Holographic Equidistribution

Hecke operators acting on modular functions arise naturally in the context of 2d conformal field theory, but in seemingly disparate areas, including permutation orbifold theories, ensembles of code CFTs, and more recently in the context of the AdS$_3$/RMT$_2$ program. We use an equidistribution theorem for Hecke operators to show that in each of these large $N$ limits, an entire heavy sector of the partition function gets integrated out, leaving only contributions from Poincar\'e series of light states. This gives an immediate holographic interpretation as a sum over semiclassical handlebody geometries. We speculate on further physical interpretations for equidistribution, including a potential ergodicity statement.