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In this paper we discuss two general models of random simplicial complexeswhich we call the lower and the upper models. We show that these models aredual to each other with respect to combinatorial Alexander duality. Thebehaviour of the Betti numbers in the lower model is characterised by thenotion of critical dimension, which was introduced by A. Costa and M. Farber:random simplicial complexes in the lower model are homologically approximatedby a wedge of spheres of dimension equal the critical dimension. In this paperwe study the Betti numbers in the upper model and introduce new notions ofcritical dimension and spread. We prove that (under certain conditions) anupper random simplicial complex is homologically approximated by a wedge ofspheres of the critical dimension.

Random Simplicial Complexes, Duality and The Critical Dimension