Problems of Information Transmission, Vol. 41, No. 4, 2005, pp. 319–330. Translated from Problemy Peredachi Informatsii, No. 4, 2005, pp. 23–35.
Original Russian Text Copyright
2005 by Blinovsky.
Random Sphere Packing
V. M. Blinovsky
Institute for Information Transmission Problems, RAS, Moscow
University of Bielefeld, Germany
Received September 26, 2003; in ﬁnal form, June 15, 2005
Abstract—We study probabilistic characteristics of random packings of a Euclidean space.
The paper was initiated by , where typical properties of random lattices and random packings
of a Euclidean space were studied. We use other (simpler and more precise) ways to obtain
estimates on parameters that characterize random packings and consider the possibility of
extending the results to L-packings.
A packing of a Euclidean space R
is a locally ﬁnite set in R
. Each packing σ ⊂ R
characterized by its minimum (inﬁmum) distance
x − x
and by its density, to be deﬁned below. In fact, we speak about a locally ﬁnite set σ ∈ R
are interested in these two parameters of σ. It is clear that, for a given packing σ with d(σ) =0
(below we consider such packings only), open balls of radius
centered at points of σ are disjoint.
is called the packing radius.
For an arbitrary locally compact set σ ⊂ R
and a cube K
with the center 0 ∈ R
of length 2N parallel to coordinate axes, we deﬁne σ
= σ ∩ K
The density ∆
(σ)ofapackingσ of K
is deﬁned as
(x, r) is a ball of radius r centered at x. We deﬁne the density ∆(σ)ofasetσ as
∆(σ) = lim sup
d(σ(d)) = d.
= − sup
Note that, by homogeneity, the quantity δ
does not depend on d, so that the last deﬁnition is
correct. In what follows, we set d =1.
2005 Pleiades Publishing, Inc.