Understanding the fate of semi-classical black hole solutions at very late times is one of the most important open questions in quantum gravity. In this paper, we provide a path integral definition of the volume of the black hole interior and study it at arbitrarily late times for black holes in various models of two-dimensional gravity. Because of a novel universal cancellation between the contributions of the semi-classical black hole spectrum and some of its non-perturbative corrections, we find that, after a linear growth at early times, the length of the interior saturates at a time, and towards a value, that is exponentially large in the entropy of the black hole. This provides a non-perturbative confirmation of the complexity equals volume proposal since complexity is also expected to plateau at the same value and at the same time.

An important open question in black hole thermodynamics is about the existence of a "mass gap" between an extremal black hole and the lightest near-extremal state within a sector of fixed charge. In this talk, I will discuss how to reliably compute the partition function of 4d Reissner-Nordstrom near-extremal black holes at temperature scales comparable to the conjectured gap. I will show that the density of states at fixed charge does not exhibit a gap in the simplest gravitational non-supersymmetric theories; rather, at the expected gap energy scale, we see a continuum of states whose meaning we will extensively discuss. Finally, I will present a similar computation for nearly-BPS black holes in 4d N=2 supergravity. As opposed to their non-supersymmetric counterparts, such black holes do in fact exhibit a gap consistent with various string theory predictions. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]