  We consider a model of a random copolymer at a selective interface which
undergoes a localization/delocalization transition. In spite of the several
rigorous results available for this model, the theoretical characterization of
the phase transition has remained elusive and there is still no agreement about
several important issues, for example the behavior of the polymer near the
phase transition line. From a rigorous viewpoint non coinciding upper and lower
bounds on the critical line are known.
  In this paper we combine numerical computations with rigorous arguments to
get to a better understanding of the phase diagram. Our main results include:
  - Various numerical observations that suggest that the critical line lies
strictly in between the two bounds.
  - A rigorous statistical test based on concentration inequalities and
super-additivity, for determining whether a given point of the phase diagram is
in the localized phase. This is applied in particular to show that, with a very
low level of error, the lower bound does not coincide with the critical line.
  - An analysis of the precise asymptotic behavior of the partition function in
the delocalized phase, with particular attention to the effect of rare atypical
stretches in the disorder sequence and on whether or not in the delocalized
regime the polymer path has a Brownian scaling.
  - A new proof of the lower bound on the critical line. This proof relies on a
characterization of the localized regime which is more appealing for
interpreting the numerical data.
