Aging properties of the voter model with long-range interactions

Abstract

We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, S i = ± 1 , positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P ( r ) ∝ r − α . Employing both analytical and numerical methods, we compute the two-time correlation function G ( r ; t , s ) ( t ⩾ s ) between the state of a variable Si at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A ( t , s ) = G ( r = 0 ; t , s ) , decays algebraically for α > 1 as [ L ( t ) / L ( s ) ] − λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ = 1 / ( α − 1 ) for 1 < α ⩽ 2 . For α ⩽ 1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α. The space-dependent correlation G ( r ; t , s ) obeys a scaling symmetry G ( r ; t , s ) = g [ r / L ( s ) ; L ( t ) / L ( s ) ] for α > 2. Similarly, for 1 < α ⩽ 2 , one has G ( r ; t , s ) = g [ r / L ( t ) ; L ( t ) / L ( s ) ] , where the length L regulating two-time correlations now differs from the coherence length as L ∝ L δ , with δ = 1 + 2 ( 2 − α ) .