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Exceptional configurations in a high-intensity random covering
Tuesday 4 November 2025, 14:00
Salle des séminaires
Audrey Chaudron
LMRS
The Boolean model of random covering was introduced by Gilbert in the 1960s as a simplified representation of a radio transmission network [2]. It is obtained by considering the union of balls of fixed radius centered at the points of a homogeneous Poisson point process in Euclidean space.
In this setting, we study the probability that a uniformly chosen connected component is exceptionally small, that is, composed of exactly k balls for some fixed integer k, in the regime where the intensity of the process, i.e. the mean number of points per unit volume, tends to infinity. In particular, we derive a two-term asymptotic expansion for this probability.
To achieve this, the talk will provide an introduction to the Boolean model in R^d with a special focus on recent work by Penrose and Yang [3]. They established an integral equivalent for the probability of interest, thereby improving upon the logarithmic equivalent obtained by Alexander in 1992 [1]. Our geometric interpretation of this integral equivalent leads to a new method of computation and yields the two-term expansion. Finally, if time allows, we will discuss the case of k tends to infinity and possible extensions of these results in the hyperbolic setting.
[1] K. S. Alexander. Finite clusters in high-density continuous percolation: compression and sphericality. Probab. Theory Related Fields, 97(1-2), 35–63, 1993.
[2] E. N. Gilbert. Random plane networks. J. Soc. Indust. Appl. Math., 9, 533–543, 1961.
[3] M. D. Penrose, X. Yang. On k-clusters of high-intensity random geometric graphs, 2022
