Relationship between long timescales and the static free-energy in the Hopfield model

The Glauber dynamics of the Hopfield model at low storage level is considered. We analytically derive the spectrum of relaxation times for large system sizes. The longest timescales are gathered in families, each family being in one-to-one correspondence with a stationary (not necessarily stable) point of the static mean-field free energy. Inside a family, the timescales are given by the reciprocals (of the absolute values) of the eigenvalues of the free-energy Hessian matrix.