Typical quasistatic compression algorithms for generating jammed packings of purely repulsive, frictionless particles begin with dilute configurations and then apply successive compressions with the relaxation of the elastic energy allowed between each compression step. It is well known that during isotropic compression these systems undergo a first-order-like jamming transition at packing fraction phi(J) from an unjammed state with zero pressure and no force-bearing contacts to a jammed, rigid state with nonzero pressure, a percolating network of force-bearing contacts, and contact number z = 2d, where d is the spatial dimension. Using computer simulations of two-dimensional systems with monodisperse and bidisperse particle size distributions, we investigate the second-order-like contact percolation transition, which precedes the jamming transition with phi(P) < phi(J) and signals the formation of a system-spanning cluster of non-force-bearing contacts between particles. By measuring the number of nonfloppy modes of the dynamical matrix, the displacement field between successive compression steps, and the overlap between the adjacency matrix, which represents the network of contacting grains, at phi and phi(J), we find that the contact percolation transition also signals the onset of a nontrivial mechanical response to applied stress. Our results show that cooperative particle motion occurs in unjammed systems significantly below the jamming transition for phi(P) < phi < phi(J), not only for jammed systems with phi > phi(J).
