  This paper investigates p-spin distributions for a generic spherical p-spin
model; we give a representation of spin distributions in terms of a stochastic
process. In order to do this, we find a novel double limit scheme that allows
us to treat the sphere as a product space and perform cavity computations. The
decomposition into a product space involves the creation of a renormalized
sphere, whose scale $R$ will be taken to infinity.
