An Ultrametric State Space with a Dense Discrete Overlap Distribution: Paperfolding Sequences

We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support (as a closed set) is the full interval [-1,+1]. The space of paperfolding sequences has an ultrametric structure. Our example provides an illustration of some properties which were suggested to occur for pure states in spin glass models.