Statistics of low energy excitations for the directed polymer in a $1+d$ random medium ($d=1,2,3$)

Abstract

We consider a directed polymer of length $L$ in a random medium of space dimension $d=1,2,3$. The statistics of low energy excitations as a function of their size $l$ is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities ${\rho }_L^{bulk}(E=0,l)$ and ${\rho }_L^{boundary}(E=0,l)$. We find that both densities follow the scaling behavior ${\rho }_L^{bulk,boundary}(E=0,l)=L^{-1-{\theta }_d}{R}^{bulk,boundary}(x=l/L)$, where ${\theta }_d$ is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value ${\theta }_1=1/3$ in one dimension). In the limit $x=l/L\to 0$, both scaling functions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^{bulk,boundary}(x)\sim x^{-1-{\theta }_d}$, leading to the droplet power law ${\rho }_L^{bulk,boundary}(E=0,l)\sim l^{-1-{\theta }_d}$ in the regime $1\ll l\ll L$. Beyond their common singularity near $x\to 0$, the two scaling functions $R^{bulk,boundary}(x)$ are very different : whereas $R^{bulk}(x)$ decays monotonically for $0<x<1$, the function $R^{boundary}(x)$ first decays for $0<x<x_{min}$, then grows for $x_{min}<x<1$, and finally presents a power law singularity $R^{boundary}(x)\sim (1-x{)}^{-{\sigma }_d}$ near $x\to 1$. The density of excitations of length $l=L$ accordingly decays as ${\rho }_L^{boundary}(E=0,l=L)\sim L^{-{\lambda }_d}$ where ${\lambda }_d=1+{\theta }_d-{\sigma }_d$. We obtain ${\lambda }_1\simeq 0.67$, ${\lambda }_2\simeq 0.53$ and ${\lambda }_3\simeq 0.39$, suggesting the possible relation ${\lambda }_d=2{\theta }_d$.

arXiv comment15 pages, 25 figures

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