  A random polymer model is a one-dimensional Jacobi matrix randomly composed
of two finite building blocks. If the two associated transfer matrices commute,
the corresponding energy is called critical. Such critical energies appear in
physical models, an example being the widely studied random dimer model. It is
proven that the Lyapunov exponent vanishes quadratically at a generic critical
energy and that the density of states is positive there. Large deviation
estimates around these asymptotics allow to prove optimal lower bounds on
quantum transport, showing that it is almost surely overdiffusive even though
the models are known to have pure-point spectrum with exponentially localized
eigenstates for almost every configuration of the polymers. Furthermore, the
level spacing is shown to be regular at the critical energy.
