Random Matrix Theory for the Black Hole Interior

In recent years a fruitful interplay has been unfolding between quantum chaos and black holes. In the first part of the talk, I provide a sampler of these developments. Next, we study the fate of the black hole interior at late times in simple models of quantum gravity that have dual descriptions in terms of Random Matrix Theory. We find that the volume of the interior grows linearly at early times and then, due to non-perturbative effects, saturates at a time and towards a value that are exponentially large in the entropy of the black hole. This provides a confirmation of the complexity equals volume proposal of Susskind, since in chaotic systems complexity is also expected to exhibit the same behavior.