Axelrod's Model in Two Dimensions

In 1997 R. Axelrod introduced a model in which individuals have one of $Q$ possible opinions about each of $F$ issues and neighbors interact at a rate proportional to the fraction of opinions they share. Thanks to work by Lanchier and collaborators there are now a number of results for the one dimensional model. Here, we consider Axelrod's model on a square subset of the two-dimensional lattice start from a randomly chosen initial state and simplify things by supposing that $Q$ and $F$ large. If $Q/F$ is large then most neighbors have all opinions different and do not interact, so by a result of Lanchier the system soon reaches a highly disordered absorbing state. In contrast if $Q/F$ is small, then there is a giant component of individuals who share at least one opinion. In this case we show that consensus develops on this percolating cluster.