$1/d$ Expansion for $k$-Core Percolation

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; otherwise they are removed from the system. Such phenomena include orientational ordering in ortho-para ${\rm H}_2$ mixtures and the onset of rigidity in a bar-joint network. 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.