Abstract
We consider long-range percolation on
Z
d
, where the probability that two vertices at distance
r
are connected by an edge is given by
p(r)=1−exp[−λ(r)]∈(0,1)
and the presence or absence of different edges are independent. Here,
λ(r)
is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the
k
-ball around the origin,
|
B
k
|
, that is, the number of vertices that are within graph-distance
k
of the origin, for
k→∞
, for different
λ(r)
. We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying
λ(r)
exist, for which, respectively:
∙
|
B
k
|
1/k
→∞
almost surely;
∙
there exist
1<
a
1
<
a
2
<∞
such that
lim
k→∞
P(
a
1
<|
B
k
|
1/k
<
a
2
)=1
;
∙
|
B
k
|
1/k
→1
almost surely. This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number,
R
0
, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.