At the jamming transition, amorphous packings are known to display anomalous vibrational modes with a density of states (DOS) that remains constant at low frequency. The scaling of the DOS at higher packing fractions remains, however, unclear. One might expect to find a simple Debye scaling, but recent results from effective medium theory and the exact solution of mean-field models both predict an anomalous, non-Debye scaling. Being mean-field in nature, however, these solutions are only strictly valid in the limit of infinite spatial dimension, and it is unclear what value they have for finite-dimensional systems. Here, we study packings of soft spheres in dimensions 3 through 7 and find, away from jamming, a universal non-Debye scaling of the DOS that is consistent with the mean-field predictions. We also consider how the soft mode participation ratio evolves as dimension increases.

When two populations of "particles" move in opposite directions, like oppositely charged colloids under an electric field or intersecting flows of pedestrians, they can move collectively, forming lanes along their direction of motion. The nature of this "laning transition" is still being debated and, in particular, the pair correlation functions, which are the key observables to quantify this phenomenon, have not been characterized yet. Here, we determine the correlations using an analytical approach based on a linearization of the stochastic equations for the density fields, which is valid for dense systems of soft particles. We find that the correlations decay algebraically along the direction of motion, and have a self-similar exponential profile in the transverse direction. Brownian dynamics simulations confirm our theoretical predictions and show that they also hold beyond the validity range of our analytical approach, pointing to a universal behavior.