Aging properties of the voter model with long-range interactions

We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent Sᵢ=± 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 mean-field. Then, at variance 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 now the length L regulating two-time correlations differs from the coherence length as L∝ L^δ, with δ=1+2(2-α).