Clustering and coexistence in the one-dimensional vectorial Deffuant model

The vectorial Deffuant model is a simple stochastic process for the dynamics of opinions that also includes a confidence threshold. To understand the role of space in this type of social interactions, we study the process on the one-dimensional lattice where individuals are characterized by their opinion - in favor or against - about $F$ different issues and where pairs of nearest neighbors potentially interact at rate one. Potential interactions indeed occur when the number of issues both neighbors disagree on does not exceed a certain confidence threshold, which results in one of the two neighbors updating her opinion on one of the issues both neighbors disagree on (if any). This paper gives sufficient conditions for clustering of the system and for coexistence due to fixation in a fragmented configuration, showing the existence of a phase transition between both regimes at a critical confidence threshold.