  Bootstrap, or $k$-core, percolation displays on the Bethe lattice a mixed
first/second order phase transition with both a discontinuous order parameter
and diverging critical fluctuations. I apply the recently introduced $M$-layer
technique to study corrections to mean-field theory showing that at all orders
in the loop expansion the problem is equivalent to a spinodal with quenched
disorder. This implies that the mean-field hybrid transition does not survive
in physical dimension. Nevertheless, its critical properties as an avoided
transition, make it a proxy of the avoided Mode-Coupling-Theory critical point
of supercooled liquids.
