Copolymers at selective interfaces: New bounds on the phase diagram

We investigate the phase diagram of disordered copolymers at the interface between two selective solvents, and in particular its weak-coupling behavior, encoded in the slope m (c) of the critical line at the origin. We focus on the directed walk case, which has turned out to be, in spite of the apparent simplicity, extremely challenging. In mathematical terms, the partition function of such a model does not depend on all the details of the Markov chain that models the polymer, but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns. This observation leads to a natural generalization of the model, in terms of arbitrary laws of return times: the most interesting case being the one of return times with power law tails (with exponent 1+alpha, alpha=1/2 in the case of the symmetric random walk). The main results we present here are: (1) the improvement of the known result 1/(1+alpha) <= m(c) <= 1, as soon as alpha > 1 for what concerns the upper bound, and down to alpha approximate to 0.65 for the lower bound. (2) a proof of the fact that the critical curve lies strictly below the critical curve of the annealed model for every non-zero value of the coupling parameter. We also provide an argument that rigorously shows the strong dependence of the phase diagram on the details of the return probability (and not only on the tail behavior). Lower bounds are obtained by exhibiting a new localization strategy, while upper bounds are based on estimates of non-integer moments of the partition function.