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Malik O, Varga M, Moussawi A, Hunt D, Szymanski BK, Toroczkai Z, Korniss G. Diffusive persistence on disordered lattices and random networks. Phys Rev E 2024; 109:024113. [PMID: 38491611 DOI: 10.1103/physreve.109.024113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 12/22/2023] [Indexed: 03/18/2024]
Abstract
To better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks, we investigated diffusive persistence in various graphs. Global diffusive persistence is defined as the fraction of nodes for which the diffusive field at a site (or node) has not changed sign up to time t (or, in general, that the node remained active or inactive in discrete models). Here we investigate disordered and random networks and show that the behavior of the persistence depends on the topology of the network. In two-dimensional (2D) disordered networks, we find that above the percolation threshold diffusive persistence scales similarly as in the original 2D regular lattice, according to a power law P(t,L)∼t^{-θ} with an exponent θ≃0.186, in the limit of large linear system size L. At the percolation threshold, however, the scaling exponent changes to θ≃0.141, as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. Moreover, studying finite-size effects for 2D lattices at and above the percolation threshold, we find that at the percolation threshold, the long-time asymptotic value obeys a power law P(t,L)∼L^{-zθ} with z≃2.86 instead of the value of z=2 normally associated with finite-size effects on 2D regular lattices. In contrast, we observe that in random networks without a local regular structure, such as Erdős-Rényi networks, no simple power-law scaling behavior exists above the percolation threshold.
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Affiliation(s)
- Omar Malik
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - Melinda Varga
- Department of Physics and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556, USA
| | - Alaa Moussawi
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - David Hunt
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - Boleslaw K Szymanski
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - Zoltan Toroczkai
- Department of Physics and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556, USA
| | - Gyorgy Korniss
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
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Latoski LCF, Dantas WG, Arenzon JJ. Curvature-driven growth and interfacial noise in the voter model with self-induced zealots. Phys Rev E 2022; 106:014121. [PMID: 35974624 DOI: 10.1103/physreve.106.014121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2022] [Accepted: 06/27/2022] [Indexed: 06/15/2023]
Abstract
We introduce a variant of the voter model in which agents may have different degrees of confidence in their opinions. Those with low confidence are normal voters whose state can change upon a single contact with a different neighboring opinion. However, confidence increases with opinion reinforcement, and above a certain threshold, these agents become zealots, irreducible agents who do not change their opinion. We show that both strategies, normal voters and zealots, may coexist (in the thermodynamical limit), leading to competition between two different kinetic mechanisms: curvature-driven growth and interfacial noise. The kinetically constrained zealots are formed well inside the clusters, away from the different opinions at the surfaces that help limit their confidence. Normal voters concentrate in a region around the interfaces, and their number, which is related to the distance between the surface and the zealotry bulk, depends on the rate at which the confidence changes. Despite this interface being rough and fragmented, typical of the voter model, the presence of zealots in the bulk of these domains induces a curvature-driven dynamics, similar to the low temperature coarsening behavior of the nonconserved Ising model after a temperature quench.
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Affiliation(s)
- Luís Carlos F Latoski
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CEP 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
| | - W G Dantas
- Departamento de Ciências Exatas, EEIMVR, Universidade Federal Fluminense, CEP 27255-125, Volta Redonda, Rio de Janeiro, Brazil
| | - Jeferson J Arenzon
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CEP 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
- Instituto Nacional de Ciência e Tecnologia-Sistemas Complexos, Rio de Janeiro, 22290-180, Rio de Janeiro, Brazil
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Saha S, Sen P. Virtual walks inspired by a mean-field kinetic exchange model of opinion dynamics. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20210168. [PMID: 35400189 DOI: 10.1098/rsta.2021.0168] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2021] [Accepted: 10/15/2021] [Indexed: 06/14/2023]
Abstract
We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange model of opinion dynamics. The walks are either Markovian or non-Markovian in nature. The opinion dynamics model is characterized by a parameter [Formula: see text] which drives an order disorder transition at a critical value [Formula: see text]. The distribution [Formula: see text] of the displacements [Formula: see text] from the origin of the walkers is computed at different times. Below [Formula: see text], two time scales associated with a crossover behaviour in time are detected, which diverge in a power law manner at criticality with different exponent values. [Formula: see text] also carries the signature of the phase transition as it changes its form at [Formula: see text]. The walks show the features of a biased random walk below [Formula: see text], and above [Formula: see text], the walks are like unbiased random walks. The bias vanishes in a power law manner at [Formula: see text] and the width of the resulting Gaussian function shows a discontinuity. Some of the features of the walks are argued to be comparable to the critical quantities associated with the mean-field Ising model, to which class the opinion dynamics model belongs. The results for the Markovian and non-Markovian walks are almost identical which is justified by considering the different fluxes. We compare the present results with some earlier similar studies. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.
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Affiliation(s)
- Surajit Saha
- Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India
| | - Parongama Sen
- Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India
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Jo HH, Masuda N. Finite-size effects on the convergence time in continuous-opinion dynamics. Phys Rev E 2021; 104:014309. [PMID: 34412253 DOI: 10.1103/physreve.104.014309] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2021] [Accepted: 06/25/2021] [Indexed: 11/07/2022]
Abstract
We study finite-size effects on the convergence time in a continuous-opinion dynamics model. In the model, each individual's opinion is represented by a real number on a finite interval, e.g., [0,1], and a uniformly randomly chosen individual updates its opinion by partially mimicking the opinion of a uniformly randomly chosen neighbor. We numerically find that the characteristic time to the convergence increases as the system size increases according to a particular functional form in the case of lattice networks. In contrast, unless the individuals perfectly copy the opinion of their neighbors in each opinion updating, the convergence time is approximately independent of the system size in the case of regular random graphs, uncorrelated scale-free networks, and complete graphs. We also provide a mean-field analysis of the model to understand the case of the complete graph.
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Affiliation(s)
- Hang-Hyun Jo
- Department of Physics, The Catholic University of Korea, Bucheon 14662, Republic of Korea
| | - Naoki Masuda
- Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260-2900, USA.,Computational and Data-Enabled Science and Engineering Program, State University of New York at Buffalo, Buffalo, New York 14260-5030, USA
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Mullick P, Sen P. Virtual walks in spin space: A study in a family of two-parameter models. Phys Rev E 2018; 97:052122. [PMID: 29906899 DOI: 10.1103/physreve.97.052122] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2017] [Indexed: 11/07/2022]
Abstract
We investigate the dynamics of classical spins mapped as walkers in a virtual "spin" space using a generalized two-parameter family of spin models characterized by parameters y and z [de Oliveira et al., J. Phys. A 26, 2317 (1993)JPHAC50305-447010.1088/0305-4470/26/10/006]. The behavior of S(x,t), the probability that the walker is at position x at time t, is studied in detail. In general S(x,t)∼t^{-α}f(x/t^{α}) with α≃1 or 0.5 at large times depending on the parameters. In particular, S(x,t) for the point y=1,z=0.5 corresponding to the Voter model shows a crossover in time; associated with this crossover, two timescales can be defined which vary with the system size L as L^{2}logL. We also show that as the Voter model point is approached from the disordered regions along different directions, the width of the Gaussian distribution S(x,t) diverges in a power law manner with different exponents. For the majority Voter case, the results indicate that the the virtual walk can detect the phase transition perhaps more efficiently compared to other nonequilibrium methods.
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Affiliation(s)
- Pratik Mullick
- Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India
| | - Parongama Sen
- Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India
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Tartaglia A, Cugliandolo LF, Picco M. Percolation and coarsening in the bidimensional voter model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042109. [PMID: 26565170 DOI: 10.1103/physreve.92.042109] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2015] [Indexed: 06/05/2023]
Abstract
We study the bidimensional voter model on a square lattice with numerical simulations. We demonstrate that the evolution takes place in two distinct dynamic regimes; a first approach towards critical site percolation and a further approach towards full consensus. We calculate the time dependence of the two growing lengths, finding that they are both algebraic but with different exponents (apart from possible logarithmic corrections). We analyze the morphology and statistics of clusters of voters with the same opinion. We compare these results to the ones for curvature driven two-dimensional coarsening.
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Affiliation(s)
- Alessandro Tartaglia
- Sorbonne Universités, Université Pierre et Marie Curie-Paris VI, Laboratoire de Physique Théorique et Hautes Energies UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
| | - Leticia F Cugliandolo
- Sorbonne Universités, Université Pierre et Marie Curie-Paris VI, Laboratoire de Physique Théorique et Hautes Energies UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
| | - Marco Picco
- Sorbonne Universités, Université Pierre et Marie Curie-Paris VI, Laboratoire de Physique Théorique et Hautes Energies UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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Hase MO, Tomé T, de Oliveira MJ. Aging and fluctuation-dissipation ratio in a nonequilibrium q-state lattice model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:011133. [PMID: 20866591 DOI: 10.1103/physreve.82.011133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2010] [Revised: 06/28/2010] [Indexed: 05/29/2023]
Abstract
A generalized version of the nonequilibrium linear Glauber model with q states in d dimensions is introduced and analyzed. The model is fully symmetric, its dynamics being invariant under all permutations of the q states. Exact expressions for the two-time autocorrelation and response functions on a d-dimensional lattice are obtained. In the stationary regime, the fluctuation-dissipation theorem holds, while in the transient the aging is observed with the fluctuation-dissipation ratio leading to the value predicted for the linear Glauber model.
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Affiliation(s)
- M O Hase
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05314-970 São Paulo, Brazil
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Maillard G, Mountford T. Large deviations for voter model occupation times in two dimensions. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2009. [DOI: 10.1214/08-aihp178] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Täuber UC, Howard M, Vollmayr-Lee BP. Applications of field-theoretic renormalization group methods to reaction–diffusion problems. ACTA ACUST UNITED AC 2005. [DOI: 10.1088/0305-4470/38/17/r01] [Citation(s) in RCA: 217] [Impact Index Per Article: 11.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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de Oliveira MJ. Linear Glauber model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:066101. [PMID: 16241298 DOI: 10.1103/physreve.67.066101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2002] [Indexed: 05/04/2023]
Abstract
We study the time-dependent and the stationary properties of the linear Glauber model in a d-dimensional hypercubic lattice. This model is equivalent to the voter model with noise. By using the Green function method, we get exact results for the two-point correlations from which the critical behavior is obtained. For vanishing noise the model becomes critical with exponents beta=0, gamma=1, and nu=1/2 for d > or =2, with logarithmic corrections at the upper critical dimension d(c)=2, and beta=0, gamma=1/2, and nu=1/2 for d=1. We show that the model can be mapped into a particular reaction-diffusion model.
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Affiliation(s)
- Mário J de Oliveira
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, São Paulo, Brazil
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Hellén EKO, Alava MJ. Persistence in cluster-cluster aggregation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:026120. [PMID: 12241250 DOI: 10.1103/physreve.66.026120] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2001] [Revised: 04/25/2002] [Indexed: 11/07/2022]
Abstract
Persistence is considered in one-dimensional diffusion-limited cluster-cluster aggregation when the diffusion coefficient of a cluster depends on its size s as D(s) approximately s(gamma). The probabilities that a site has been either empty or covered by a cluster all the time define the empty and filled site persistences. The cluster persistence gives the probability of a cluster remaining intact. The empty site and cluster persistences are universal whereas the filled site depends on the initial concentration. For gamma>0 the universal persistences decay algebraically with the exponent 2/(2-gamma). For the empty site case the exponent remains the same for gamma<0 but the cluster persistence shows a stretched exponential behavior as it is related to the small s behavior of the cluster size distribution. The scaling of the intervals between persistent regions demonstrates the presence of two length scales: the one related to the distances between clusters and that between the persistent regions.
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Affiliation(s)
- E K O Hellén
- Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
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Albano EV, Muñoz MA. Numerical study of persistence in models with absorbing states. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:031104. [PMID: 11308627 DOI: 10.1103/physreve.63.031104] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2000] [Indexed: 05/23/2023]
Abstract
Extensive Monte Carlo simulations are performed in order to evaluate both the local (straight theta(l)) and global (straight theta(g)) persistence exponents in the Ziff-Gulari-Barshad (ZGB) [Phys. Rev. Lett. 56, 2553 (1986)] irreversible reaction model. At the second-order irreversible phase transition (IPT) we find that both the local and the global persistence exhibit power-law behavior with a crossover between two different time regimes. On the other hand, at the ZGB first-order IPT, active sites are short lived and the persistence decays more abruptly; it is not clear whether it shows power-law behavior or not. In order to analyze universality issues, we have also studied another model with absorbing states, the contact process, and evaluated the local persistence exponent in dimensions from 1 to 4. A striking apparent superuniversality is reported: the local persistence exponent seems to coincide in both one- and two-dimensional systems. Some other aspects of persistence in systems with absorbing states are also analyzed.
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Affiliation(s)
- E V Albano
- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA), CONICET, UNLP, CIC, Buenos Aires, Sucursal 4, Casilla de Correo 16, (1900) La Plata, Argentina
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Chave J. Spatial Patterns and Persistence of Woody Plant Species in Ecological Communities. Am Nat 2001; 157:51-65. [DOI: 10.1086/317003] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
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Sire C, Majumdar SN, Rudinger A. Analytical results for random walk persistence. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:1258-69. [PMID: 11046403 DOI: 10.1103/physreve.61.1258] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/1998] [Indexed: 04/15/2023]
Abstract
In this paper, we present a detailed calculation of the persistence exponent straight theta for a nearly Markovian Gaussian process X(t), a problem initially introduced elsewhere in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. Resummed perturbative and nonperturbative expressions for straight theta are derived, which suggest a connection with the result of the alternative independent interval approximation. The perturbation theory is extended to the calculation of straight theta for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and nonperturbative expressions for the persistence exponent straight theta(X0), describing the probability that the process remains larger than X(0)sqrt[<X2(t)>].
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Affiliation(s)
- C Sire
- Laboratoire de Physique Quantique (UMR C5626 du CNRS), Universite Paul Sabatier, 31062, Toulouse Cedex, France
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Doussal PL, Monthus C. Reaction diffusion models in one dimension with disorder. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:1212-38. [PMID: 11969881 DOI: 10.1103/physreve.60.1212] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/1999] [Indexed: 11/07/2022]
Abstract
We study a large class of one-dimensional reaction diffusion models with quenched disorder using a real space renormalization group method (RSRG) which yields exact results at large time. Particles (e.g., of several species) undergo diffusion with random local bias (Sinai model) and may react upon meeting. We obtain a detailed description of the asymptotic states (i.e., attractive fixed points of the RSRG), such as the large time decay of the density of each specie, their associated universal amplitudes, and the spatial distribution of particles. We also derive the spectrum of nontrivial exponents which characterize the convergence towards the asymptotic states. For reactions which lead to several possible asymptotic states separated by unstable fixed points, we analyze the dynamical phase diagram and obtain the critical exponents characterizing the transitions. We also obtain a detailed characterization of the persistence properties for single particles as well as more complex patterns. We compute the decay exponents for the probability of no crossing of a given point by, respectively, the single particle trajectories (theta) or the thermally averaged packets (theta). The generalized persistence exponents associated to n crossings are also obtained. Specifying to the process A+A--> or A with probabilities (r,1-r), we compute exactly the exponents delta(r) and psi(r) characterizing the survival up to time t of a domain without any merging or with mergings, respectively, and the exponents deltaA(r) and psiA(r) characterizing the survival up to time t of a particle A without any coalescence or with coalescences, respectively. theta, psi, and delta obey hypergeometric equations and are numerically surprisingly close to pure system exponents (though associated to a completely different diffusion length). The effect of additional disorder in the reaction rates, as well as some open questions, are also discussed.
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Affiliation(s)
- P L Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole, Normale Supérieure, 24 rue Lhomond, F-75231 Paris, France
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Drouffe JM, Godrèche C. Phase ordering and persistence in a class of stochastic processes interpolating between the Ising and voter models. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/2/003] [Citation(s) in RCA: 45] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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17
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Dornic I, Godrèche C. Large deviations and nontrivial exponents in coarsening systems. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/31/24/004] [Citation(s) in RCA: 57] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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