Diffusion near absorbing fractals: Harmonic measure exponents for polymers

We study the depletion of a diffusing substance [described by a scalar Laplacian field u(r)] in the vicinity of an absorbing fractal, consisting of a Gaussian random walk (RW) or a self-avoiding walk (SAW) in d dimensions. The moments 〈[u(r${\left)\right]}^n$〉 of the field u(r) at a distance r from a point on an absorber of linear size R obey 〈[u(r${\left)\right]}^n$〉∼(r/R${)}^{\lambda \left(n\right)}$, where the λ(n) are a set of independent exponents. The incident flux φ of the field u defines a fractal measure on the absorber. The scaling dimensions D(n) describing this measure obey (1-n)[D(n)-D]=nλ(1)-λ(n), with D the fractal dimension; moreover, λ(1)=D+2-d. These relations apply for any fractal absorber. For the RW and SAW, we observe that the exponents λ(n) are the same as those describing the statistics of linear chains and star polymers with selective excluded-volume interactions. This allows us to calculate λ(n) and D(n) explicitly to order ${\epsilon }^2$, where ε=4-d. We find D(n)=D-n${\epsilon }^2$/4 for the RW and D(n)=D-9n${\epsilon }^2$/64 for the SAW. We also study nonperturbatively the limit of high n. For n large, D(n)-D(∞)∼$n^{\left(3\mathrm{}\mathrm{-}d\right)/\left(d\mathrm{-}2\right)}$ for 3<d<4, whereas in d=3 dimensions, D(n)-D(∞)∼[ln(n${\left)\right]}^{\mathrm{-}1}$; here D(∞)=d-2. These high-n results apply to the RW and SAW alike. We construct for the absorbing RW part of the scaling function f(α), defined by Halsey et al. [Phys. Rev. A 33, 1141 (1986)], and find a range of α for which f(α) is negative. Identifying f(α) with the histogram of the measure expressed in logarithmic variables, we discuss the meaning of negative f(α).