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

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 $\rho^{bulk,boundary}_L(E=0,l) = L^{-1-\theta_d} R^{bulk,boundary}(x=l/L)$,where$\theta_d$ is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value ${\theta }_1=1\mathrm{/}3$ in one dimension). In the limit $x=l\mathrm{/}L\to 0$, both scaling functions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^{bulk,boundary}(x) \sim x^{-1-\theta_d}$,leadingtothedropletpowerlaw$\rho^{bulk,boundary}_L(E=0,l)\sim l^{-1-\theta_d} $intheregime$1 \ll l \ll L$\mathrm{.}Beyondtheircommonsingularitynear$x \to 0$,thetwoscalingfunctions$R^{bulk,boundary}(x)$areverydifferent: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$\lambda_3 \simeq 0.39$,suggestingthepossiblerelation$\lambda_d= 2 \theta_d$.