  We study the Glauber dynamics for heavy-tailed spin glasses, in which the
couplings are in the domain of attraction of an $\alpha$ stable law for
$\alpha\in (0,1)$. We show a sharp description of metastability on exponential
timescales, in a form that is believed to hold for Glauber/Langevin dynamics
for many mean-field spin glass models, but only known rigorously for the Random
Energy Models. Namely, we establish a decomposition of the state space into
sub-exponentially many wells, and show that the projection of the Glauber
dynamics onto which well it resides in, asymptotically behaves like a Markov
chain on wells with certain explicit transition rates. In particular, mixing
inside wells occurs on much shorter timescales than transit times between
wells, and the law of the next well the Glauber dynamics will fall into depends
only on which well it currently resides in, not its full configuration. We can
deduce consequences like an exact expression for the two-time autocorrelation
functions that appear in the activated aging literature.
