  The physics of $k$-core percolation pertains to those systems whose
constituents require a minimum number of $k$ connections to each other in order
to participate in any clustering phenomenon. Examples of such a phenomenon
range from orientational ordering in solid ortho-para ${\rm H}_2$ mixtures to
the onset of rigidity in bar-joint networks to dynamical arrest in
glass-forming liquids. Unlike ordinary ($k=1$) and biconnected ($k=2$)
percolation, the mean field $k\ge3$-core percolation transition is both
continuous and discontinuous, i.e. there is a jump in the order parameter
accompanied with a diverging length scale. To determine whether or not this
hybrid transition survives in finite dimensions, we present a $1/d$ expansion
for $k$-core percolation on the $d$-dimensional hypercubic lattice. We show
that to order $1/d^3$ the singularity in the order parameter and in the
susceptibility occur at the same value of the occupation probability. This
result suggests that the unusual hybrid nature of the mean field $k$-core
transition survives in high dimensions.
