Title: Liu-Nagel phase diagrams in infinite dimension,
Abstract: We study Harmonic Soft Spheres as a model of thermal structural glasses in
the limit of infinite dimensions. We show that cooling, compressing and
shearing a glass lead to a Gardner transition and, hence, to a marginally
stable amorphous solid as found for Hard Spheres systems. A general outcome of
our results is that a reduced stability of the glass favors the appearance of
the Gardner transition. Therefore using strong perturbations, e.g. shear and
compression, on standard glasses or using weak perturbations on weakly stable
glasses, e.g. the ones prepared close to the jamming point, are the generic
ways to induce a Gardner transition. The formalism that we discuss allows to
study general perturbations, including strain deformations that are important
to study soft glassy rheology at the mean field level.