Local state space geometry and thermal relaxation in complex landscapes: the spin-glass case

A simple geometrical characterization of configuration space neighborhoods of local energy minima in spin glass landscapes is found by exhaustive search. Combined with previous Monte Carlo investigations of thermal domain growth, it allows a discussion of the connection between real and configuration space descriptions of low temperature relaxational dynamics. We argue that the part of state-space corresponding to a single growing domain is adequately modeled by a hierarchically organized set of states and that thermal (meta)stability in spin glasses is related to the nearly exponential local density of states present within each trap.