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We consider a directed polymer of length $L$ in a random medium of spacedimension $d=1,2,3$. The statistics of low energy excitations as a function oftheir size $l$ is numerically evaluated. These excitations can be divided intobulk and boundary excitations, with respective densities $\rho^{bulk}_L(E=0,l)$and $\rho^{boundary}_L(E=0,l)$. We find that both densities follow the scalingbehavior $\rho^{bulk,boundary}_L(E=0,l) = L^{-1-\theta_d}R^{bulk,boundary}(x=l/L)$, where $\theta_d$ is the exponent governing theenergy 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 scalingfunctions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^{bulk,boundary}(x)\sim x^{-1-\theta_d}$, leading to the droplet power law$\rho^{bulk,boundary}_L(E=0,l)\sim l^{-1-\theta_d} $ in the regime $1 \ll l \llL$. Beyond their common singularity near $x \to 0$, the two scaling functions$R^{bulk,boundary}(x)$ are very different : whereas $R^{bulk}(x)$ decaysmonotonically 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 lawsingularity $R^{boundary}(x)\sim (1-x)^{-\sigma_d}$ near $x \to 1$. The densityof excitations of length $l=L$ accordingly decays as$\rho^{boundary}_L(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$\lambda_d= 2 \theta_d$.Comment: 15 pages, 25 figure

physical review. e, statistical, nonlinear and soft matter physics

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Statistics of low energy excitations for the directed polymer in a1+drandom medium(d=1,2,3)