Mathew Penrose

University of Bath University of Bath

Coverage and connectivity in stochastic geometry

Probability Seminar
17th February 2023, 3:30 pm – 4:30 pm
Fry Building, 2.04

Consider a random uniform sample of size n over a bounded region A in R^d, d ≥ 2, having a smooth boundary. The coverage threshold T_n is the smallest r such that the union Z of Euclidean balls of radius r centred on the sample points covers A. The connectivity threshold K_n is twice the smallest r required for Z to be connected. These thresholds are random variables determined by the sample, and are of interest, for example, in wireless communications, set estimation, and topological data analysis.

We discuss new results on the large-n limiting distributions of T_n and K_n. When A has unit volume, with v denoting the volume of the unit ball in R^d and |dA| the perimiter of A, these take the form of weak convergence of nv(T_n)^d− (2 − 2/d) log n − (a_d) log(log n) to a Gumbel-type random variable with cumulative distribution function
F(x) = exp(−(b_d)e^(−x) − (c_d)|dA|e^(−x/2)),
for suitable constants a_d, c_d with b_2 = 1, b_d = 0 for d > 2. The corresponding result for K_n takes the same form with different constants a_d, c_d.

If time permits, we may also discuss extensions and related results concerning
(i) taking A to be a polytope rather than having a smooth boundary;
(ii) taking A to be a d-dimensional manifold with boundary embedded in a higher-dimensional Euclidean space;
(iii) strong laws of large numbers for T_n and K_n for non-uniform random samples of points.

Some of the work described here is joint work with Xiaochuan Yang.

Organisers: Jessica Jay, Elnur Emrah