Fractal geometry of spin-glass models

The topology of the phase space of spin glasses can be conveniently condensed into a tree structure, termed a barrier tree, whose tips are the local minima and whose internal nodes are the lowest-energy saddles connecting the local minima. For the mean-field Ising spin glass with p-spin interactions and sizes up to N = 24 spins we compute exactly the weight distribution psi (w), where w = w (s) is the fraction of minima that are connected through the saddle s. For low-energy saddles we find a power law psi (w) similar to w(-D), where D is the fractal dimension of the phase space, and barrier trees which are qualitatively similar to balanced random trees. For higher energy levels, on the other hand, the barrier trees become highly unbalanced resulting in a flat weight distribution.