Consider one of our percolation models G
∗
n, d
π
, and construct it using explosions and an inter- mediate random graph G
∗
˜ n, ˜
d as described in the introduction. Recall that ˜ d is random, while d
and the limiting probabilities p
j
and ˜ p
j
are not. Let C
j
:= C
j
G
∗
n, d
π
and ˜ C
j
:= C
j
G
∗
˜ n, ˜
d
denote the components of G
∗
n, d
π
, and G
∗
˜ n, ˜
d, respectively.
As remarked in Section 2, we may assume that G
∗
˜ n, ˜
d too satisfies Condition 2.1, with p
j
replaced by ˜
p
j
. At least a.s.; recall that ˜ d is random. Hence, assuming ˜
p
1
0, if we first condition on ˜
d, then Theorem 3.1 applies immediately to the exploded graph G
∗
˜ n, ˜
d. We also have to remove
n
+
randomly chosen “red” vertices of degree 1, but luckily this will not break up any component. Consequently, if E ˜
D ˜ D
− 2 0, then G
∗
˜ n, ˜
d w.h.p. has a giant component ˜ C
1
, with v ˜ C
1
, v
j
˜ C
1
and e ˜ C
1
given by Theorem 3.1 with p
j
replaced by ˜ p
j
, and after removing the red vertices, the remainder of ˜
C
1
is still connected and forms a component C in G
∗
n, d
π
. Furthermore, since E
˜ D ˜
D − 2 0, ˜p
j
0 for at least one j 2, and it follows by 3.2 that ˜ C
1
contains cn + o
p
n vertices of degree j, for some c 0; all these belong to C although possibly with smaller degrees, so
C contains w.h.p. at least cn2 vertices. Moreover, all other components of G
∗
n, d
π
are contained in components of G
∗
˜ n, ˜
d different from ˜ C
1
, and thus at most as large as ˜ C
2
, which by Theorem 3.1 has o
p
˜ n = o
p
n vertices. Hence, w.h.p. C is the largest component C
1
of G
∗
n, d
π
, and this is the unique giant component in G
∗
n, d
π
. Since we remove a fraction n
+
˜ n
1
of all vertices of degree 1, we remove by the law of large numbers for a hypergeometric distribution about the same fraction of the vertices of degree 1 in the giant
component ˜ C
1
. More precisely, by 3.2, ˜ C
1
contains about a fraction 1 − ξ of all vertices of degree
1, where g
′ ˜
D
ξ = ˜ λξ; hence the number of red vertices removed from ˜
C
1
is 1 − ξn
+
+ o
p
n. 3.4
By 3.1 and 3.4, v
C
1
= v ˜ C
1
− 1 − ξn
+
+ o
p
n = ˜ n 1
− g
˜ D
ξ − n
+
+ n
+
ξ + o
p
n. 3.5
Similarly, by 3.3 and 3.4, since each red vertex that is removed from C
1
also removes one edge with it,
e C
1
= e ˜ C
1
− 1 − ξn
+
+ o
p
n =
1 2
˜ λ˜
n1 − ξ
2
− 1 − ξn
+
+ o
p
n. 3.6
The case E ˜ D ˜
D − 2 ≤ 0 is even simpler; since the largest component C
1
is contained in some component ˜
C
j
of G
∗
˜ n, ˜
d, it follows that v C
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