Abstract

Random Matrix models naturally arise in relation to a great variety of problems in mathematics and physics, and constitute a paradigm for the modelling of complex phenomena. We focus on the Hermitian Matrix Ensemble (HME) and Symmetric Matrix Ensemble (SME) and their relationship with integrable hierarchies of nonlinear equations. This relationship is fundamental in the investigation of the features of these models, and in the study of their thermodynamic limit. This talk includes results from C.B, A. Moro (2020) and C.B, M.Dell’Atti, A. Moro (2021). In particular, we show that the HME supports a novel type of phase transition, signalled by a dispersive shock of the relevant order parameter, propagating in the space of coupling constants. On the other hand, in the thermodynamic limit, the order parameters of the SME satisfy a novel integrable hydrodynamic chain of PDEs.