Deffuant opinion dynamics with attraction and repulsion

In the Deffuant model, individuals are located on the vertices of a graph, and are characterized by their opinion, a number in $[-1, 1]$. The dynamics depends on two parameters: a confidence threshold $\theta < 2$ and a convergent parameter $\mu_- \leq 1/2$. Neighbors on the graph interact at rate one, which results in no changes if the neighbors disagree by more than $\theta$, and a compromise with the opinions moving toward each other by a factor $\mu_-$ if they disagree by less than $\theta$ (attraction). The main conjecture about the Deffuant model, which was proved for the process on the integers, states that, for all $\mu_- > 0$ and starting from the product measure in which the opinions are uniformly distributed in the interval $[-1, 1]$, there is a phase transition from discordance to consensus at the confidence threshold one. In this paper, we study a natural variant of the model in which neighbors who disagree by more than $\theta$ feel more strongly about their own opinion, which is modeled by assuming that the opinions move away from each other by a divergent parameter $\mu_+$ (repulsion). We prove, for the process on the integers, the absence of a phase transition even for arbitrarily small $\mu_+ > 0$, in the sense that, for every nontrivial choice of $\theta$, there is always discordance.