Ordering Kinetics of the two-dimensional voter model with long-range interactions

Abstract

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance $r$ with probability $P(r) \propto r^{-\al}$. The model is characterized by different regimes, as $\al$ is varied. For $\al > 4$ the behaviour is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as $L(t)\propto \sqrt{t}$, until consensus is reached in a time or order $N\ln N$, $N$ being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slow as $\rho (t)\propto 1/\ln t$. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbors interactions. For $0<\al \leq 4$ standard scaling is reinstated, and the correlation length increases algebraically as $L(t)\propto t^{1/z}$, with $1/z=2/\al$ for $3<\al<4$ and $1/z=2/3$ for $0<\al<3$. In addition, for $\al \le 3$, $L(t)$ depends on $N$ at any time $t>0$. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the $N\to \infty$ limit. In finite systems consensus is reached in a time of order $N$ for any $\al <4$.

arXiv comment16 pages, 6 figures

References in corpus (37)

• Statistical physics of social dynamics
• Theory of Phase Ordering Kinetics
• Statistical mechanics and dynamics of solvable models with long-range interactions
• Voter Model on Heterogeneous Graphs
• Voter Models on Heterogeneous Networks
• Critical Coarsening without Surface Tension: the Voter Universality Class
• Does a Single Zealot Affect an Infinite Group of Voters ?
• On the Role of Zealotry in the Voter Model
• The non-linear q-voter model
• Evolutionary dynamics on degree-heterogeneous graphs
• Reality Inspired Voter Models: A Mini-Review
• Exact Results for Kinetics of Catalytic Reactions
• Coarsening and persistence in the voter model
• Growth Laws for Phase Ordering
• Decelerating microdynamics can accelerate macrodynamics in the voter model
• Ultimate Fate of Constrained Voters
• Zealots in the mean-field noisy voter model
• Phase Ordering Kinetics of One-Dimensional Non-Conserved Scalar Systems
• Spatial Scaling in Model Plant Communities
• Voting and Catalytic Processes with Inhomogeneities
• Dynamics and Thermodynamics of Systems with Long Range Interactions: an Introduction
• Effective surface-tension in the noise-reduced voter model
• Phase ordering kinetics of the long-range Ising model
• Mean-field analysis of the q-voter model on networks
• Consensus time in a voter model with concealed and publicly expressed opinions
• One dimensional phase-ordering in the Ising model with space decaying interactions
• Kinetics of the Two-dimensional Long-range Ising Model at Low Temperatures
• Zero-Temperature Coarsening in the Two-Dimensional Long-Range Ising Model
• Coexistence of coarsening and mean field relaxation in the long-range Ising chain
• Effective mobility and diffusivity in coarsening processes
• Kinetic theory for non-equilibrium stationary states in long-range interacting systems
• Quasi-deterministic dynamics, memory effects, and lack of self-averaging in the relaxation of quenched ferromagnets
• Analytical and numerical treatment of continuous ageing in the voter model
• Ordering Dynamics of the Random Field Long-range Ising Model in One Dimension
• Domain statistics in the relaxation of the one-dimensional Ising model with strong long-range interactions
• Voter models with conserved dynamics
• Fractal character of the phase ordering kinetics of a diluted ferromagnet

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• Coarsening in the Persistent Voter Model: analytical results
• Schrödinger-invariance in the voter model