On the 1/D expansion for directed polymers

We present a variational approach for directed polymers in $D$ transversal dimensions which is used to compute the corrections to the mean field theory predictions with broken replica symmetry. The trial function is taken to be a symmetrized version of the mean-field solution, which is known to be exact for $D=\infty$. We compute the free energy corresponding to that function and show that the finite-$D$ corrections behave like $D^{-4/3}$. It means that the expansion in powers of 1/D should be used with great care here. We hope that the techniques developed in this note will be useful also in the study of spin glasses.