Particle pinning as a method to manipulate marginal stability

We study the critical behavior of low-frequency vibrations of packings with pinned particles near the jamming point. Soft modes form a plateau in the density of states and its frequency is controlled by the contact number as the ordinary jamming transition. The spatial structure of these modes is not largely affected by pins. Below the plateau, the non-Debye scaling predicted by mean-field theories and quasi-localized modes breaks down depending on the pinning procedures. We comprehensively explain these behaviors by the impact of pinning operations on the marginal stability of the packings.