Self Avoiding Surfaces in the 3D Ising Model

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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 Tc, the number of surfaces of genus g and area A behaves as Aˣ⁽ᵍ⁾e⁻μ⁽ᵍ⁾ᵃ, with x approximately linear in g and μ 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 Tc decompose into a distribution N(l) of loops of length l that scales as l⁻τ with τ 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.