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The state-following technique allows the study of metastable glassy statesunder external perturbations. Here we show how this construction can be used tostudy the behavior of glassy states of Hard Spheres in infinite dimensionsunder compression or shear strain. In a preceding work it has been shown thatin both cases, when the external perturbation is sufficiently strong, glassystates undergo a second-order transition, called the Gardner transition,whereupon a hierarchical structure of marginal micro-states manifests withinthe original glass state. The purpose of this work is to study the solution ofthe state-following construction in this marginal phase. We show that uponcompression, close to the jamming transition, the metastable states aredescribed by a scaling solution characterized by a set of non-trivial criticalexponents that agree with the results obtained in a preceding work, and wecompute the value of the final jamming density for various glassy states.Moreover we show that under the action of the shear strain, beyond the Gardnerpoint, the metastable states can be followed in the marginal phase and wedetect an overshoot in the stress-strain curve in agreement with numerical andexperimental observations. Finally we further characterize the Gardnertransition point by computing both the critical 4-point susceptibility and theexponents that characterize the critical slowing down of the dynamics within aglassy state close to the transition.

Following the evolution of glassy states under external perturbations: the full replica symmetry breaking solution