Spectral Gap Estimates in Mean Field Spin Glasses

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 (Panchenko in Ann Probab 46(2):865-896, 2018) 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 (Franz et al. in J Phys I 2(10):1869-1880, 1992; Kurchan et al. J Phys I 3(8):1819-1838, 1993). This condition holds in a wider range of temperatures.