Statistical field theory for dialectology

Is it possible to develop a "physics of language" which can explain the spatial, temporal, and social patterns we see and which can predict future change like we forecast the weather? Such a theory is likely to involve ideas from statistical physics. A substantial literature already applies these ideas to language. However, we lack a model which can match the spatial-temporal detail of historical changes at the level of individual linguistic features and which offers a principled mechanism to predict the future. Here we present a statistical field theory for the evolution of linguistic variables which takes steps to fill this gap. Linguistic variant frequencies are represented as a stochastic state field with spatial interaction and social conformity, coupled to a latent bias field with Onsager-Machlup action that reduces overfitting to data. We derive parameter inference procedures and demonstrate them using examples of large-scale dialect survey data from the 20th-century United States. The bias field has a characteristic half-life, which determines the horizon over which linguistic change can be predicted. Inferred model parameters provide evidence for surface-tension-driven coarsening of dialect regions, with population-density gradients exerting systematic forces on interfaces.