Earthquake statistical properties: an explanation for the distribution of magnitude and for the existence of aftershocks

Earthquakes in nature follow several statistical properties. In particular, the distribution of energy released by an earthquake (Gutenberg-Richter's law) and the frequency of aftershocks after a large event (Omori's law) are both power-laws.

By studying several earthquake models, we show that these properties result from  the spatial distribution of the stress field. This field can be described as a random curve for one-dimensional models and a random surface for two-dimensional models. Using this analogy, a series of predictions is made that includes the Gutenberg-Richter law and, for two-dimensional models, the existence of aftershocks and their temporal distribution following Omori's law.