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 $S_i=\pm 1$ positioned at a lattice vertex $i$, copies the state of another one located at a distance $r$, selected randomly with a probability $P(r) \propto r^{-\alpha}$. Employing both analytical and numerical methods, we compute the two-time correlation function $G(r;t,s)$ ($t\ge s$) between the state of a variable $S_i$ 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 {\it autocorrelation function} $A(t,s)=G(r=0;t,s)$, decays algebraically for $\alpha >1$ as $[L(t)/L(s)]^{-\lambda}$, where $L$ is a time-increasing coherence length and $\lambda $ is the Fisher-Huse exponent. We find $\lambda =1$ for $\alpha >2$, and $\lambda =1/(\alpha-1)$ for $1<\alpha \le 2$. For $\alpha \le 1$, 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 $\lambda $ increases upon decreasing $\alpha$. 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 $\alpha >2$. Similarly, for $1<\alpha \le 2$ one has $G(r;t,s)=g[r/{\cal L}(t);{\cal L}(t)/{\cal L}(s)] $, where now the length ${\cal L}$ regulating two-time correlations differs from the coherence length as ${\cal L}\propto L^\delta$, with $\delta=1+2(2-\alpha)$.