In this talk we focus on the distance function from a random set of points P in the Euclidean space. The distance function is continuous, however, it is not everywhere differentiable. Nevertheless, one can accurately define critical points and then apply Morse theory to it.We study the number of critical points in small neighborhoods around P.

Specifically, we are interested in the case where the number of points in P goes to infinity, and the size of the neighborhoods goes to zero. We present limit theorems for the number of critical points and show that it exhibits a phase transition, depending on how fast the size of the neighborhoods goes to zero. A similar phase transition was presented recently by Kahle & Meckes who studied the Betti-numbers of random geometric complexes.

We show that this is more than just a coincidence, and discuss the connection between the distance function and geometric complexes.