  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 $\rho^{bulk}_L(E=0,l)$
and $\rho^{boundary}_L(E=0,l)$. We find that both densities follow the scaling
behavior $\rho^{bulk,boundary}_L(E=0,l) = L^{-1-\theta_d}
R^{bulk,boundary}(x=l/L)$, 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^{bulk,boundary}_L(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^{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
$\lambda_d= 2 \theta_d$.
