Statistics of low energy excitations for the directed polymer in a random medium

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 --> 0, both scaling functions R(bulk)(x) and R(boundary)(x) behave as R(bulk,boundary)(x) approximately x(-1-theta)(d), leading to the droplet power law rho(bulk, boundary)(L) (E = 0,l) approximately l(-1-theta)(d) in the regime 1 << l << L. Beyond their common singularity near x --> 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) approximately (1-x)(-sigma)(d) near x -->1. The density of excitations of length l = L accordingly decays as rho(boundary)(L)(E = 0,l = L) approximately L(-lambda)(d) where gamma(d) = 1+ theta(d) - lambda(d). We obtain lambda(1) approximately 10.67, lambda(2) = 0.53, and lambda(3) approximately 0.39, suggesting the possible relation lambda(d) = 2theta(d).