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 ρLbulk(E=0,l)
and ρLboundary(E=0,l). We find that both densities follow the scaling
behavior ρLbulk,boundary(E=0,l)=L−1−θdRbulk,boundary(x=l/L), where θd is the exponent governing the
energy fluctuations at zero temperature (with the well-known exact value
θ1=1/3 in one dimension). In the limit x=l/L→0, both scaling
functions Rbulk(x) and Rboundary(x) behave as Rbulk,boundary(x)∼x−1−θd, leading to the droplet power law
ρLbulk,boundary(E=0,l)∼l−1−θd in the regime 1≪l≪L. Beyond their common singularity near x→0, the two scaling functions
Rbulk,boundary(x) are very different : whereas Rbulk(x) decays
monotonically for 0<x<1, the function Rboundary(x) first decays for
0<x<xmin, then grows for xmin<x<1, and finally presents a power law
singularity Rboundary(x)∼(1−x)−σd near x→1. The density
of excitations of length l=L accordingly decays as
ρLboundary(E=0,l=L)∼L−λd where
λd=1+θd−σd. We obtain λ1≃0.67, λ2≃0.53 and λ3≃0.39, suggesting the possible relation
λd=2θd.Comment: 15 pages, 25 figure