Ageing is an ubiquitous mechanism in nature. It has different meanings, depending on the strand of research considered. Classical examples range from non-equilibrium statistical mechanics, where its effects are studied on spin glasses, to biology, considered as the increase of mortality with age of a species, to chemistry, where the properties of a material change over time without any external forces. In any case, ageing can be seen as the dependence of the dynamics of a system on an internal time, often heterogeneously distributed, of the individual components that form such a system. Specifically, we take the approach of considering ageing as the influence that persistence times have on the state transitions: it constrains the transitions in a way that the longer an element remains in a given state, the smaller is the probability to change it. The inclusion of this effect in the modeling part adds a realistic component into the description of a given problem and unveils new and rich phenomenology.

In this work we investigate the effects of including heterogeneous time-dependent transitions on the critical properties of a stochastic model. We add ageing into the noisy voter model (also known as the Kirman model), a paradigmatic model appeared in several contexts: percolation, surface-catalytic reactions, probability theory, opinion dynamics or economics. The agents modify their binary state by means of noise and pair-wise interactions. Interestingly, due to ageing the system passes from a finite-size discontinuous transition between ordered (ferromagnetic) and disordered (paramagnetic) phases to a second order phase transition, well-defined in the thermodynamic limit, belonging to the Ising universality class (see Figure). We characterize it analytically by finding the stationary solution of an infinite set of mean field equations. The theoretical predictions are tested with extensive numerical simulations in low dimensional lattices and complex networks. In addition, the ageing properties are employed to understand the symmetries broken in the phase transition.

In summary, by adding a realistic ingredient in the modeling framework, we prove that ageing plays a central role in modifying the critical properties of a stochastic model. The studied phenomenology can occur in other complex systems beyond opinion dynamics models and it is certainly worth exploring in the future.