We show that Hermitian Matrix Models with arbitrary degree of nonlinearity support the occurrence of a new type of phase transition described by a dispersive shock solution of a nonlinear dispersive hydrodynamic system. The order parameters, defined as derivatives of the free energy with respect to the coupling constants can be obtained as a solution of the Toda lattice equations. 

 

The thermodynamic limit corresponds to the continuum limit of the Toda system where the order parameter is a solution of a nonlinear partial differential equation in a small dispersion regime. The order parameter evolves in the space of coupling constants as a nonlinear wave that develops a dispersive shock for a suitable choice of the couplings. Our analysis explains the origin and the mechanisms leading to the emergence of chaotic behaviours observed by Jurkiewicz in M^6 matrix models.

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