Self-avoiding surfaces in the 3d Ising model Original Research Article

We examine the geometrical and topological properties of surfaces surrounding clusters in the 3d 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 3d 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 2d self-avoiding loops that we compare with analytic predictions. We address the prospects for a string-theoretic description of cluster boundaries.