  Models for RNA secondary structures (the topology of folded RNA) without
pseudo knots are disordered systems with a complex state-space below a critical
temperature. Hence, a complex dynamical (glassy) behavior can be expected, when
performing Monte Carlo simulation. Interestingly, in contrast to most other
complex systems, the ground states and the density of states can be computed in
polynomial time exactly using transfer matrix methods. Hence, RNA secondary
structure is an ideal model to study the relation between static/thermodynamic
properties and dynamics of algorithms. Also they constitute an ideal benchmark
system for new Monte Carlo methods.
  Here we considered three different recent Monte Carlo approaches: entropic
sampling using flat histograms, optimized-weights ensembles, and ParQ, which
estimates the density of states from transition matrices.
  These methods were examined by comparing the obtained density of states with
the exact results. We relate the complexity seen in the dynamics of the Monte
Carlo algorithms to static properties of the phase space by studying the
correlations between tunneling times, sampling errors, amount of meta-stable
states and degree of ultrametricity at finite temperature.
