Counting regions with bounded surface area


Define a cubical complex to be a collection of integer-aligned unit cubes in d dimensions. Lebowitz and Mazel (1998) proved that there are between (C_1d)n/2d and (C_2d)64n/d complexes containing a fixed cube with connected boundary of (d - 1)-volume n. In this paper we narrow these bounds to between (C_3d)n/d and (C_4d)2n/d . We also show that there are n{n/(2d(d-1))+o(1) connected complexes containing a fixed cube with (not necessarily connected) boundary of volume n. © Springer-Verlag 2007.

Publication Title

Communications in Mathematical Physics