Abstract
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.
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Communicated by J.L. Lebowitz
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Balister, P.N., Bollobás, B. Counting Regions with Bounded Surface Area. Commun. Math. Phys. 273, 305–315 (2007). https://doi.org/10.1007/s00220-007-0231-5
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DOI: https://doi.org/10.1007/s00220-007-0231-5