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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 T c , the number of surfaces of genus g and area A behaves as A x(g) e −μ(g)A , with x approximately linear in g and μ constant. 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 -τ with τ ∼ 2.2. We address the prospects for a string—theoretic description of cluster boundaries.

The Phenomenology of Strings and Clusters in the 3-d Ising Model