  We uncover a nontrivial signature of the hierarchical structure of
quasi-degenerate random directed polymers (RDPs) at zero temperature in 1+1
dimensional lattices. Using a cylindrical geometry with circumference $8 \leq W
\leq 512$, we study the differences in configurations taken by RDPs forced to
pass through points displaced successively by one unit lattice mesh. The
transition between two successive configurations (interpreted as an avalanche)
defines an area $S$. The distribution of moderatly sized avalanches is found to
be a power-law $P(S) dS \sim S^{-(1+\mu)} dS$. Using a hierarchical formulation
based on the length scales $W^{2\over 3}$ (transverse excursion) and the
distance $W^{{2\over 3}\alpha}$ between quasi-degenerate ground states (with
$0<\alpha\le 1$), we determine $\mu = {2\over 5}$, in excellent agreement with
numerical simulations by a transfer matrix method. This power-law is valid up
to a maximum size $S_{5\over 3} \sim W^{5\over 3}$. There is another population
of avalanches which, for characteristic sizes beyond $S_{5\over 3}$, obeys
$P(S) dS \sim \exp(-(S/S_{5\over 3})^3) dS$ also confirmed numerically. The
first population corresponds to almost degenerate ground states, providing a
direct evidence of ``weak replica symmetry breaking'', while the second
population is associated with different optimal states separated by the typical
fluctuation $W^{2\over 3}$ of a single RDP.
