  We consider symmetric trap models in the d-dimensional hypercube whose
ordered mean waiting times, seen as weights of a measure in the natural
numbers, converge to a finite measure as d diverges, and show that the models
suitably represented converge to a K process as d diverges. We then apply this
result to get K processes as the scaling limits of the REM-like trap model and
the Random Hopping Times dynamics for the Random Energy Model in the hypercube
in time scales corresponding to the ergodic regime for these dynamics.
