 
Summary: Submitted to the Annals of Applied Probability
DISTANCE FUNCTIONS, CRITICAL POINTS, AND
TOPOLOGY FOR SOME RANDOM COMPLEXES
By Omer Bobrowski and Robert J. Adler
Technion
For a finite set of points P in Rd
, the function dP : Rd
R+
measures Euclidean distance to the set P. We study the number of
critical points of dP when P is random. In particular, we study the
limit behavior of Nk the number of critical points of dP with Morse
index k as the number of points in P goes to infinity. We present
explicit computations for the normalized, limiting, expectations and
variances of the Nk, as well as distributional limit theorems. We link
these results to recent results in [10, 11] in which the Betti numbers
of the random Cech complex based on P were studied.
1. Introduction. For a finite set P of points in Rd, of size P, let
dP : Rd R+ be the distance function for P, so that
dP (x) := min
pP
