5 The growth exponent
5.1 Introduction
Recall that W
t
denotes standard complex Brownian motion and γ denotes radial SLE
2
in D started uniformly on
∂ D. In this chapter we will consider random walks and independent LERWs. We will view them as being
defined on different probability spaces so that b
P {.} and bE [.] denote probabilities and expectations
with respect to the LERW, while P
{.} and E [.] will denote probabilities and expectations with respect
to the random walk. For m ≤ n, we define Esm, n, Esn and e
Esn as follows. Esm, n =
b
E
h
P
n S[1,
σ
n
] ∩ η
2 m,n
b S
n
[0, b
σ
n
] = ; oi
, Esn =
b
E
P ¦
S[1, σ
n
] ∩ b S
n
[0, b
σ
n
] = ; ©
= Es0, n, e
Esn = b
E
P ¦
S[1, σ
n
] ∩ b S[0,
b σ
n
] = ; ©
. Esm, n is the probability that a random walk from the origin to
∂ B
n
and the terminal part of an independent LERW from m to n do not intersect. Esn is the probability that a random walk from
the origin to ∂ B
n
and the loop-erasure of an independent random walk from the origin to ∂ B
n
do not intersect. e
Esn is the probability that a random walk from the origin to ∂ B
n
and an infinite LERW from the origin to
∂ B
n
do not intersect. In section 5.2, we prove that for m
n, Esn can be decomposed as Esn
≍ Esm Esm, n. In section 5.3, we use the convergence of LERW to SLE
2
Theorem 2.6 and the intersection expo- nent 3
4 for SLE
2
Theorem 2.5 to show that Esm, n
≍ m
n
3 4
. We then combine these two results to show that Esn
≈ n
−34
. Finally, in section 5.4, we show how the fact that Esn
≈ n
−34
implies that Grn ≈ n
5 4
. Before proceeding, we prove the following lemma which shows that e
Esn and Es4n are on the same order of magnitude.
Lemma 5.1.
e Esn
≍ Es4n. Proof. By Corollary 4.5, it suffices to show that
b
E
P ¦
S[1, σ
n
] ∩ b S
4n
[0, b
σ
n
] = ; ©
≍ bE
P
¦ S[1,
σ
4n
] ∩ b S
4n
[0, b
σ
4n
] = ; ©
. It is clear that the left hand side is greater than or equal to the right hand side. To prove the other
direction, we will use the separation lemma Theorem 4.7. Given a point z ∈ ∂ B
n
, let W z be the half-wedge
W z = {w ∈ Λ : 1 − c
1
n ≤ |w| ≤ 4n, argw − argz
c
1
2}, 1059
where c
1
is as in the statement of the separation lemma. We also let A
n
= {S[1, σ
n
] ∩ b S
4n
[0, b
σ
n
] = ;}, z
= b S
4n
b σ
n
, and
D
n
= n
−1
min {distSσ
n
, b S
4n
[0, b
σ
n
], distz , S[0,
σ
n
]}. By the strong Markov property for random walk,
b
E
P ¦
S[1, σ
4n
] ∩ b S
4n
[0, b
σ
4n
] = ; ©
≥ cbE
1 {b
S
4n
[ b
σ
n
, b
σ
4n
] ⊂ W z }P
A
n
; D
n
≥ c
1
.
By Lemma 2.4 and Corollary 3.7, b
E
1 {b
S
4n
[ b
σ
n
, b
σ
4n
] ⊂ W z }P
A
n
; D
n
≥ c
1
≥ cbE P
A
n
; D
n
≥ c
1
. Finally, by the separation lemma,
b
E P
A
n
; D
n
≥ c
1
≥ cbE P A
n
, and therefore,
b
E
P ¦
S[1, σ
4n
] ∩ b S
4n
[0, b
σ
4n
] = ; ©
≥ cbE
P
¦ S[1,
σ
n
] ∩ b S
4n
[0, b
σ
n
] = ; ©
.
5.2 Proof that Esn
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