Parisi function for two spin glass models

Abstract

The probability distribution function P(q) for the overlap of pairs of metastable states and the associated Parisi order function q(x) are calculated exactly at zero temperature for two simple models. The first is a chain in which each spin interacts randomly with the sum of all the spins between it and one end of the chain; the second is an infinite-range limit of a spin glass version of Dyson's hierarchical model. Both have non-trivial overlap distributions. In the first case the problem reduces to a variable-step-length random-walk problem, leading to q(x)=sin( pi x). In the second model, P(q) can be calculated by a simple recursion relation which generates devil's staircase structure in q(x). If the fraction p of antiferromagnetic bonds is less than (2)-12/, the staircase is complete and the fractal dimensionality of the complement of the domain where q(x) is flat is ln 2/ln(1/p2). In both models the space of metastable states can be described in terms of Cayley trees, which have, however, a different physical interpretation than in the SK model.