  Coarse-grained (lattice-) models have a long tradition in aiding efforts to
decipher the physical or biological complexity of proteins. Despite the
simplicity of these models, however, numerical simulations are often
computationally very demanding and the quest for efficient algorithms is as old
as the models themselves. Expanding on our previous work [T. W\"ust and D. P.
Landau, Phys. Rev. Lett. 102, 178101 (2009)], we present a complete picture of
a Monte Carlo method based on Wang-Landau sampling in combination with
efficient trial moves (pull, bond-rebridging and pivot moves) which is
particularly suited to the study of models such as the hydrophobic-polar (HP)
lattice model of protein folding. With this generic and fully blind Monte Carlo
procedure, all currently known putative ground states for the most difficult
benchmark HP sequences could be found. For most sequences we could also
determine the entire energy density of states and, together with suitably
designed structural observables, explore the thermodynamics and intricate
folding behavior in the virtually inaccessible low-temperature regime. We
analyze the differences between random and protein-like heteropolymers for
sequence lengths up to 500 residues. Our approach is powerful both in terms of
robustness and speed, yet flexible and simple enough for the study of many
related problems in protein folding.
