The critical value of the Deffuant model equals one half

Similarly to the popular voter model, the Deffuant model describes opinion dynamics taking place in spatially structured environments represented by a connected graph. Pairs of adjacent vertices interact at a constant rate. If the opinion distance between the interacting vertices is larger than some confidence threshold ǫ > 0, then nothing happens, otherwise, the vertices’ opinions get closer to each other. It has been conjectured based on numerical simulations that this process exhibits a phase transition at the critical value ǫc = 1/2. For confidence thresholds larger than one half, the process converges to a global consensus, whereas coexistence occurs for confidence thresholds smaller than one half. In this article, we develop new geometrical techniques to prove this conjecture.