General properties of the response function in a class of solvable non-equilibrium models

We study the non-equilibrium response function $R_{ij}(t,t')$, namely the variation of the local magnetization $\langle S_i(t)\rangle$ on site $i$ at time $t$ as an effect of a perturbation applied at the earlier time $t'$ on site $j$, in a class of solvable spin models characterized by the vanishing of the so-called {\it asymmetry}. This class encompasses both systems brought out of equilibrium by the variation of a thermodynamic control parameter, as after a temperature quench, or intrinsically out of equilibrium models with violation of detailed balance. The one-dimensional Ising model and the voter model (on an arbitrary graph) are prototypical examples of these two situations which are used here as guiding examples. Defining the fluctuation-dissipation ratio $X_{ij}(t,t')=\beta R_{ij}/(\partial G_{ij}/\partial t')$, where $G_{ij}(t,t')=\langle S_i(t)S_j(t')\rangle$ is the spin-spin correlation function and $\beta$ is a parameter regulating the strength of the perturbation (corresponding to the inverse temperature when detailed balance holds), we show that, in the quite general case of a kinetics obeying dynamical scaling, on equal sites this quantity has a universal form$X_{ii}(t,t') = (t+t')/(2t)$, whereas $\lim _{t\to \infty}X_{ij}(t,t')=1/2$ for any $ij$ couple. The specific case of voter models with long-range interactions is thoroughly discussed.