  A directed polymer is considered on a flat substrate with randomly located
parallel ridges. It prefers to lie inside wide regions between the ridges. When
the transversel width $W=\exp(\lambda L^{1/3})$ is exponential in the
longitudinal length $L$, there can be a large number $\sim \exp L^{1/3}$ of
available wide states. This ``complexity'' causes a phase transition from a
high temperature phase where the polymer lies in the widest lane, to a glassy
low temperature phase where it lies in one of many narrower lanes. Starting
from a uniform initial distribution of independent polymers, equilibration up
to some exponential time scale induces a sharp dynamical transition. When the
temperature is slowly increased with time, this occurs at a tunable
temperature. There is an asymmetry between cooling and heating. The structure
of phase space in the low temperature non-equilibrium glassy phase is of a
one-level tree.
