For l ≥ 2, B
l
is a event which involves the random variables u
k
, v
k
, V
k
for k = τ
l−1
+ 1, . . . , τ
l
. Using the Markov property, we have that the probability P B
l
| ∩
l−1 j=1
B
j
depends only on
u
τ
l−1
, v
τ
l−1
, V
τ
l−1
. Furthermore, on the set ∩
l−1 j=1
B
j
, we note that n
2
l−1
1−ε
≤ ku
τ
l−1
− v
τ
l−1
k
1
≤ n
2
l−1
1+ε
, 0 ≤ u
τ
l−1
4 − v
τ
l−1
4 log n
2
l−1
and V
τ
l−1
= ;. Therefore we have that, for l ≥ 2, P
B
l
| ∩
l−1 j=1
B
j
≥ inf
n
2l−11− ε
≤kz
1
−z
2
k
1
≤n
2l−11+ ε
, 0≤
z
1
4−z
2
4log n
P B
l
| u
τ
l−1
, v
τ
l−1
, V
τ
l−1
= z
1
, z
2
, ; =
inf
n
2l−11− ε
≤kz
1
−z
2
k
1
≤n
2l−11+ ε
, 0≤
z
1
4−z
2
4log n
P A
n
2l−1
, ε
| u
, v
, V
= z
1
, z
2
, ; ≥ 1 − C
1
n
−2
l−1
β
33 and since n
1− ε
≤ kuk
1
≤ n
1+ ε
, 0 ≤ u4
log n. P
B
1
= P A
n, ε
| u ,
v , V
= u, 0, ;
≥ 1 − C
1
n
−β
34 Therefore, from 32, 33 and 34, we have,
P {G is disconnected} ≥ P
u, 0 are both open × lim
i→∞ i
Y
l=1
1 − C
1
n
−2
l−1
β
0. This completes the proof of the claim.
We will work towards the proof of Lemma 3.1. Towards that, we introduce an independent version of the above process. In the same probability space, starting with two vertices
u and v, and the same
set of uniformly distributed random variables, define u
I
= max{u ,
v } and v
I
= min{u ,
v }.
As in the construction at the beginning of the Section, we define u
I n+1
= max{R
I
u
I n
, v
I n
} and
v
I n+1
= min{R
I
u
I n
, v
I n
} where R
I
is similar to R defined in the earlier construction except that the history part is completely ignored.
The independent version tracks the two trees, emanating from the vertices u and v, with the condi-
tion that the trees do not depend on the information history carried. The only constraint is that while growing the tree from a vertex, it waits for the tree from the other vertex to catch up, before
taking the next step. Note that if the history set is empty, then both constructions match exactly.
We define an event similar to A
n, ε
but in terms of { u
I k
, v
I k
: 1 ≤ k ≤ n
4
}. Fix n ≥ 1, 0 ε 13 and two open vertices
u, v ∈ Z and define the event,
B
n, ε
u
I
, v
I
:=
ku
I k
− v
I k
k
1
≥ logn
2
for 1 ≤ k ≤ n
4
− 1, 0 ≤
u
I k
4 − v
I k
4 logn
2
for 1 ≤ k ≤ n
4
, n
21− ε
≤ ku
I n
4
− v
I n
4
k
1
≤ n
21+ ε
.
We will show that the following Lemma holds:
Lemma 3.2. For 0
ε 13 there exist constants C
2
, γ 0 and n
≥ 1 such that, for all n ≥ n ,
inf
n
1− ε
≤ku−vk
1
≤n
1+ ε
, 0≤
u4−v4 log n
P B
n, ε
u, v
≥ 1 − C
2
n
−γ
.
2180
First we prove Lemma 3.1, assuming Lemma 3.2.
Proof of Lemma 3.1: Given 0 ε 13, fix n
≥ 1 from Lemma 3.2. Now fix n ≥ n and
u, v ∈ Z
