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. Recently, there has been a renewal of interest for the long-time dynamics of spin-glass models, which may already exhibit nontrivial features, for example, violation of the fluctuation–dissipation relations, ageing phenomena, at the mean-field level [1]. In this context, it is natural to study the spectrum of relaxation times occuring in the Glauber dynamics of disordered Ising spin models [2]. Unfortunately, numerical investigations have so far been limited to very small sizes [4]. Moreover, to the best of our knowledge, no analytical studies of the relaxation times spectra have yet been performed owing to the technical difficulties arising in the diagonalization of the Glauber matrix. In this letter, we focus on the Glauber dynamics of the Hopfield model at low storage [6]. This system is simple enough to be analytically solved along the lines of [5]. Our main result is that 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. As a consequence, our study strengthens the close relationship between long-time dynamics and static properties in disordered mean-field models [1]. We consider a Hopfield model including N Ising spins S i , i = 1,. .. , N and p quenched patterns ξ µ i , µ = 1,. .. , p. In addition to its intrinsic interest as a paradigm for attractor neural networks [7], the Hopfield model may be seen as a spin-glass system smoothly interpolating between the Mattis model [8] (when p = 1) and the much more involved Sherrington–Kirpatrick model (for infinite p) [3]. Its Hamiltonian depends on the spin configuration S = (S 1 ,. .. , S N) through the set of p overlaps m µ (S) = 1 N N i=1 S i ξ µ i and reads H (S) = − …