UVictoria Probability and Dynamics Seminar: Mathew Penrose

Consider a random uniform sample of size $n$ over a bounded region $A$ in $R^d$, $d \geq 2$, 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 recent 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 $ n v 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) Other domains $A$ such as polytopes or manifolds; (ii) coverage by balls of random radii; (iii) strong laws of large numbers for $T_n$ and $K_n$ for non-uniform random samples of points. Some of the work mentioned here is joint work with Xiaochuan Yang and Frankie Higgs.