  We study the problem of predictability, or "nature vs. nurture", in several
disordered Ising spin systems evolving at zero temperature from a random
initial state: how much does the final state depend on the information
contained in the initial state, and how much depends on the detailed history of
the system? Our numerical studies of the "dynamical order parameter" in
Edwards-Anderson Ising spin glasses and random ferromagnets indicate that the
influence of the initial state decays as dimension increases. Similarly, this
same order parameter for the Sherrington-Kirkpatrick infinite-range spin glass
indicates that this information decays as the number of spins increases. Based
on these results, we conjecture that the influence of the initial state on the
final state decays to zero in finite-dimensional random-bond spin systems as
dimension goes to infinity, regardless of the presence of frustration. We also
study the rate at which spins "freeze out" to a final state as a function of
dimensionality and number of spins; here the results indicate that the number
of "active" spins at long times increases with dimension (for short-range
systems) or number of spins (for infinite-range systems). We provide
theoretical arguments to support these conjectures, and also study analytically
several mean-field models: the random energy model, the uniform Curie-Weiss
ferromagnet, and the disordered Curie-Weiss ferromagnet. We find that for these
models, the information contained in the initial state does not decay in the
thermodynamic limit-- in fact, it fully determines the final state. Unlike in
short-range models, the presence of frustration in mean-field models
dramatically alters the dynamical behavior with respect to the issue of
predictability.
