  In this mostly numerical study, we revisit the statistical properties of the
ground state of a directed polymer in a $d=1+1$ "hilly" disorder landscape,
i.e. when the quenched disorder has power-law tails. When disorder is Gaussian,
the polymer minimizes its total energy through a collective optimization, where
the energy of each visited site only weakly contributes to the total.
Conversely, a hilly landscape forces the polymer to distort and explore a
larger portion of space to reach some particularly deep energy sites. As soon
as the fifth moment of the disorder diverges, this mechanism radically changes
the standard "KPZ" scaling behaviour of the directed polymer, and new exponents
prevail. After confirming again that the Flory argument accurately predicts
these exponent in the tail-dominated phase, we investigate several other
statistical features of the ground state that shed light on this unusual
transition and on the accuracy of the Flory argument. We underline the
theoretical challenge posed by this situation, which paradoxically becomes even
more acute above the upper critical dimension.
