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The vectorial Deffuant model is a simple stochastic process for the dynamicsof opinions that also includes a confidence threshold. To understand the roleof space in this type of social interactions, we study the process on theone-dimensional lattice where individuals are characterized by their opinion -in favor or against - about $F$ different issues and where pairs of nearestneighbors potentially interact at rate one. Potential interactions indeed occurwhen the number of issues both neighbors disagree on does not exceed a certainconfidence threshold, which results in one of the two neighbors updating heropinion on one of the issues both neighbors disagree on (if any). This papergives sufficient conditions for clustering of the system and for coexistencedue to fixation in a fragmented configuration, showing the existence of a phasetransition between both regimes at a critical confidence threshold.

Clustering and coexistence in the one-dimensional vectorial Deffuant model