  We study classical percolation models in Fock space as proxies for the
quantum many-body localisation (MBL) transition. Percolation rules are defined
for two models of disordered quantum spin-chains using their microscopic
quantum Hamiltonians and the topologies of the associated Fock-space graphs.
The percolation transition is revealed by the statistics of Fock-space cluster
sizes, obtained by exact enumeration for finite-sized systems. As a function of
disorder strength, the typical cluster size shows a transition from a volume
law in Fock space to sub-volume law, directly analogous to the behaviour of
eigenstate participation entropies across the MBL transition. Finite-size
scaling analyses for several diagnostics of cluster size statistics yield
mutually consistent critical properties. We show further that local observables
averaged over Fock-space clusters also carry signatures of the transition, with
their behaviour across it in direct analogy to that of corresponding eigenstate
expectation values across the MBL transition. The Fock-space clusters can be
explored under a mapping to kinetically constrained models. Dynamics within
this framework likewise show the ergodicity-breaking transition via Monte Carlo
averaged local observables, and yield critical properties consistent with those
obtained from both exact cluster enumeration and analytic results derived in
our recent work [arXiv:1812.05115]. This mapping allows access to system sizes
two orders of magnitude larger than those accessible in exact enumerations.
Simple physical pictures based on freezing of local real-space segments of
spins are also presented, and shown to give values for the critical disorder
strength and correlation length exponent $\nu$ consistent with numerical
studies.
