Focusing of the scan statistic and geometric clique number

Abstract

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

Random channel assignment in the plane

“…See [29] for the behavior of (G n ) in this case, and for further related results. Also, see [28] for very precise results on scan statistics with a fixed scanning set.…”

Two-point concentration in random geometric graphs

“…We remark that our notion of a clustering rule differs from the notion used in [13], so that we can accommodate a wider variety of random variables. and h n (A) = 0 otherwise.…”

“…He also conjectured that the same holds for the clique number ω(G n ). In the paper [13] he had already shown this to be true under the stronger assumption that nr d is bounded. We remark here that it is natural to state our results in terms of the quantity nr d , because this is a good measure of the average degree of G n (see appendix A for a precise result).…”

“…We remark that often a sequence of (shrinking) sets (W n ) n is considered, but we have chosen to formulate things in terms of a sequence r(n) and a fixed set W because this fits better with the presentation. Månsson ([9]) noticed that the scan statistic becomes two-point concentrated when (translated into our r(n), fixed W setting) nr d is bounded by n − for some > 0, and this result was later extended by Penrose whose results in [13] show that two-point concentration also holds under the weaker condition that nr d is bounded (he also imposes some regularity conditions on the probability density function of ν). In this paper we will prove that two-point concentration occurs when nr d = o(ln n) for a range of random variables including the scan statistic and the clique number ω(G n ) and chromatic number χ(G n ) of the random geometric graph.…”

Localization in random geometric graphs with too many edges

“…In our context, the natural combinatorial structure for creating many edges with a small number of edges is a "giant clique". The clique number, the (typical) size of the largest clique of the random geometric graph, falls under the general class of scan statistics, and has been shown to focus on two values with high probability for certain values of r (see [31], [29]); however, these works do not explore the large deviation regime. Our work uses techniques from large deviations, concentration inequalities, convex analysis, and geometric measure theory.…”