The Lush World of Random Matrices

The Lush World of Random Matrices

August 2025 - December 2025

Random Matrix Theory (RMT) stands out as a unifying framework at the core of seemingly unrelated fields ranging from signal processing, theoretical ecology, economics and finance, to the theory of learning and the statistical physics of disordered classical and quantum systems. Far from being a mere set of computational tools, RMT provides deep theoretical insights and a foundational understanding of various complex systems. It is key to understand the stability or fragility of complex systems with numerous interacting agents, providing a theoretical framework to capture their responses to perturbations, that may lead to significant changes in the system’s configuration with substantial rearrangements in the state of the individual components (thus describing scenarios of crises or extinctions).
In the theory of learning, it helps describing the robustness of the solutions reached by the optimization algorithms exploited in the training of artificial neural networks, as well as in characterizing how information is progressively encoded during the training process. RMT plays also a crucial role in understanding how correlations spread in quantum systems, leading to chaotic dynamics and eventual thermalization.