Why earthquakes stop: Growth and arrest in stochastic fields

Classical theory predicts that earthquakes growing in a homogeneous stress field should not arrest until the fault boundaries are encountered. However, earthquakes on natural faults, where stresses are heterogeneous, are observed over a wide range of sizes. Here we suggest that heterogeneity should be characterized by the Hausdorff dimension H of the walk associated with a stochastic stress difference field. The critical value for arrest, H = 0.5, corresonds to a Brownian walk through the stress difference field.