MEAN-FIELD THEORY OF DIRECTED POLYMERS WITH RANDOM COMPLEX WEIGHTS

We show that for the problem of directed polymers on a tree with i.i.d . random complex weights on each bond, three possible phases can exist ; the phase of a particular system is determined by the distribution r ho of the random weights. For each of these three phases, we give the expression of the free energy per unit length in the limit of infinite ly long polymers. Our proofs require several hypotheses on the distrib ution rho, most importantly, that the amplitude and the phase of each complex weight be statistically independent. The main steps of our pro ofs use bounds on noninteger moments of the partition function and sel f averaging properties of the free energy. We illustrate our results b y some examples and discuss possible generalizations to a larger class of distributions, to Random Energy Models, and to the finite dimensio nal case. We note that our results are not in agreement with the predi ctions of a recent replica approach to a similar problem.