  We examine the degree to which fluctuating dynamics on exponentially
expanding spaces remember initial conditions. In de Sitter space, the global
late-time configuration of a free scalar field always contains information
about early fluctuations. By contrast, fluctuations near the boundary of
Euclidean Anti-de Sitter may or may not remember conditions in the center, with
a transition at \Delta=d/2. We connect these results to literature about
statistical mechanics on trees and make contact with the observation by Anninos
and Denef that the configuration space of a massless dS field exhibits
ultrametricity. We extend their analysis to massive fields, finding that
preference for isosceles triangles persists as long as \Delta_- < d/4.
