Counting regions with bounded surface area (Q2384759)

An \(r\)-dimensional cubical complex is a finite union of \(r\)-cubes in \(\mathbb R^d\) , where \(r\)-cube is an \(r\)-dimensional unit cube in \(\mathbb R^d\) with vertices in \(\mathbb Z^d.\) The authors prove that there are between \((C_3d)^{n/d}\) and \((C_4d)^{2n/d}\) complexes containing a fixed cube with connected boundary of \((d-1)\)-volume \(n.\) A similar result with bonds between \((C_1d)^{n/2d}\) and \((C_2d)^{64n/d}\) was proved by \textit{J. L. Lebowitz} and \textit{A. E. Mazel} [J. Stat. Phys. 90, No.~3--4, 1051--1059 (1998; Zbl 0921.60084)].