  Directed polymers on 1+1 dimensional lattices coupled to a heat bath at
temperature $T$ are studied numerically for three ensembles of the site
disorder. In particular correlations of the disorder as well as fractal
patterning are considered. Configurations are directly sampled in perfect
thermal equilibrium for very large system sizes with up to $N=L^2= 32768 \times
32768 \approx 10^{9}$ sites. The phase-space structure is studied via the
distribution of overlaps and hierarchical clustering of configurations. One
ensemble shows a simple behavior like a ferromagnet. The other two ensembles
exhibit indications for complex behavior reminiscent of multiple
replica-symmetry breaking. Also results for the ultrametricity of the phase
space and the phase transition behavior of $P(q)$ when varying the temperature
$T$ are studied. In total, the present model ensembles offer convenient
numerical accesses to comprehensively studying complex behavior.
