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The complex dynamics of an increasing number of systems is attributed to theemergence of a rugged energy landscape with an exponential number of metastablestates. To develop this picture into a predictive dynamical theory I discusshow to compute the exponentially small probability of a jump from onemetastable state to another. This is expressed as a path integral that can beevaluated by saddle-point methods in mean-field models, leading to a boundaryvalue problem. The resulting dynamical equations are solved numerically bymeans of a Newton-Krylov algorithm in the paradigmatic spherical $p$-spin glassmodel that is invoked in diverse contexts from supercooled liquids tomachine-learning algorithms. I discuss the solutions in the asymptotic regimeof large times and the physical implications on the nature of theergodicity-restoring processes.

Path Integral Approach Unveils the Role of Complex Energy Landscape for Activated Dynamics of Glassy Systems