By the use of traveling wave equations we calculate the finite-size corrections to
the free energy of random energy models in their low-temperature phases and
in the neighborhood of the transition temperature. We find that although the
extensive part of the free energy does not show any critical behavior when the
temperature approaches its transition value, the finite*size corrections signal the
transition by becoming singular. We obtain a scaling form for these finite-size
corrections valid in the limit N — * oe and T ~ T c. By considering a generalized
random energy model in the limit of a very large number of steps, we obtain
results for the finite-size corrections in the problem of a polymer in a random
medium.