  We demonstrate through two case studies, one on the p-spin interaction model
and the other on the random K-satisfiability problem, that a heterogeneity
transition occurs to the ground-state configuration space of a random
finite-connectivity spin glass system at certain critical value of the
constraint density. At the transition point, exponentially many configuration
communities emerge from the ground-state configuration space, making the
entropy density s(q) of configuration-pairs a non-concave function of
configuration-pair overlap q. Each configuration community is a collection of
relatively similar configurations and it forms a stable thermodynamic phase in
the presence of a suitable external field. We calculate s(q) by the
replica-symmetric and the first-step replica-symmetry-broken cavity methods,
and show by simulations that the configuration space heterogeneity leads to
dynamical heterogeneity of particle diffusion processes because of the entropic
trapping effect of configuration communities. This work clarifies the fine
structure of the ground-state configuration space of random spin glass models,
it also sheds light on the glassy behavior of hard-sphere colloidal systems at
relatively high particle volume fraction.
