Dynamical Mean-Field Theory: from glassy systems to large ecological communities with asymmetric interactions

The emergence of numerous metastable states, slow relaxation dynamics, and aging are hallmarks of glassiness. By focusing on systems with soft disordered interactions, I will show how to incorporate aging within Dynamical Mean-Field Theory and deduce the main quantities of interest – such as effective temperatures, correlations, and responses – in the slow regime. I will present both the case with only one slow timescale and that with an infinite hierarchy of slower and slower timescales. Remarkably, for the latter, aging dynamics is governed by a self-consistent stochastic equation, which can be shown to precisely correspond to the ultrametric protocol proposed by Parisi for the static counterpart.

These findings extend the realm of out-of-equilibrium mean-field scenarios to a broader spectrum of contexts where DMFT is applicable.

As a final highlight, I will discuss a promising approach to theoretical ecology aiming to capture the complexity of large ecosystems. I will then stick to a Generalized Lotka-Volterra model with random interactions between species and finite demographic fluctuations.