  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. 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) smaller or equal to m_c
smaller or equal to 1, as soon as alpha >1 for what concerns the upper bound,
and down to alpha = 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.
