: 26606143 PMCID: PMC4659656 DOI: 10.1371/journal.pone.0142685] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2013] [Accepted: 10/26/2015] [Indexed: 12/05/2022] Open
Abstract
The main finding of this paper is a novel avalanche-size exponent τ ≈ 1.87 when the generalised sandpile dynamics evolves on the real-world Japanese inter-firm network. The topology of this network is non-layered and directed, displaying the typical bow tie structure found in real-world directed networks, with cycles and triangles. We show that one can move from a strictly layered regular lattice to a more fluid structure of the inter-firm network in a few simple steps. Relaxing the regular lattice structure by introducing an interlayer distribution for the interactions, forces the scaling exponent of the avalanche-size probability density function τ out of the two-dimensional directed sandpile universality class τ = 4/3, into the mean field universality class τ = 3/2. Numerical investigation shows that these two classes are the only that exist on the directed sandpile, regardless of the underlying topology, as long as it is strictly layered. Randomly adding a small proportion of links connecting non adjacent layers in an otherwise layered network takes the system out of the mean field regime to produce non-trivial avalanche-size probability density function. Although these do not display proper scaling, they closely reproduce the behaviour observed on the Japanese inter-firm network.
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Affiliation(s)
- Nicky Zachariou
- Department of Physics, Blackett Laboratory, Imperial College London, Prince Consort Road, South Kensington Campus, London SW7 2AZ, United Kingdom
- Centre for Complexity Science, Electrical & Electronic Engineering Building, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
| | - Paul Expert
- Department of Physics, Blackett Laboratory, Imperial College London, Prince Consort Road, South Kensington Campus, London SW7 2AZ, United Kingdom
- Centre for Complexity Science, Electrical & Electronic Engineering Building, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
- Centre for Neuroimaging Sciences, Institute of Psychiatry, Psychology and Neuroscience, De Crespigny Park, King’s College London, London SE5 8AF, United Kingdom
| | - Misako Takayasu
- Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan
| | - Kim Christensen
- Department of Physics, Blackett Laboratory, Imperial College London, Prince Consort Road, South Kensington Campus, London SW7 2AZ, United Kingdom
- Centre for Complexity Science, Electrical & Electronic Engineering Building, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
- * E-mail:
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Jo HH, Ha M. Universality classes and crossover behaviors in non-Abelian directed sandpiles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:041101. [PMID: 21230232 DOI: 10.1103/physreve.82.041101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2010] [Indexed: 05/30/2023]
Abstract
We study universality classes and crossover behaviors in non-Abelian directed sandpile models in terms of the metastable pattern analysis. The non-Abelian property induces spatially correlated metastable patterns, characterized by the algebraic decay of the grain density along the propagation direction of an avalanche. Crossover scaling behaviors are observed in the grain density due to the interplay between the toppling randomness and the parity of the threshold value. In the presence of such crossovers, we show that the broadness of the grain distribution plays a crucial role in resolving the ambiguity of the universality class. Finally, we claim that the metastable pattern analysis is important as much as the conventional analysis of avalanche dynamics.
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Affiliation(s)
- Hang-Hyun Jo
- School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
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Bunzarova NZ. Statistical properties of directed avalanches. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:031116. [PMID: 21230034 DOI: 10.1103/physreve.82.031116] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2010] [Revised: 05/28/2010] [Indexed: 05/30/2023]
Abstract
A two-dimensional directed stochastic sandpile model is studied both numerically and analytically. One of the known analytical approaches is extended by considering general stochastic toppling rules. The probability density distribution for the first-passage time of stochastic process described by a nonlinear Langevin equation with power-law dependence of the diffusion coefficient is obtained. Large-scale Monte Carlo simulations are performed with the aim to analyze statistical properties of the avalanches, such as the asymmetry between the initial and final stages, scaling of voids and the width of the thickest branch. Comparison with random walks description is drawn and different plausible scenarios for the avalanche evolution and the scaling exponents are suggested.
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Affiliation(s)
- N Zh Bunzarova
- Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia.
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Jo HH, Ha M. Relevance of Abelian symmetry and stochasticity in directed sandpiles. PHYSICAL REVIEW LETTERS 2008; 101:218001. [PMID: 19113452 DOI: 10.1103/physrevlett.101.218001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2008] [Indexed: 05/27/2023]
Abstract
We provide a comprehensive view of the role of Abelian symmetry and stochasticity in the universality class of directed sandpile models, in the context of the underlying spatial correlations of metastable patterns and scars. It is argued that the relevance of Abelian symmetry may depend on whether the dynamic rule is stochastic or deterministic, by means of the interaction of metastable patterns and avalanche flow. Based on the new scaling relations, we conjecture critical exponents for an avalanche, which is confirmed reasonably well in large-scale numerical simulations.
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Affiliation(s)
- Hang-Hyun Jo
- School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
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Bonachela JA, Muñoz MA. Confirming and extending the hypothesis of universality in sandpiles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:041102. [PMID: 18999374 DOI: 10.1103/physreve.78.041102] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2008] [Indexed: 05/27/2023]
Abstract
Stochastic sandpiles self-organize to an absorbing-state critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called the C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the DP class despite the existence of a conservation law. We show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what was previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self-organized criticality in systems with conservation. In passing, we obtain a number of results for the C-DP class and introduce a strategy to easily discriminate between DP and C-DP scaling.
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Affiliation(s)
- Juan A Bonachela
- Departamento de Electromagnetismo y Física de la Materia and Instituto de Física Teórica y Computacional Carlos I, Facultad de Ciencias, Universidad de Granada, Granada, Spain
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Santra SB, Chanu SR, Deb D. Characteristics of deterministic and stochastic sandpile models in a rotational sandpile model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:041122. [PMID: 17500880 DOI: 10.1103/physreve.75.041122] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2007] [Indexed: 05/15/2023]
Abstract
Rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena. In order to study the effect of rotational bias in an out-of-equilibrium situation like self-organized criticality, a two state "quasideterministic" rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains. An extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model. The probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves. The model exhibits characteristics of both deterministic and stochastic sandpile models.
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Affiliation(s)
- S B Santra
- Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India
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Karmakar R, Manna SS. Directed fixed energy sandpile model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:067107. [PMID: 15244786 DOI: 10.1103/physreve.69.067107] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2003] [Indexed: 05/24/2023]
Abstract
We numerically study the directed version of the fixed energy sandpile. On a closed square lattice, the dynamical evolution of a fixed density of sand grains is studied. The activity of the system shows a continuous phase transition around a critical density. While the deterministic version has the set of nontrivial exponents, the stochastic model is characterized by mean field like exponents.
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Affiliation(s)
- R Karmakar
- Satyendra Nath Bose National Centre for Basic Sciences Block-JD, Sector-III, Salt Lake, Kolkata-700098, India
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Chen CC, den Nijs M. Directed avalanche processes with underlying interface dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:011306. [PMID: 12241353 DOI: 10.1103/physreve.66.011306] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2002] [Indexed: 05/23/2023]
Abstract
We describe a directed avalanche model; a slowly unloading sandbox driven by lowering a retaining wall. The directness of the dynamics allows us to interpret the stable sand surfaces as world sheets of fluctuating interfaces in one lower dimension. In our specific case, the interface growth dynamics belongs to the Kardar-Parisi-Zhang (KPZ) universality class. We formulate relations between the critical exponents of the various avalanche distributions and those of the roughness of the growing interface. The nonlinear nature of the underlying KPZ dynamics provides a nontrivial test of such generic exponent relations. The numerical values of the avalanche exponents are close to the conventional KPZ values, but differ sufficiently to warrant a detailed study of whether avalanche-correlated Monte Carlo sampling changes the scaling exponents of KPZ interfaces. We demonstrate that the exponents remain unchanged, but that the traces left on the surface by previous avalanches give rise to unusually strong finite-size corrections to scaling. This type of slow convergence seems intrinsic to avalanche dynamics.
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Affiliation(s)
- Chun-Chung Chen
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
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Biham O, Milshtein E, Malcai O. Evidence for universality within the classes of deterministic and stochastic sandpile models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:061309. [PMID: 11415094 DOI: 10.1103/physreve.63.061309] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2000] [Indexed: 05/23/2023]
Abstract
Recent numerical studies have provided evidence that within the family of conservative, undirected sandpile models with short range dynamic rules, deterministic models such as the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic models such as the Manna model [S. S. Manna, J. Phys. A 24, L363 (1991)] belong to different universality classes. In this paper we examine the universality within each of the two classes in two dimensions by numerical simulations. To this end we consider additional deterministic and stochastic models and use an extended set of critical exponents, scaling functions, and geometrical features. Universal behavior is found within the class of deterministic Abelian models, as well as within the class of stochastic models (which includes both Abelian and non-Abelian models). In addition, it is observed that deterministic but non-Abelian models exhibit critical exponents that depend on a parameter, namely they are nonuniversal.
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Affiliation(s)
- O Biham
- Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
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