Mean and Variance of Vacancy for Hard‐Core Disc Processes and Applications

. Hard‐core Strauss disc processes with inhibition distance r and disc radius R are considered. The points in the Strauss point process are thought of as trees and the discs as crowns. Formulas for the mean and the variance of the vacancy (non‐covered area) are derived. This is done both for the case of a fixed number of points and for the case of a random number of points. For tractability, the region is assumed to be a torus or, in one dimension, a circle in which case the discs are segments. In the one‐dimensional case, the formulas are exact for all r. This case, although less important in practice than the two‐dimensional case, has provided a lot of inspiration. In the two‐dimensional case, the formulas are only approximate but rather accurate for r