Manifolds in random media: two extreme cases

Abstract

We discuss two special cases of directed manifolds in random media. One is the zero dimensional model of one particle in a random potential, the other is the limit where the manifold is embedded in a space of large dimensionality. We use the recently introduced Gaussian variational approach in replica space, with replica symmetry breaking, and compare it to known results and simulations of the toy model. In large dimensions (N - co) the variational approach is exact. We compute one of the diagrams contributing to the O(1IN) correction.