The onset of jamming as the sudden emergence of an infinite $\mth{k}$-core cluster

A theory is constructed to describe the zero-temperature jamming transition of repulsive soft spheres as the density 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 reminiscent of the jamming transition. In particular, the exponents characterizing the continuous parts of both transitions appear to be the same. Finally, numerical simulations suggest that in finite dimensions the k-core transition can be discontinuous with a nontrivial diverging correlation length.