Thermodynamic limit and dispersive regularization in matrix models

We show that Hermitian matrix models support the occurrence of a new type of phase transition characterised by dispersive regularisation of the order parameter near the critical point. Using the identification of the partition function with a solution of a reduction of the Toda hierarchy, known as Volterra system, we argue that the singularity is resolved by the onset of a multi-dimensional dispersive shock of the order parameter in the space of coupling constants. This analysis explains
the origin and mechanism leading to the emergence of chaotic behaviours observed in M6 matrix models and extends its validity to even nonlinearity of arbitrary order.