DIRECTED POLYMERS IN A RANDOM MEDIUM - 1/D EXPANSION AND THE N-TREE APPROXIMATION

The authors develop a method which allows one to obtain 1/d expansions for the problem of directed polymers in a random medium. Using this method, they expand thermal properties (such as the free energy, the specific heat), the overlaps and the fluctuations of the transverse displacement. Their results are consistent with the existence of a finite upper critical dimension above which the low-temperature phase is mean field like, i.e. characterised by a broad distribution of overlaps (broken replica symmetry) and transverse fluctuations which scale as the length of the polymer.