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 $S_i=\pm 1$ positioned at a lattice vertex $i$, copies the state of another one located at a distance $r$, selected randomly with a probability $P(r)\propto r^{-\hat{I}\pm }$. 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 $\hat{I}\pm >1$ as $[L(t)/L(s){]}^{-\hat{I}»}$, where $L$ is a time-increasing coherence length and $\hat{I}»$ is the Fisher-Huse exponent. We find $\hat{I}»=1$ for $\hat{I}\pm >2$, and $\hat{I}»=1/(\hat{I}\pm -1)$ for $1<\hat{I}\pm \le 2$. For $\hat{I}\pm \le 1$, 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 $\hat{I}»$ increases upon decreasing $\hat{I}\pm$. 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 $\hat{I}\pm >2$. Similarly, for $1<\hat{I}\pm \le 2$ one has $G(r;t,s)=g[r/\mathcal{L}(t);\mathcal{L}(t)/\mathcal{L}(s)]$, where now the length $\mathcal{L}$ regulating two-time correlations differs from the coherence length as $\mathcal{L}\propto L^{\hat{I}}´$, with $\hat{I}´=1+2(2-\hat{I}\pm )$.