Convergence of the driving functions

4.3 Convergence of LERW to HRLC 2

4.3.1 Convergence of the driving functions

For a < b, letA_a,b be the annulus bounded by C_a and C_b. For any 0< q < p, there is a smallest l(p, q) ∈ (0, p) such that if K is a hull in A_p on C₀ with the capacity (w.r.t. C_p) less than q, thenK does not intersect C_l(p,q). Using the fact that for any 0< s≤r, ReS_r attains its unique maximum and minimum on A_s,r ate^−s and −e^−s, respectively, it is not hard to derive the following Lemma.

Lemma 4.3.1 Fix 0 < q < p, let r ∈ (l(p, q), p). There are ι ∈ (0,1/2) and M > 0 depending onp,q andr, which satisfy the following properties. Supposeϕ_t,0≤t < p,

are some modulus p standard annulus LE maps driven by ξ on [0, p). Then we have

|∂_zS_p−t(ϕ_t(z)/e^iξ(t))| ≤M, for all t ∈[0, q] and z ∈A_r,p. Moreover, Aι(p−t),p−t⊃ϕt(Ar,p)⊃A(1−ι)(p−t),p−t, ∀t∈[0, q].

We may lift the ϕ_t to the covering space, and find a conformal mapϕe_t such that eⁱ◦ϕe_t=ϕ_t◦eⁱ,ϕe₀ is an identity map, and ϕe_t is continuous in t. Then we have

∂_tϕe_t(z) =Se_p−t(ϕ_t(z)−ξ(t)),

where eS_r(z) = ¹_iS_r(e^iz). If we let Ae_a,b := (eⁱ)⁻¹(A_a,b), then with the assumption of the above lemma, we have

Ae_{ι(p−t),p−t}⊃ϕe_t(Ae_r,p)⊃Ae(1−ι)(p−t),p−t, ∀t ∈[0, q].

It is clear that Ser has period 2π, is meromorphic in C with poles {2kπ+i2mr : k, m ∈ Z}, ImSe_r ≡ 0 on R\ {poles}, and ImSe_r ≡ −1 on Ce_r := ri+R. It is also easy to check that eSr is an odd function, and the principal part ofSer at 0 is 2/z. So Se_r(z) = 2/z+az+O(z³) near 0, for some a∈R. It is possible to explicit this kernel using classical functions in [2]:

Se_r(z) = 2ζ(z)− 2

πζ(π)z= 1 π

∂_vθ θ ( z

2π,ir π),

whereζ is the Weierstrass zeta function with basic periods (2π, i2r), andθ =θ(v, τ) is Jacobi’s theta function. The following lemma is a direct consequence of the heat-type differential equation satisfied by θ: (∂_v² −4iπ∂_τ)θ = 0. The symbols ⁰ and ⁰⁰ in the lemma denote the first and second derivatives w.r.t. z.

Lemma 4.3.2 ∂_reS_r−eS_reS⁰_r−eS⁰⁰_r ≡0.

Let K_t^δ = β^δ(0, t], for 0 ≤ t < p. Suppose W maps D conformally onto Ap so that W(0₊) = 1. Then t 7→ W(K_t^δ) is a Loewner chain in A_p on C₀ such that CAp,Cp(W(K_t^δ)) =t. By Proposition 2.4.1, (W(K_t^δ),0≤t < p) is a family of modulus pstandard annulus LE hulls driven by some real continuous functionξ_t^δ on [0, p) with ξ^δ(0) = 0. Let ϕ^δ_t be the corresponding LE maps. We want to prove that as δ →0, the law ofξ^δconverges to the law of√

2B, whereB(t) is a standard Brownian motion.

Define

E₋₁^δ =∂_VD^δ∩I₁, F^δ =∂_VD^δ∩I₂, N₋₁^δ =V(D^δ)∩D,

and

E_k^δ =E₋₁^δ ∪ {X^δ(0), . . . , X^δ(k)}, N_k^δ =N₋₁^δ \ {X^δ(0), . . . , X^δ(k)},

for 0 ≤k < υ. Let f_k be the f in Lemma 4.2.1 with G =D^δ, A =F^δ and B =E_k^δ; let gk be the g in Lemma 4.2.3 with G = D^δ, A = F^δ, B = E_k−1^δ , and x = X^δ(k), for 0 ≤k < υ. Note that one ofI₁ and I₂ must be bounded, so one of F^δ and E_k−1^δ must be finite, which implies L(E_k−1^δ , F^δ)<∞. And since X^δ(k+·) is a path on D^δ from X^δ(k) to F^δ without passing through E_k−1^δ , we have f_k−1(X^δ(k))> 0, so g_k is well defined. From Theorem 4.2.1, (gk) is a {Fk} martingale, where Fk denotes the σ-algebra generated byX^δ(0), X^δ(1), . . ., X^δ(k∧υ).

Now fix q0 ∈ (0, p). Let q1 = (q0 + p)/2. Choose p1 ∈ (l(p, q1), p), and let p₂ = (p₁ +p)/2. Denote α_j = W⁻¹(C_pj), j = 1,2. Then α₁ and α₂ are disjoint Jordan curves in D such that αj disconnects α3−j fromIj, j = 1,2. For j = 1,2, let U_j be the subdomain ofD bounded byα_j and I_j, and V_j^δ =V(D^δ)∩U_j. Let L^δ be the set of simple lattice pathswonD^δof finite length such thatw(0)∈I1,w(n)∈V₁^δ for all n >0, and there is a path on D^δ from the last vertex P(w) of w toI₂ without passing through I1 or other vertices ofw. For w∈L^δ of length k, denote

E_w^δ =E₋₁^δ ∪ {w(0), . . . , w(k)}, and N_w^δ =N₋₁^δ \ {w(0), . . . , w(k)}.

Let gw be the g in Lemma 4.2.3 with G = D^δ, A = F^δ, B = E_w^δ \ {P(w)}, and x = P(w) = w(k). Now define D_w = D\ ∪^k_j=1[w(j −1), w(j)]. Let u_w be the non- negative harmonic function inDw whose continuation is constant 1 onI2, constant 0 on∪^k_j=0[w(j−1), w(j)]∪I₁ except at P(w), and R

I2∂_nu_w(z)ds(z) = 0. It is intuitive to guess that gw should be close to uw. In fact, we have the following proposition.

The proof is postponed to the next section.

Proposition 4.3.1 Given any ε >0, there is δ(ε)>0 such that if 0< δ < δ(ε) and w∈L^δ, then |gw(v)−uw(v)|< ε, for any v ∈V₂^δ.

Let n∞=dS(q0)e, where dxe is the smallest integer that is not less thanx. Then n_∞ is a {F_k} stopping time. For 0 ≤ k ≤ n_∞− 1, T(k) ≤ q₀ < q₁, so from the choice of p1, we see that W(X^δ(k)) lies in the domain bounded by Cp1 and C0, so X^δ(k) lies in the domain bounded by I₁ and α₁. Note that X^δ(−1) = 0∈ I₁. So for

−1 ≤ k ≤ n∞−1, if δ is small, then [X^δ(k), X^δ(k + 1)] can be disconnected from I₂ by an annulus centered at X^δ(k) with inner radius δ and outer radius dist(α₁, I₂).

So as δ → 0, the conjugate extremal distance between I2 and [X^δ(k), X^δ(k+ 1)] in D\ ∪_0≤j≤k[X^δ(j −1), X^δ(j)] tends to 0, uniformly in −1 ≤ k ≤ n_∞−1. It follows that T(k+ 1)−T(k) and max{|ξ^δ(t)−ξ^δ(T(k))| : T(k) ≤ t ≤ T(k+ 1)} tend to 0 as δ → 0, uniformly in −1 ≤ k ≤ n_∞ −1. Since T(n_∞ −1) ≤ q₀, we may choose δ small enough such that T(n∞) < q1. Since p1 ∈ (l(p, q1), p), and α1 = W⁻¹(Cp1), so X^δ[−1, k]∩α₁ = ∅ for 0 ≤ k ≤ n_∞. So for 0 ≤ k ≤ n_∞, w_k := X^δ(· −1)|_Nk+1 is contained in L^δ, andgwk =gk. Since ϕ^δ_T(k)◦W maps (Dwk, P(wk)) conformally onto (A_p−T(k), e^iξδ(T(k))), so

uwk(z) = Sp−T(k)(ϕ^δ_T(k)◦W(z)/e^iξδ(T(k))).

Now fix d >0. Define a non-decreasing sequence (n_j)_j≥0 inductively. Let n₀ = 0.

Let n_j+1 be the first integer n ≥ n_j such that T(n)−T(n_j) ≥ d², or |ξ^δ(T(n))− ξ^δ(T(n_j))| ≥d, orn =n_∞, whichever comes first. Thenn_j’s are stopping times w.r.t.

{F_k}, and they are bounded above byn_∞. If we let δ be smaller than some constant

depending on d, then T(nj+1)−T(nj)≤ 2d² and |ξ^δ(T(s))−ξ^δ(T(nj))| ≤2d for all s ∈[n_j, n_j+1] and j ≥0. LetF_j⁰ =F_nj. Then for any v ∈ V₂^δ, (g_nj(v),0≤j <∞) is an{F_j⁰} martingale. By Proposition 4.3.1 for anyz ∈W(V₂^δ) and 0≤j ≤k,

E[ReS_p−T(nk)(ϕ^δ_T(nk)(z)/e^iξδ(T(nk)))|F_j⁰] = ReS_p−T(nj)(ϕ^δ_T(nj)(z)/e^iξ(T(nj))) +o_δ(1).

Asδtends to 0, the setW(V₂^δ) tends to be dense inAp2,p. So for anyz ∈Ap2,p, there is somez₀ ∈W(V₂^δ) such that |z−z₀|=o_δ(1). Note thatT(n_j)≤T(n_k)≤T(n_∞)≤q₁ for 0 ≤j ≤k. Using the boundedness of the derivative in Lemma 4.3.1 with q =q1

and r =p₂, we then have that for all z ∈A_p2,p,

E[ReS_p−T(nk)(ϕ^δ_T(nk)(z)/e^iξδ(T(nk)))|F_j⁰] = ReS_p−T(nj)(ϕ^δ_T(nj)(z)/e^iξδ(T(nj))) +o_δ(1).

Then we have for all z ∈Ae_p2,p,

E[ImSe_p−T(nk)(ϕe^δ_T(nk)(z)−ξ^δ(T(n_k))|F_j⁰] = ImSe_p−T(nj)(ϕe^δ_T(nj)(z)−ξ^δ(T(n_j))) +o_δ(1).

(4.3.1) In Lemma 4.3.1, let q=q₁ and r=p₂, then we have some ι∈(0,1/2) such that

Ae_{ι(p−t),p−t}⊃ϕe^δ_t(Aep2,p)⊃Ae(1−ι)(p−t),p−t, (4.3.2)

for 0≤t≤q₁.

Proposition 4.3.2 There are an absolute constant C > 0 and a constant δ(d) > 0 such that if δ < δ(d), then for all j ≥0,

|E[ξ^δ(T(n_j+1))−ξ^δ(T(n_j))|F_j⁰]| ≤Cd³, and

|E[(ξ^δ(T(n_j+1))−ξ^δ(T(n_j)))²/2−(T(n_j+1)−T(n_j))|F_j⁰]| ≤Cd³.

Proof. Fix somej ≥0. Let a=T(nj) and b=T(nj+1). Then 0≤ a≤b≤ q1. And if δ is less than some δ₁(d), we have |b−a| ≤ 2d² and |ξ^δ(c)−ξ^δ(a)| ≤ 2d, for any c∈[a, b]. Now supposez ∈Aep2,p, and consider

F :=Se_p−b(ϕe^δ_b(z)−ξ^δ(b))−eS_p−a(ϕe^δ_a(z)−ξ^δ(a)).

Then F =F1+F2, where

F₁ :=Se_p−b(ϕe^δ_b(z)−ξ^δ(b))−eS_p−b(ϕe^δ_a(z)−ξ^δ(a)),

F₂ :=Se_p−b(ϕe^δ_a(z)−ξ^δ(a))−Se_p−a(ϕe^δ_a(z)−ξ^δ(a)).

Then for some c₁ ∈[a, b],F₁ =F₃+F₄+F₅, where

F₃ :=Se⁰_p−b(ϕe^δ_a(z)−ξ^δ(a))[(ϕe^δ_b(z)−ϕe^δ_a(z))−(ξ^δ(b)−ξ^δ(a))],

F₄ :=Se⁰⁰_p−b(ϕe^δ_a(z)−ξ^δ(a))[(ϕe^δ_b(z)−ϕe^δ_a(z))−(ξ^δ(b)−ξ^δ(a))]²/2, F₅ :=Se⁰⁰⁰_p−b(ϕe^δ_c1(z)−ξ^δ(c₁))[(ϕe^δ_b(z)−ϕe^δ_a(z))−(ξ^δ(b)−ξ^δ(a))]³/6.

And for some c₂ ∈[a, b], we have

F₂ =−∂_rSe_p−b(ϕe^δ_a(z)−ξ^δ(a))(b−a) +∂_r²eS_p−c2(ϕe^δ_a(z)−ξ^δ(a))(b−a)²/2. (4.3.3) Now for some c₃ ∈[a, b], we have

e

ϕ^δ_b(z)−ϕe^δ_a(z) =∂rϕe^δ_c3(z)(b−a) =Sep−c3(ϕe^δ_c3(z)−ξ^δ(c3))(b−a). (4.3.4) For somec₄ ∈[c₃, b], we have

Se_p−c3(ϕe^δ_c3(z)−ξ^δ(c₃)) =Se_p−b(ϕe^δ_c3(z)−ξ^δ(c₃))+∂_reS_p−c4(ϕe^δ_c3(z)−ξ^δ(c₃))(b−c₃). (4.3.5)

For somec5 ∈[a, c3], we have

Se_p−b(ϕe^δ_c3(z)−ξ^δ(c₃)) =Se_p−b(ϕe^δ_a(z)−ξ^δ(a))

+Se⁰_p−b(ϕe^δ_c5(z)−ξ^δ(c5))[(ϕe^δ_c3(z)−ϕe^δ_a(z))−(ξ^δ(c3)−ξ^δ(a))]. (4.3.6) Once again, there is c6 ∈[a, c3] such that

e

ϕ^δ_c3(z)−ϕe^δ_a(z) =∂_rϕe^δ_c6(z)(c₃−a) = eS_p−c6(ϕe^δ_c6(z)−ξ^δ(c₆))(c₃−a). (4.3.7) We have the freedom to choose d arbitrarily small. Now suppose d <(1−ι)(p− q₁)/2. Then

p−a≤p−b+ 2d ≤(p−b) + (1−ι)(p−q₁)≤(2−ι)(p−b).

Thus for anym ≤M ∈[a, b], p−m ≤(2−ι)(p−M). By formula (4.3.2),

e

ϕ^δ_m(z)−ξ^δ(m)∈Ae_{ι(p−m),p−m} ⊂Aeι(p−M),(2−ι)(p−M).

So the values of Se_p−M, ∂_reS_p−M, ∂_r²Se_p−M, eS⁰_p−M, Se⁰⁰_p−M and Se⁰⁰⁰_p−M at ϕe^δ_m(z)−ξ^δ(m) are uniformly bounded. In formula (4.3.3), consider m = a and M = c₂. Since

|b−a| ≤2d², we have

F₂ =−∂_rSe_p−b(ϕe^δ_a(z)−ξ^δ(a))(b−a) +O(d⁴).

Similarly, formula (4.3.7) implies

e

ϕ^δ_c3(z)−ϕe^δ_a(z) =O(c3−a) =O(d²).

This together with formulae (4.3.5),(4.3.6) and ξ^δ(c₃)−ξ^δ(a) =O(d) implies that Sep−c3(ϕe^δ_c3(z)−ξ^δ(c3)) = eSp−b(ϕe^δ_a(z)−ξ^δ(a)) +O(d).

By formula (4.3.4), we have

e

ϕ^δ_b(z)−ϕe^δ_a(z) = eS_p−b(ϕe^δ_a(z)−ξ^δ(a))(b−a) +O(d³) = O(d²).

ThusF5 =O(d³),

F₄ =Se⁰⁰_p−b(ϕe^δ_a(z)−ξ^δ(a))(ξ^δ(b)−ξ^δ(a))²/2 +O(d³), and

F₃ =eS⁰_p−b(ϕe^δ_a(z)−ξ^δ(a))[eS_p−b(ϕe^δ_a(z)−ξ^δ(a))(b−a)−(ξ^δ(b)−ξ^δ(a))] +O(d³).

Note thatF =F₂+F₃+F₄ +F₅. Using Lemma 4.3.2, we get

F =eS⁰⁰_p−b(ϕe^δ_a(z)−ξ^δ(a))[(ξ^δ(b)−ξ^δ(a))²/2−(b−a)]

−Se⁰_p−b(ϕe^δ_a(z)−ξ^δ(a))(ξ^δ(b)−ξ^δ(a)) +O(d³).

By formula (4.3.1), if δ is smaller than some δ₂(d), then the conditional expectation of

ImSe⁰⁰_p−b(ϕe^δ_a(z)−ξ^δ(a))[(ξ^δ(b)−ξ^δ(a))²/2−(b−a)]−ImSe⁰_p−b(ϕe^δ_a(z)−ξ^δ(a))[ξ^δ(b)−ξ^δ(a)]

w.r.t. F_j⁰ is bounded by C1d³.

By formula (4.3.2), for any w∈Ae(1−ι)(p−a),p−a, the conditional expectation of

ImSe⁰⁰_p−b(w)[(ξ^δ(b)−ξ^δ(a))²/2−(b−a)]−ImSe⁰_p−b(w)[ξ^δ(b)−ξ^δ(a)] (4.3.8) w.r.t F_j⁰ is bounded by C₁d³, if δ is small enough (depending on d).

Now suppose d <(p−q₁)ι/(4−4ι). Then

(1−ι)(p−a)<(1−ι/2)(p−b)< p−a.

Thusi(1−ι/2)(p−b)∈Ae(1−ι)(p−a),p−a. We may check

ImeS⁰⁰_p−b(i(1−ι/2)(p−b))>0, and ImeS⁰_p−b(i(1−ι/2)(p−b)) = 0.

So we can find C2 > 0 such that for all b ∈ [0, q1], ImSe⁰⁰_p−b(i(1−ι/2)(p−b)) > C2. Letw=i(1−ι/2)(p−b) in formula (4.3.8), then we get

|E[(ξ^δ(b)−ξ^δ(a))²/2−(b−a)|F_j⁰]| ≤C₃d³.

Since ImeS⁰⁰_p−b(w) is uniformly bounded on Ce(1−ι/2)(p−b), so for allw∈Ce(1−ι/2)(p−b), ImSe⁰_p−b(w)|E[ξ^δ(b)−ξ^δ(a)|F_j⁰]| ≤C₄d³. (4.3.9) We may check that

x_b := ImSe_p−b(π+i(1−ι/2)(p−b))−ImeS_p−b(i(1−ι/2)(p−b))>0.

Sox_b is greater than some absolute constantC₅ >0 for b∈[0, q₁]. Then there exists w_b ∈Ce(1−ι/2)(p−b) such that

|ImSe⁰_p−b(w_b)|=|∂_xImeS_p−b(w_b)|=x_b/π≥C₅/π.

Pluggingw=wb in formula (4.3.9), we then have|E[ξ^δ(b)−ξ^δ(a)|F_j⁰]| ≤C6d³. 2 The following Theorem about the convergence of the driving process can be de- duced from Proposition 4.3.2 by using the Skorokhod Embedding Theorem. It is very similar to Theorem 3.6 in [7]. So we omit the proof.

Theorem 4.3.2 For every q₀ ∈ (0, p) and ε > 0 there is a δ₀ > 0 depending on q₀ and ε such that for δ < δ0 there is a coupling of the processes ξ^δ and √

2B such that

Pr[sup{|ξ^δ(t)−√

2B(t)}|:t∈[0, q₀]}> ε]< ε.