  By defining a spatially varying replica overlap parameter for a supercooled
liquid referenced to an ensemble of fiducial liquid state configurations we
explicitly construct a constrained replica free energy functional that maps
directly onto an Ising Hamiltonian with both random fields and random
interactions whose statistics depend on liquid structure. Renormalization group
results for random magnets when combined with these statistics for the
Lennard-Jones glass suggest that discontinuous replica symmetry breaking would
occur if a liquid with short range interactions could be equilibrated at a
sufficiently low temperature where its mean field configurational entropy would
vanish, even though the system strictly retains a finite configurational
entropy.
