Large Deviations of the Finite-Time Magnetization of the Curie-Weiss Random Field Ising Model

We study the large deviations of the magnetization at some finite time in the Curie-Weiss Random Field Ising Model with parallel updating. While relaxation dynamics in an infinite time horizon gives rise to unique dynamical trajectories (specified by initial conditions and governed by first-order dynamics of the form $m_{t+1}=f(m_t)$), we observe that the introduction of a finite time horizon and the specification of terminal conditions can generate a host of metastable solutions obeying \textit{second-order} dynamics. We show that these solutions are governed by a Newtonian-like dynamics in discrete time which permits solutions in terms of both the first order relaxation ("forward") dynamics and the backward dynamics $m_{t+1} = f^{-1}(m_t)$. Our approach allows us to classify trajectories for a given final magnetization as stable or metastable according to the value of the rate function associated with them. We find that in analogy to the Freidlin-Wentzell description of the stochastic dynamics of escape from metastable states, the dominant trajectories may switch between the two types (forward and backward) of first-order dynamics.