The onset of jamming as the sudden emergence of an infinite $k$-core cluster

A theory is constructed to describe the zero-temperature jamming transition as the density of repulsive soft spheres is increased. Local mechanical stability imposes a constraint on the minimum number of bonds per particle; we argue that this constraint suggests an analogy to $k$-core percolation. The latter model can be solved exactly on the Bethe lattice, and the resulting transition has a mixed first-order/continuous character. The exponents characterizing the continuous part appear to be the same as for the jamming transition. Finally, numerical simulations suggest that in finite dimensions the $k$-core transition can be discontinuous with a nontrivial diverging correlation length.