Self-organized critical random directed polymers

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.