  In this paper we revisit and extend the mapping between two apparently
different classes of models. The first class contains the prototypical models
described --at the mean-field level-- by the Random First Order Transition
(RFOT) theory of the glass transition, called either "random XORSAT problem"
(in the information theory community) or "diluted $p$-spin model" (in the spin
glass community), undergoing a single-spin flip Glauber dynamics. The models in
the second class are Kinetically Constrained Models (KCM): their Hamiltonian is
that of independent spins in a constant magnetic field, hence their
thermodynamics is completely trivial, but the dynamics is such that only groups
of spin can flip together, thus implementing a kinetic constraint that induces
a non-trivial dynamical behavior. A mapping between some representatives of
these two classes has been known for long. Here we formally prove this mapping
at the level of the master equation, and we apply it to the particular case of
Bethe lattice models. This allows us to show that a RFOT model can be mapped
exactly into a KCM. However, the natural order parameter for the RFOT model,
namely the spin overlap, is mapped into a very complicated non-local function
in the KCM. Therefore, if one were to study the KCM without knowing of the
mapping onto the RFOT model, one would guess that its physics is quite
different from the RFOT one. Our results instead suggest that these two
apparently different descriptions of the glass transition are, at least in some
case, closely related.
