Loop expansion around the Bethe-Peierls approximation for lattice models

We develop an effective field theory for lattice models, in which the only non-vanishing diagrams exactly reproduce the topology of the lattice. The Bethe-Peierls approximation appears naturally as the saddle point approximation. The corrections to the saddle-point result can be obtained systematically. We calculate the lowest loop corrections for magnetisation and correlation function.Comment: 8 page

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 correction 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