Finite-size effects in random energy models and in the problem of polymers in a random medium

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 limitN andTT _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.