Thermodynamic limit and dispersive regularization in matrix models

We show that Hermitian matrix models support the occurrence of a phase transition characterized by dispersive regularization of the order parameter near the critical point. Using the identification of the partition function with a solution of the reduction of the Toda hierarchy known as the Volterra system, we argue that the singularity is resolved by the onset of a multidimensional 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 behaviors observed in M^{6} matrix models and extends its validity to even nonlinearity of arbitrary order.