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

We study the non-equilibrium response function Rᵢⱼ(t,t'), namely the variation of the local magnetization Sᵢ(t) 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 { 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 Xᵢⱼ(t,t')=β Rᵢⱼ/(∂ Gᵢⱼ/∂ t'), where Gᵢⱼ(t,t')= Sᵢ(t)Sⱼ(t') is the spin-spin correlation function and β 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 Xᵢᵢ(t,t') = (t+t')/(2t), whereas lim t→ ∞Xᵢⱼ(t,t')=1/2 for any $ij$ couple. The specific case of voter models with long-range interactions is thoroughly discussed.