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On the Capacity Functional of the Infinite Cluster of a Boolean Model

Guenter Last ; Mathew D. Penrose ; Sergei Zuyev (Institutionen för matematiska vetenskaper, Analys och sannolikhetsteori)
ANNALS OF APPLIED PROBABILITY (1050-5164). Vol. 27 (2017), 3, p. 1678-1701.
[Artikel, övrig vetenskaplig]

Consider a Boolean model in R-d with balls of random, bounded radii with distribution F-0, centered at the points of a Poisson process of intensity t > 0. The capacity functional of the infinite cluster Z(infinity) is given by theta(L) (t) = P{Z(infinity) boolean AND L not equal empty set L not equal phi}, defined for each compact L subset of R-d. We prove for any fixed L and F-0 that theta(L) (t) is infinitely differentiable in t, except at the critical value t(c); we give a Margulis-Russo-type formula for the derivatives. More generally, allowing the distribution F-0 to vary and viewing theta(L), as a function of the measure F := t F-0, we show that it is infinitely differentiable in all directions with respect to the measure F in the supercritical region of the cone of positive measures on a bounded interval. We also prove that theta(L) (.) grows at least linearly at the critical value. This implies that the critical exponent known as beta is at most 1 (if it exists) for this model. Along the way, we extend a result of Tanemura [J. AppL Probab. 30 (1993) 382-396], on regularity of the supercritical Boolean model in d >= 3 with fixed-radius balls, to the case with bounded random radii.

Nyckelord: continuum percolation, supercritical phase, poisson processes



Denna post skapades 2017-08-29. Senast ändrad 2017-08-30.
CPL Pubid: 251481

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, Analys och sannolikhetsteoriInstitutionen för matematiska vetenskaper, Analys och sannolikhetsteori (GU)

Ämnesområden

Sannolikhetsteori och statistik

Chalmers infrastruktur