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#### Abstract

Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in R^n, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and non-singular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we study general properties of the discrete gradient and obtain precise expressions for the expected numbers in low dimensions. In particular, we get the expected numbers of simplices in the Poisson--Delaunay mosaic in dimension n <= 4.