Particle pinning as a method to manipulate marginal stability

We investigate the critical behaviour of low-frequency vibrations in jammed packings with a fraction of pinned particles near the jamming transition. Soft modes form a plateau in the vibrational density of states, with the plateau frequency controlled by the contact number, as in ordinary jamming. The spatial structure of these modes is largely unaffected by pinning. Below the plateau, the non-Debye scaling predicted by mean-field theories and the presence of quasi-localised modes break down, depending on how particles are pinned. We show that these behaviours can be comprehensively understood in terms of the impact of pinning on the marginal stability of the packings.