k-core percolation in four dimensions

The k-core percolation on the Bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first-order-second-order nature, We investigate numerically k-core percolation on the four-dimensional regular lattice. For k=4, the presence of a discontinuous transition is clearly established but its nature is strictly first-order. In particular, the k-core density displays no singular behavior before the jump and its correlation length remains finite. For k=3, the transition is continuous.