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We consider the multi-parameter random simplicial complex as a higherdimensional extension of the classical Erd\"os-R\'enyi graph. We investigateappearance of "unusual" topological structures in the complex from the point ofview of large deviations. We first study upper tail large deviationprobabilities for subcomplex counts, deriving the order of magnitude of suchprobabilities at the logarithmic scale precision. The obtained results are thenapplied to analyze large deviations for the number of simplices at the criticaldimension and below. Finally, these results are also used to deduce largedeviation estimates for Betti numbers of the complex in the critical dimension.

Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes