Average Domain Size Scales Like Population Size in the Absorbing Configurations of the One-Dimensional Axelrod Model with Three Features and Three States

The Axelrod model for the evolution of cultural domains is a stochastic spatial process, with parameters the number of cultural features f and their states q, that has been studied primarily by numerical simulations in social sciences and statistical physics. It may also be viewed as an asynchronous cellular automaton that exhibits a phase transition or, for a certain range of its parameters, as a distributed consensus (albeit, non-optimal) algorithm. Recently, rigorous results on this model were achieved, which are useful to benchmark simulations or for comparison with the numerical findings. We review these results in one and two dimensions and we also offer a heuristic for an open problem. Namely, our heuristic indicates that consensus is reached in one dimension if f = q = 3, conditional on the existence of an appropriately defined infinite open path in the initial configuration.