  We show that mixing for local, reversible dynamics of mean field spin glasses
is exponentially slow in the low temperature regime. We introduce a notion of
free energy barriers for the overlap, and prove that their existence imply that
the spectral gap is exponentially small, and thus that mixing is exponentially
slow. We then exhibit sufficient conditions on the equilibrium Gibbs measure
which guarantee the existence of these barriers, using the notion of replicon
eigenvalue and 2D Guerra Talagrand bounds. We show how these sufficient
conditions cover large classes of Ising spin models for reversible
nearest-neighbor dynamics and spherical models for Langevin dynamics. Finally,
in the case of Ising spins, Panchenko's recent rigorous calculation [79] of the
free energy for a system of "two real replica" enables us to prove a quenched
LDP for the overlap distribution, which gives us a wider criterion for slow
mixing directly related to the Franz-Parisi-Virasoro approach [43,60]. This
condition holds in a wider range of temperatures.
