Kinetics of the one-dimensional voter model with long-range interactions

Abstract

Abstract The voter model is an extremely simple yet nontrivial prototypical model of ordering dynamics, which has been studied in great detail. Recently, a great deal of activity has focused on long-range statistical physics models, where interactions take place among faraway sites, with a probability slowly decaying with distance. In this paper, we study analytically the one-dimensional long-range voter model, where an agent takes the opinion of another at distance r with probability ∝ r − α . The model displays rich and diverse features as α is changed. For α > 3 the behavior is similar to the one of the nearest-neighbor version, with the formation of ordered domains whose typical size grows as R ( t ) ∝ t 1 / 2 until consensus (a fully ordered configuration) is reached. The correlation function C ( r , t ) between two agents at distance r obeys dynamical scaling with sizeable corrections at large distances r > r ∗ ( t ) , slowly fading away in time. For 2 < α ⩽ 3 violations of scaling appear, due to the simultaneous presence of two lengh-scales, the size of domains growing as t ( α − 2 ) / ( α − 1 ) , and the distance L ( t ) ∝ t 1 / ( α − 1 ) over which correlations extend. For α ⩽ 2 the system reaches a partially ordered stationary state, characterised by an algebraic correlator, whose lifetime diverges in the thermodynamic limit of infinitely many agents, so that consensus is not reached. For a finite system escape towards the fully ordered configuration is finally promoted by development of large distance correlations. In a system of N sites, global consensus is achieved after a time T ∝ N 2 for α > 3, T ∝ N α − 1 for 2 < α ⩽ 3 , and T ∝ N for α ⩽ 2 .