Self Avoiding Surfaces in the 3D Ising Model

We examine the geometrical and topological properties of surfaces surrounding clusters in the 3--$d$ Ising model. For geometrical clusters at the percolation temperature and Fortuin--Kasteleyn clusters at $T_c$, the number of surfaces of genus $g$ and area $A$ behaves as $A^{x(g)}e^{-\mu(g)A}$, with $x$ approximately linear in $g$ and $\mu$ constant. These scaling laws are the same as those we obtain for simulations of 3--$d$ bond percolation. We observe that cross--sections of spin domain boundaries at $T_c$ decompose into a distribution $N(l)$ of loops of length $l$ that scales as $l^{-\tau}$ with $\tau \sim 2.2$. We also present some new numerical results for 2--$d$ self-avoiding loops that we compare with analytic predictions. We address the prospects for a string--theoretic description of cluster boundaries.