Doubling versus non-doubling of equations and phase space structure in one-hermitean-matrix models

In the framework of the saddle-point approximation, we analyse the phase structure of one-hermitean-matrix models with non-even interaction potential. This enables us to reconsider the issue of the doubling phenomenon: the boundaries of the phase space are shown to correspond to critical behaviours with doubling of equations, when the eigenvalue density ρ{variant}(λ) vanishes as the (m - 1 2)th power of λ at both edges of its support; if this occurs at only one edge, the same critical behaviour is realized without doubling phenomenon. Critical domains involving the Painlevé II equation arise also in this non-even regime. As a corollary it appears that pure gravity could be realized within a quartic potential bounded below and leading unambiguously to a one-arc distribution.