Emergence of polarization in a voter model with personalized information

The flourishing of fake news is supported by recommendation algorithms of online social networks, which, based on previous user activity, provide content adapted to their preferences and so create filter bubbles. We introduce an analytically tractable voter model with personalized information, in which an external field tends to align the agent's opinion with the one she held more frequently in the past. Our model shows a surprisingly rich dynamics despite its simplicity. An analytical mean-field approach, confirmed by numerical simulations, allows us to build a phase diagram and to predict if and how consensus is reached. Remarkably, polarization can be avoided only for weak interaction with personalized information and if the number of agents is below a threshold. We compute analytically this critical size, which depends on the interaction probability in a strongly nonlinear way.