The flourishing of fake news is favored by recommendation algorithms of
online social networks which, based on previous users 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 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 the personalized information and if the number of agents is
below a threshold. We analytically compute this critical size, which depends on
the interaction probability in a strongly non linear way.Comment: 14 pages, 9 figure
