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.