  We introduce two new concepts, frictional rigidity percolation and minimal
rigidity proliferation, to help identify the nature of the frictional jamming
transition as well as significantly broaden the scope of rigidity percolation.
For frictional rigidity percolation, we construct rigid clusters in two
different lattice models using a $(3,3)$ pebble game, while taking into account
contacts below and at the Coulomb threshold. The first lattice is a honeycomb
lattice with next-nearest neighbors, the second, a hierarchical lattice. For
both, we generally find a continuous rigidity transition. Our numerical results
suggest that, for the honeycomb lattice, the exponents associated with the
transition found with the frictional $(3,3)$ pebble game are distinct from
those of a central-force $(2,3)$ pebble game. We propose that localized motifs,
such as hinges, connecting rigid clusters that are allowed only with friction
could give rise to this new frictional universality class. However, the
closeness of the order parameter exponent between the two cases hints at
potential superuniversality. To explore this possibility, we construct a
bespoke cluster generating algorithm invoking generalized Henneberg moves,
dubbed minimal rigidity proliferation. The minimally rigid clusters the
algorithm generates appear to be in the same universality class as connectivity
percolation, suggesting superuniversality between all three types of
transitions. Finally, the hierarchical lattice is analytically tractable and we
find that the exponents depend both on the type of force and on the fraction of
contacts at the Coulomb threshold. These combined results allow us to compare
two universality classes on the same lattice via rigid clusters for the first
time to highlight unifying and distinguishing concepts within the set of all
possible rigidity transitions in disordered systems.
