and by Lemma 3.3 applied to the ball Bz, r
1
, Gz; B
l
z ≥ cGz; B
r
1
z ≥ cGz; B
n
\ η
2
. Finally, by the reverse separation lemma Theorem 4.10,
b
E
P
y
¦ ξ
m
ξ
η
2
∧ ξ
W
∗
|ξ
z
σ
n
©
≥ cbE
P
y
¦ ξ
m
ξ
η
2
|ξ
z
σ
n
© ,
and thus b
E
Y η
2
≥ cbE
1
{η
2
∈K
2
}
P
z
¦ S[1,
σ
n
] ∩ η
2
∪ A = ; ©
≥ cbE
P
z
¦ S[1,
σ
n
] ∩ η
2
∪ A = ; ©
≥ c Gz; B
l
Gz; B
n
f n, mb E
X
y ∈∂ B
n
P
y
¦ ξ
m
ξ
η
2
∧ ξ
W
∗
|ξ
z
σ
n
© P
z
S σ
n
= y
≥ cbE
Gz; B
n
\ η
2
Gz; B
n
f n, m X
y ∈∂ B
n
P
y
¦ ξ
m
ξ
η
2
|ξ
z
σ
n
© P
z
S σ
n
= y
≥ cbE
Gz; B
n
\ η
2
Gz; B
n
X
y ∈∂ B
n
P
y
¦ ξ
z
ξ
η
2
|ξ
z
σ
n
© P
z
S σ
n
= y
.
However, by applying Lemma 3.1 again, Gz; B
n
\ η
2
Gz; B
n
X
y ∈∂ B
n
P
y
¦ ξ
z
ξ
η
2
|ξ
z
σ
n
© P
z
S σ
n
= y =
P
z
¦ S[1,
σ
n
] ∩ η
2
= ; ©
. and thus
b
E
Y η
2
≥ c Esm, n.
5.3 Intersection exponents for SLE
2
and LERW
In this section, we use the convergence of LERW to SLE
2
to show that for 0 r 1, Esr n, n ≍ r
3 4
. We combine this result with the decomposition
Esn ≍ Esrn Esrn, n
from the previous section to obtain that Esn ≈ n
−34
. We recall the notation introduced in Section 2.6. Let Γ denote the set of continuous curves
α : [0, t
α
] → D we allow t
α
to be ∞ such that α0 ∈ ∂ D, α0, t
α
] ⊂ D and αt
α
= 0. We can make Γ into a metric space as follows. If α, β ∈ Γ, we let
d α, β = inf sup
≤t≤t
α
αt − βθt ,
1064
where the infimum is taken over all continuous, increasing bijections θ : [0, t
α
] → [0, t
β
]. Note that d is a pseudo-metric on Γ, and is a metric if we consider two curves to be equivalent if they are the
same up to reparametrization. Recall Theorem 2.6 that LERW converges weakly to SLE
2
on the space Γ, d. We want to apply this result to the functions f
r
defined as follows. Given 0 r 1 and α ∈ Γ, we let
f
r
α = P
W [0, τ
D
] ∩ α[0, ρ
r
] = ; ,
where ρ
r
= inf{t : |αt| = r}. We also define f
r
to be identically 1 for r ≥ 1 think of ρ
r
= 0 in that case, so that the above probability is 1. Recall that Theorem 2.5 states that if
γ is SLE
2
then
E f
r
γ ≍ r
3 4
. Unfortunately, the f
r
are not continuous on the space Γ, d. However, the following lemma shows that they can be approximated by continuous functions.
Lemma 5.4. For all 0 r 1, there exists a function e
f
r
that is uniformly continuous on the space Γ, d such that for all α ∈ Γ
f
r 2
α ≤ ef
r
α ≤ f
2r
α. Proof. We define
e f
r
α = 2
3r Z
2r r
2
f
s
α ds. Note that for a fixed
α, f
s
α is increasing, and therefore f
s
α is integrable. In addition, the second assertion in the statement of the lemma follows immediately. It remains to show that e
f
r
is uniformly continuous.
First of all, we claim that for all ε 0, there exists δ 0 such that for all 0 r 1 and all α, β
with d α, β δ,
f
r
α ≤ f
r+ δ
β + ε. 19
To prove this note that f
r
α − f
r+ δ
β ≤ P W [0,
τ
D
] ∩ β[0, ρ
r+ δ
] 6= ;; W [0, τ
D
] ∩ α[0, ρ
r
] = ; .
By Lemma 3.11, there exists ν 0 depending only on ε such that
P ¦
W [0, τ
D
] ∩ α[0, ρ
r
] = ;; W τ
D
∈ e C
ν
α ©
ε. Furthermore, if d
α, β δ, then for every z ∈ β[0, ρ
r+ δ
], there exists y ∈ α[0, ρ
r
] such that z − y
δ. Therefore, by the Beurling estimates Theorem 2.3, one can make δ small enough so that
P §
W τ
∗ α
− W τ
∗ β
ν ª
ε where
τ
∗ α
= τ
α[0,ρ
r
]
∧ τ
D
and τ
∗ β
= τ
β[0,ρ
r+ δ
]
∧ τ
D
. 1065
Therefore, for such a δ,
P W [0,
τ
D
] ∩ β[0, ρ
r+ δ
] 6= ;; W [0, τ
D
] ∩ α[0, ρ
r
] = ; ≤ P E
1
+ P E
2
+ P E
3
, where
E
1
= {W τ
∗ α
∈ C
ν
α} E
2
= {W [0, τ
D
] ∩ α[0, ρ
r
] = ;; W τ
∗ β
∈ D
1 −ν
} E
3
= {W τ
∗ α
∈ ∂ D \ C
ν
α; W τ
∗ β
∈ A
ν
} recall that A
ν
denotes the annulus D \ D
1 −ν
. By our choice of
ν, P E
1
ε. Provided we take δ ν2, the events E
2
and E
3
are subsets of the event
§ W τ
∗ α
− W τ
∗ β
ν 2
ª ,
and therefore P E
2
and P E
3
can be made less than ε. This proves the claim 19.
Fix 0 r 1. Given ε 0, let δ 0 be such that 19 holds recall that δ depends only on ε and
not on r and suppose that d α, β δ. Then since f
s
β ≤ 1 for all s and β, e
f
r
α − ef
r
β = 2
3r Z
2r r
2
f
s
α ds − 2
3r Z
2r r
2
f
s
β ds ≤
2 3r
Z
2r r
2
f
s+ δ
β ds + ε − 2
3r Z
2r r
2
f
s
β ds ≤
2 3r
δ + δ + ε. The latter can be made arbitrarily small by choosing
δ small enough. By reversing the roles of α and
β, one gets a similar lower bound, proving that e f
r
is uniformly continuous.
Lemma 5.5. There exists C
∞ such that the following holds. Given a random walk S and an independent LERW b
S
n
, we extend them to continuous curves S
t
and b S
n t
by linear interpolation. Then for all 0
r 1, there exists N = N r such that for n ≥ N, 1
C r
3 4
≤ bE
h
P
n S[0,
σ
n
] ∩ η
2 r n,n
b S
n
= ; oi
≤ C r
3 4
. Proof. We’ll prove the upper bound. The lower bound is proved in exactly the same fashion.
Fix 0 r 1. Recall that S
n t
= n
−1
S
n
2
t
. By Proposition 3.12, there exists N
1
such that for n ≥ N
1
, and any realization of b
S
n t
, P
n S[0,
σ
n
] ∩ η
2 r n,n
b S
n
= ; o
= P
n S
n
[0, σ
D
] ∩ n
−1
η
2 r n,n
b S
n
= ; o
≤ P
n W [0,
τ
D
] ∩ n
−1
η
2 r n,n
b S
n
= ; o
+ r
3 4
= f
r
n
−1
b S
n
+ r
3 4
. 1066
By Lemma 5.4, f
r
n
−1
b S
n
≤ ef
2r
n
−1
b S
n
, and e
f
2r
is continuous in the metric Γ, d. Therefore, by the weak convergence of LERW to SLE
2
described at the beginning of this section, there exists N
2
such that for n ≥ N
2
, b
E
e f
2r
n
−1
b S
n
≤ E
e
f
2r
γ
+ r
3 4
where γ denotes SLE
2
. Furthermore, applying first Lemma 5.4, and then Theorem 2.5,
E
e f
2r
γ
≤ E f
4r
γ ≍ r
3 4
. Therefore, the upper bound holds for N = max
{N
1
, N
2
}. The lower bound is proved in the same fashion.
We now prove the analogue of the previous lemma for the case where S and b S
n
are discrete pro- cesses. The reason why the discrete case does not follow immediately from the continuous case is
that we allow random walks that “jump”, and therefore it’s possible for two realizations of S and b S
n
to avoid each other on the lattice Λ but to intersect after they are made continuous curves by linear interpolation.
Theorem 5.6. There exists a constant C such that the following holds. For all 0 r 1, there exists
N = N r such that for n ≥ N,
1 C
r
3 4
≤ Esrn, n ≤ C r
3 4
. Proof. Fix 0
r 1. The lower bound follows immediately from Lemma 5.5 and the fact that if the discrete processes intersect each other so too will the continuous curves.
To prove the upper bound we introduce some notation that will be used only in this proof. Let S[0, . . . ,
σ
n
] denote the discrete set of points in Λ visited by S between S
and S σ
n
. We will write S[0,
σ
n
] to denote the continuous set of points in C visited by the continuous curve S
t
from S to S
σ
n
. We use similar notation for b
S
n
. In addition, we let η
2
= η
2 r n,n
b
S
n
[0, . . . , b
σ
n
]
be the terminal part of the discrete LERW curve and e
η
2
= η
2 r n,n
b
S
n
[0, b
σ
n
]
be the terminal part of the continuous LERW curve. As in the proof of Lemma 3.11, one can choose
δ 0 small enough so that for all n sufficiently large, and for all z
∈ ∂ B
n
, P
S[0, σ
n
] ∩ B
δn
z 6= ; r
3 4
. 1067
Furthermore, given such a δ, we can choose ε 0 and N such that for all n ≥ N, and all z ∈ ∂ B
n
, the following holds. Let y
∈ Λ be the closest point to 1 − εz. Then,
P
y
S[0, σ
n
] ⊂ B
δn
z 1 − r
3 4
. Since the LERW path b
S
n
is a subset of a random walk path, one can combine the previous two observations to show that there exists
ε 0 and N such that for all n ≥ N, b
E
P ¦
S[0, σ
n
] ∩
b S
n
[0, b
σ
n
] ∩ A
εn
6= ;
© 2r
3 4
, where A
εn
denotes the annulus B
n
\ B
1−εn
. By the Beurling estimates Theorem 2.3, if R is the range of S, then for any realization of b
S
n
, P
¦ S[0,
σ
n
] ∩
e η
2
∩ B
1−εn
6= ;; S[0, . . . , σ
n
] ∩ η
2
= ; ©
≤ C R
εn
1 2
. Therefore, we can select N large enough so that for all n
≥ N, Esr n, n =
b
E
P ¦
S[0, . . . , σ
n
] ∩ η
2
= ; ©
= b
E
P ¦
S[0, σ
n
] ∩ e η
2
= ; ©
+ b E
P
¦ S[0,
σ
n
] ∩
e η
2
∩ A
εn
6= ;
© +
b
E
P ¦
S[0, σ
n
] ∩
e η
2
∩ B
1−εn
6= ;; S[0, . . . , σ
n
] ∩ η
2
= ; ©
≤ C r
3 4
.
Theorem 5.7.
Esn ≈ n
−34
. Proof. We prove the upper bound using Proposition 5.2. One gets the lower bound in exactly the
same way using Proposition 5.3. Let δ 0 be given. Let C denote the larger of the constants in
Theorem 5.6 and Proposition 5.2. Let 0 r 14 be small enough so that
ln C ln1
r δ.
By Theorem 5.6 and our choice of r, there exists N such that for n ≥ N,
ln Esr n, n ln 1
r +
3 4
δ. Any n
≥ N can be written uniquely as n = r
− j
s for some j = 0, 1, 2, . . . and N ≤ s r
−1
N . Therefore, by Proposition 5.2,
ln Esn = ln Esr
− j
s ≤ ln
C
j
Ess
j
Y
k=1
Esr
k − j
s, r
k −1− j
s
≤ j ln C + ln Ess + j− 3
4 + δ ln1r.
1068
Thus, ln Esn
ln n ≤
j ln C + ln Ess + j −
3 4
+ δ ln 1r j ln 1
r + ln s ≤
j ln C + ln EsN + j −
3 4
+ δ ln 1r j ln 1
r + ln N .
Therefore, lim sup
n →∞
ln Esn ln n
≤ lim sup
j →∞
j ln C + ln EsN + j −
3 4
+ δ ln 1r j ln 1
r + ln N =
ln C ln 1
r + −
3 4
+ δ ≤ −
3 4
+ 2δ. This proves the upper bound, since
δ was arbitrary. As mentioned before, an identical proof will work for the lower bound.
5.4 Deriving the growth exponent from the intersection exponent
