Critical and topological properties of cluster boundaries in the 3D Ising model

We analyze the ensemble of surfaces surrounding critical clusters at T=${\mathit{T}}_{\mathit{c}}$ in the 3D Ising model. We find that ${\mathit{N}}_{\mathit{g}}$(A), the number of surfaces of genus g and area A, behaves as ${\mathit{A}}^{\mathit{x}\left(\mathit{g}\right)}$${\mathit{e}}^{\mathrm{-}\mathrm{\mu }\mathit{A}}$. We show that μ is constant and x(g) is approximately linear; the sum ${\mathit{tsum}}_{\mathit{g}}$ ${\mathrm{N}}_{\mathit{g}}$(A) scales as a power of A. The cluster volume is proportional to its surface area. We discuss similar reuslts for the ordinary spin clusters of the 3D Ising model and for 3D bond percolation.