Self-organized critical random directed polymers

Abstract

We uncover a nontrivial signature of the hierarchical structure of quasidegenerate random directed polymers (RDPs) at zero temperature in $(1+1)$-dimensional lattices. Using a cylindrical geometry with circumference $8<~W<~512$, 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 moderately sized avalanches is found to be a power law $P(S)\mathrm{dS}\ensuremath{\sim}{S}^{\ensuremath{-}(1+\ensuremath{\mu})}\mathrm{dS}$. Using a hierarchical formulation based on the length scales ${W}^{2/3}$ (transverse excursion) and the distance ${W}^{(2/3)\ensuremath{\alpha}}$ between quasidegenerate ground states (with $0<\ensuremath{\alpha}<~1)$, we determine $\ensuremath{\mu}=$$\frac{2}{5}$, in excellent agreement with numerical simulations by a transfer matrix method. This power law is valid up to a maximum size ${S}_{5/3}\ensuremath{\sim}{W}^{5/3}$. There is another population of avalanches that for characteristic sizes beyond ${S}_{5/3},$ obeys $P(S)\mathrm{dS}\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(S/S}_{5/3}{)}^{3}]\mathrm{dS}$, 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 ${W}^{2/3}$ of a single RDP.