Family of Exponents for Laplace's Equation near a Polymer

We study the depletion of a diffusing substance (i.e., of a scalar Laplacian field) near an absorbing fractal, consisting of a random or a self-avoiding walk. We establish a mapping between the moments $〈U{\mathrm{}\left(r\right)\mathrm{}}^n〉$ of the field $U\left(r\right)\mathrm{}$ at a distance $r$ from a point on the absorber and the partition functions of certain star polymers. The scaling with $r$ of each moment is governed by an independent exponent $\lambda \mathrm{}\left(n\right)\mathrm{}$, which we calculate to order ${\epsilon }^2(\epsilon =4-d)$. Nonperturbative results for the limit of high $n$ are also given. We relate the $\lambda \mathrm{}\left(n\right)\mathrm{}$ to the exponents $D\left(n\right)\mathrm{}$ of a fractal measure.