De (baby) Sitter overlaps

In this note we employ methods borrowed from spin glass theory to study the phase space structure of fields in an inflating universe. In particular, we compute the overlap distribution of a suitably coarse-grained, massless scalar on a (1+1)-dimensional (hence baby) de Sitter background, and find that (after an appropriate shift and rescaling) it is given by a Gumbel distribution. We also calculate the triple overlap distribution of this system, whose characteristic function turns out to be a product of two Gumbel factors.