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

TL;DR

This paper investigates the large deviations of finite-time magnetization in the Curie-Weiss Random Field Ising Model, revealing metastable solutions governed by second-order dynamics and classifying trajectories based on stability and rate functions.

Contribution

It introduces a novel analysis of finite-time magnetization deviations, highlighting the role of second-order dynamics and metastability in the model.

Findings

Metastable solutions obey second-order dynamics.

Trajectories can switch between forward and backward dynamics.

Classification of trajectories by stability and rate functions.

Abstract

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.