Effective multidimensional crossover behavior in a one-dimensional voter model with long-range probabilistic interactions

Abstract

A variant of the standard voter model, where a randomly selected site of a one-dimensional lattice (d=1) adopts the state of another site placed at a distance r from the previous one, is proposed and studied by means of numerical simulations that are rationalized with the aid of dynamical and finite-size scaling arguments. The distance between the two sites is also selected randomly with a probability given by P(r)∝r-(d+σ), where σ is a control parameter. In this way one can study how the introduction of these long-range interactions influences the dynamic behavior of the standard voter model with nearest-neighbor interactions. It is found that the dynamics strongly depends on the range of the interactions, which is parameterized by σ, leading to an interesting effective multidimensional crossover behavior, as follows. (a) For σ<1 ordering is no longer observed and the average interface density [ρ(t)] assumes a steady state in the thermodynamic limit. Instead, for finite-size systems an exponential decay with a characteristic time (τ) that increases with the size is observed. This behavior resembles the scenario corresponding to the short-range voter model for d>2, as well as the case of both scale-free and small-world networks. (b) For σ>1, an ordering dynamics is observed, such that ρ(t)∝t-α, where the exponent α increases with σ until it reaches the value α=1/2 for σ⩾5, which corresponds to the behavior of the standard voter model with short-range interactions in d=1. (c) Finally, for σ≈1 we show evidence of a critical-type behavior as in the case of the critical dimension (dc=2) of the standard voter model.