Green’s functions for chordal SLE curves

Abstract

For a chordal SLE$_\kappa$ ($\kappa \in (0,8)$) curve in a domain D, the n-point Green’s function valued at distinct points $z_1,\dots ,z_n\in D$ is defined to be

$$\begin{aligned} G(z_1,\dots ,z_n)=\lim _{r_1,\dots ,r_n\downarrow 0} \prod _{k=1}^n r_k^{d-2} \mathbb {P}[{{\mathrm{dist}}}(\gamma ,z_k)<r_k,1\le k\le n], \end{aligned}$$

where $d=1+\frac{\kappa }{8}$ is the Hausdorff dimension of SLE$_\kappa$, provided that the limit converges. In this paper, we will show that such Green’s functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green’s functions as well. Finally, we give up-to-constant bounds for them.

Appendices

Appendices

A Proof of Theorem 3.1

In order to prove Theorem 3.1, we need some lemmas. The proof of Theorem 3.1 will be given after the proof of Lemma A.4. We still let $\gamma$ be a chordal SLE$_\kappa$ curve in $\mathbb {H}$ from 0 to $\infty$. Throughout the appendix, we use C (without subscript) to denote a positive constant depending only on $\kappa$, and use $C_x$ to denote a positive constant depending only on $\kappa$ and some variable x. The value of a constant may vary between occurrences.

First, let’s recall the one-point estimate and the boundary estimate for chordal SLE$_\kappa$. (see [17, Lemma 2.6, Lemma 2.5]).

Lemma A.1

(One-point Estimate) Let T be a stopping time for $\gamma$. Let $z_0\in \overline{\mathbb {H}}$, $y_0={{\mathrm{Im}}}z_0\ge 0$, and $R\ge r>0$. Then

$$\begin{aligned} \mathbb {P}\big [\tau ^{z_0}_r<\infty |{\mathcal {F}}_T,{{\mathrm{dist}}}(z_0,K_T)\ge R\big ]\le C \frac{P_{y_0}(r)}{P_{y_0}(R)}. \end{aligned}$$

Lemma A.2

(Boundary Estimate) Let T be a stopping time. Let $\xi _1$ and $\xi _2$ be a disjoint pair of crosscuts of $H_T$ such that

• 1. either $\xi _1$ disconnects $\gamma (T)$ from $\xi _2$ in $H_T$, or $\gamma (T)$ is an end point of $\xi _1$;

• 2. among the three bounded components of $H_T{\setminus }(\xi _1\cup \xi _2)$, the boundary of the unbounded component does not contain $\xi _2$.

Then

$$\begin{aligned} \mathbb {P}[\tau _{\xi _2}<\infty |{\mathcal {F}}_T]\le C e^{-\alpha \pi d_{H_T}(\xi _1,\xi _2)}. \end{aligned}$$

Lemma A.3

Let $m\in \mathbb {N}$. Let $z_j\in \overline{\mathbb {H}}$, $y_j={{\mathrm{Im}}}z_j$, and $R_j\ge r_j>0$ be such that $|z_j|>R_j$, $1\le j\le m$. Let $D_j=\{|z-z_j|<r_j\}$ and $\widehat{D}_j=\{|z-z_j|<R_j\}$, $1\le j\le m$. Let $\widehat{J}_0,J_0,J_0'$ be three mutually disjoint Jordan curves in ${\mathbb {C}}$, which bound Jordan domains $\widehat{D}_0,D_0,D_0'$, respectively, such that $\widehat{D}_0\supset D_0\supset D_0'$ and $0\not \in \overline{D_0}$. Let $A=\widehat{D}_0{\setminus }\overline{D_0}$ be the doubly connected domain bounded by $\widehat{J}_0$ and $J_0$. Suppose that $A\cap \widehat{D}_j=\emptyset$, $1\le j\le m$, and there is some $n_0\in \{1,\dots ,m\}$ such that $\widehat{D}_0\cap \widehat{D}_{n_0}=\emptyset$. Let $\xi _j=\partial D_j\cap \mathbb {H}$, $\widehat{\xi }_j=\partial \widehat{D}_j\cap \mathbb {H}$, $0\le j\le m$, and $\xi _0'=\partial D_0'\cap \mathbb {H}$. Let

$$\begin{aligned} E=\{\tau _{\xi _0}<\tau _{\widehat{\xi }_1}\le \tau _{\xi _1}<\tau _{\widehat{\xi }_2}\le \tau _{\xi _2}< \cdots< \tau _{\widehat{\xi }_m}\le \tau _{\xi _m}<\tau _{\xi _0'}<\infty \}. \end{aligned}$$

Then

$$\begin{aligned} \mathbb {P}[{E}|{\mathcal {F}}_{\tau _{\xi _0}}]\le C^m e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2} \prod _{j=1}^m \frac{P_{y_j}(r_j)}{P_{y_j}(R_j)}. \end{aligned}$$

Remark

The lemma is similar to and stronger than [17, Theorem 3.1], which has the same conclusion but stronger assumption: $\widehat{D}_j$, $1\le j\le m$, are all assumed to be disjoint from $\widehat{D}_0$. Here we only require that $\widehat{D}_j$, $1\le j\le m$, are disjoint from A, and at least one of them: $\widehat{D}_{n_0}$ is disjoint from $\widehat{D}_0$. The condition that $\widehat{D}_0\cap \widehat{D}_{n_0}=\emptyset$ can not be removed. The proof is similar to that of [17, Theorem 3.1]. The symbols such as $z_j,R_j,r_j$ in the statement of this lemma and the proof below are not related with the symbols with the same names in other parts of this paper, but are related with the symbols in [17].

Proof

We write $\tau _0=\tau _{\xi _0}$, $\widehat{\tau }_j=\tau _{\widehat{\xi }_j}$ and $\tau _j=\tau _{\xi _j}$, $1\le j\le m$, and $\tau _{m+1}=\tau _{\xi _0'}$.

From the one-point estimate, we have

$$\begin{aligned} \mathbb {P}\big [\tau _j<\infty |{\mathcal {F}}_{\widehat{\tau }_{j}}\big ]\le C\frac{P_{y_j}(r_j)}{P_{y_j}(R_j)},\quad 1\le j\le m. \end{aligned}$$

(A.1)

Thus, $\mathbb {P}[E|{\mathcal {F}}_{\tau _0}]\le C^m\prod _{j=1}^m \frac{P_{y_j}(r_j)}{P_{y_j}(R_j)}$. Now we need to derive the factor $e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2}$.

By mapping A conformally onto an annulus, we see that there is a Jordan curve $\rho$ in A that disconnects $J_0$ from $\widehat{J}_0$, such that

$$\begin{aligned} d_{{\mathbb {C}}}(\rho ,J_0 )=d_{{\mathbb {C}}}(\rho ,\widehat{J}_0 )= d_{{\mathbb {C}}}(J,\widehat{J}_0)/2. \end{aligned}$$

(A.2)

Let $T=\inf \{t\ge 0:\xi _0'\not \subset H_t\}$. Let $t\in [\tau _0,T)$. Each connected component $\eta$ of $\rho \cap H_t$ is a crosscut of $H_t$, and $H_t{\setminus }\eta$ is the disjoint union of a bounded domain and an unbounded domain. We use $H_t^*(\eta )$ to denote the bounded domain. First, consider the connected components $\eta$ of $\rho \cap H_t$ such that $\xi _0'\subset H_t^*(\eta )$. If such $\eta$ is unique, we denote it by $\rho _t$. Otherwise, applying [17, Lemma 2.1], we may find the unique component $\eta _0$, such that $H_t^*(\eta _0)$ is the smallest among all of these $H_t^*(\eta )$. Again we use $\rho _t$ to denote this $\eta _0$. Let $\widehat{U}^\rho _t=H_t^*(\rho _t)$. Then $\xi _0'\subset \widehat{U}^\rho _t$. Next, consider the connected components $\eta$ of $\rho \cap H_t$ such that $H_t^*(\eta )\subset \widehat{U}^\rho _t{\setminus } \xi _0'$. Let the union of $H_t^*(\eta )$ for these $\eta$ be denoted by $U^\rho _t$. Then we have $U^\rho _t\subset \widehat{U}^\rho _t$ and $U^\rho _t\cap \xi _0'=\emptyset$.

Now we define a family of events.

• Let $A_{(0,1)}$ be the event that $\tau _0<\widehat{\tau }_1\wedge T$ and $D_1\cap \mathbb {H}\subset U^\rho _{\tau _0}$.

• For $1\le j\le n_0-1$, let $A_{(j,j)}$ be the event that $\tau _{j-1}<\tau _j<T$, and $D_j\cap \mathbb {H}\not \subset U^\rho _{\tau _{j-1}}$, but $D_j\cap \mathbb {H}\subset U^\rho _{\tau _{j}}$.

• For $1\le j\le n_0-1$, let $A_{(j,j+1)}$ be the event that $\tau _j<\widehat{\tau }_{j+1}\wedge T$, and $D_j\cap \mathbb {H}\not \subset U^\rho _{\tau _{j}}$, but $D_{j+1}\cap \mathbb {H}\subset U^\rho _{\tau _{j}}$.

• For $n_0\le j\le m$, let $A_{(j,j)}$ be the event that $\tau _{j-1}<\tau _{j}<T$, and $D_j\cap \mathbb {H}\not \subset \widehat{U}^\rho _{\tau _{j-1}}$, but $D_j\cap \mathbb {H}\subset \widehat{U}^\rho _{\tau _{j}}$.

• For $n_0\le j\le m-1$, let $A_{(j,j+1)}$ be the event that $\tau _{j}<\widehat{\tau }_{j+1}\wedge T$, and $D_j\cap \mathbb {H}\not \subset \widehat{U}^\rho _{\tau _{j}}$, but $D_{j+1}\cap \mathbb {H}\subset \widehat{U}^\rho _{\tau _{j}}$.

• Let $A_{(m,m+1)}$ be the event that $\tau _m<\tau _{m+1}\wedge T$ and $D_m\cap \mathbb {H}\not \subset \widehat{U}^\rho _{\tau _{m}}$.

Let $I=\{(j,j+1):0\le j\le m\}\cup \{(j,j):1\le j\le m\}$. We claim that $E\subset \bigcup _{\iota \in I} A_\iota$. To see this, note that, if none of the events $A_{(j,j+1)}$, $0\le j\le n_0-1$, and $A_{(j,j)}$, $1\le j\le n_0-1$, happens, then $D_{n_0}\cap \mathbb {H}\not \subset U^\rho _{\tau _{n_0}}$. Since $D _{n_0}$ is disjoint from $\widehat{D}_0$, we can conclude that $D _{n_0}\cap \mathbb {H}\not \subset \widehat{U}^\rho _{\tau _{n_0}}$. In fact, if $D _{n_0}\cap \mathbb {H}\subset \widehat{U}^\rho _{\tau _{n_0}}$, then from $D _{n_0}\cap \widehat{D}_0=\emptyset$, $\rho \subset \widehat{D}_0$, and $\rho$ surrounds $\xi _0'$, we may find a connected component $\eta$ of $\rho \cap H_{\tau _{n_0}}$ that disconnects $D _{n_0}\cap \mathbb {H}$ from $\xi _0'$ in $H_{\tau _{n_0}}$. Since $D _{n_0}\cap \mathbb {H},\xi _0' \subset \widehat{U}^\rho _{\tau _{n_0}}$, we have $\eta \subset \widehat{U}^\rho _{\tau _{n_0}}$. From the definitions of $\rho _{n_0}$ and $\widehat{U}^\rho _{n_0}$, we see that $\eta$ does not disconnect $\xi _0'$ from $\infty$ in $H_{\tau _{n_0}}$. Thus, $D _{n_0}\cap \mathbb {H}\subset H_{\tau _{n_0}}^*(\eta )\subset \widehat{U}^\rho _{\tau _{n_0}}$, and $\xi _0'\cap H_{\tau _{n_0}}^*(\eta )=\emptyset$. This shows that $D _{n_0}\cap \mathbb {H}\subset U^\rho _{\tau _{n_0}}$, which is a contradiction. Since $D_{n_0}\cap \mathbb {H}\not \subset \widehat{U}^\rho _{\tau _{n_0}}$, one of the events $A_{(j,j)}$ and $A_{(j,j+1)}$, $n_0\le j\le m$, must happen. So the claim is proved. We will finish the proof by showing that

$$\begin{aligned} \mathbb {P}[E\cap A_{\iota }|{\mathcal {F}}_{\tau _0}] \le C^m e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2} \prod _{j=1}^m \frac{P_{y_j}(r_j)}{P_{y_j}(R_j)}, \quad \iota \in I. \end{aligned}$$

(A.3)

Case 1 Suppose $A_{(0,1)}$ occurs. Then at time $\tau _0$, there is a connected component, denoted by $\widetilde{\rho }_{\tau _0}$, of $\rho \cap H_{\tau _0}$, that disconnects $\widehat{\xi }_1$ from both $\xi _0'$ and $\infty$ in $H_{\tau _0}$. Since $\xi _0'\subset D_0\cap \mathbb {H}\subset H_{\tau _0}$ and $\gamma (\tau _0)\in \partial D_0$, we see that $\widetilde{\rho }_{\tau _0}$ disconnects $\widehat{\xi }_1$ also from $\gamma (\tau _0)$ in $H_{\tau _0}$. Since $\widehat{\xi }_1$ is disjoint from A, it is contained in either $D_0$ or ${\mathbb {C}}{\setminus }\widehat{D}_0$. If $\widehat{\xi }_1$ is contained in $D_0$ (resp. ${\mathbb {C}}{\setminus }\widehat{D}_0$), then $J_0\cap H_{\tau _0}$ (resp. $\widehat{J}_0\cap H_{\tau _0}$) contains a connected component, denoted by $\eta _{\tau _0}$, which disconnects $\widehat{\xi }_1$ from $\widetilde{\rho }_{\tau _0}$ and $\infty$ in $H_{\tau _0}$. Using the boundary estimate and (A.2), we get

$$\begin{aligned} \mathbb {P}[\widehat{\tau }_1<\infty |{\mathcal {F}}_{\tau _0},A_{(0,1)}] \le C e^{-\alpha \pi d_{H_{\tau _0}}(\widetilde{\rho }_{\tau _0}, \eta _{\tau _0})}\le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2}, \end{aligned}$$

which together with (A.1) implies that (A.3) holds for $\iota =(0,1)$.

Case 2 Suppose for some $1\le j\le n_0-1$, $A_{(j,j+1)}$ occurs. See Fig. 3. Then at time $\tau _j$, there is a connected component, denoted by $\widetilde{\rho }_{\tau _j}$, of $\rho \cap H_{\tau _j}$, that disconnects $\widehat{\xi }_{j+1}$ from both $\xi _j$ and $\infty$ in $H_{\tau _j}$. Since $\gamma (\tau _j)\in \xi _j$, we see that $\widetilde{\rho }_{\tau _j}$ disconnects $\widehat{\xi }_{j+1}$ also from $\gamma (\tau _j)$ in $H_{\tau _j}$. According to whether $\xi _{j+1}$ belongs to $D_0$ or ${\mathbb {C}}{\setminus }\widehat{D}_0$, we may find a connected component $\eta _{\tau _j}$ of $J_0\cap H_{\tau _0}$ or $\widehat{J}_0\cap H_{\tau _0}$ that disconnects $\widehat{\xi }_{j+1}$ from $\widetilde{\rho }_{\tau _j}$ and $\infty$ in $H_{\tau _j}$. Using the boundary estimate and (A.2), we get

$$\begin{aligned} \mathbb {P}\big [\widehat{\tau }_{j+1}<\infty |{\mathcal {F}}_{\tau _j},A_{(j,j+1)},\tau _j<\widehat{\tau }_{j+1}\big ] \le C e^{-\alpha \pi d_{H_{\tau _j}}(\widetilde{\rho }_{\tau _j}, \eta _{\tau _j})}\le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2}, \end{aligned}$$

which together with (A.1) implies that (A.3) holds for $\iota =(j,j+1)$, $1\le j\le n_0-1$.

Case 3 Suppose for some $n_0\le j\le m$, $A_{(j,j+1)}$ occurs. See Fig. 3. We write $\xi _{m+1}=\xi _0'$. Then $\rho _{\tau _j}$ disconnects $\xi _{j+1}$ from $\gamma (\tau _j)$ and $\infty$ in $H_{\tau _j}$. According to whether $\xi _{j+1}$ belongs to $D_0$ or ${\mathbb {C}}{\setminus }\widehat{D}_0$, we may find a connected component $\eta _{\tau _j}$ of $J_0\cap H_{\tau _0}$ or $\widehat{J}_0\cap H_{\tau _0}$ that disconnects $\widehat{\xi }_{j+1}$ from $\rho _{\tau _j}$ and $\infty$ in $H_{\tau _j}$. Using the boundary estimate and (A.2), we get

$$\begin{aligned} \mathbb {P}\big [\widehat{\tau }_{j+1}<\infty |{\mathcal {F}}_{\tau _j},A_{(j,j+1)},\tau _j<\widehat{\tau }_{j+1}\big ] \le C e^{-\alpha \pi d_{H_{\tau _j}}(\rho _{\tau _j}, \eta _{\tau _j})}\le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2}, \end{aligned}$$

which together with (A.1) implies that (A.3) holds for $\iota =(j,j+1)$, $n_0\le j\le m$.

Fig. 3

The two pictures above illustrate Case 2 (left) and Case 3 (right), respectively. In both pictures, the zigzag curve is $\gamma$ up to $\tau _j$, and the three big arcs are $\widehat{J}_0$, $\rho$ and $J_0$ restricted to $\mathbb {H}$. But the positions of the two pairs of concentric circles $(\widehat{\xi }_j,\xi _j)$ and $(\widehat{\xi }_{j+1},\xi _{j+1})$ are swapped. In both pictures, the pairs of acs that contribute the factors from the boundary estimate ($\widetilde{\rho }_{\tau _j}$ and $\eta _{\tau _j}$ on the left, $\rho _{\tau _j}$ and $\eta _{\tau _j}$ on the right) are labeled and colored red. We also labeled $\rho _{\tau _j}$ on the left and $\widetilde{\rho }_{\tau _{j}}$ on the right, and colored both of them green. One can see the difference between $\widehat{U}_{\tau _j}$ and $U_{\tau _j}$ as they are bounded by $\rho _{\tau _{j}}$ and $\widetilde{\rho }_{\tau _{j}}$, respectively

Case 4 Suppose for some $n_0\le j\le m-1$, $A_{(j,j)}$ occurs. Define a stopping time

$$\begin{aligned} \sigma _j=\inf \big \{t\ge \tau _{j-1}:D_j\cap \mathbb {H}\subset \widehat{U}^\rho _{t}\big \}. \end{aligned}$$

Then $\tau _{j-1}\le \sigma _j\le \tau _j$. From [17, Lemma 2.2], we know that

• $\gamma (\sigma _j)$ is an endpoint of $\rho _{\sigma _j}$;

• $D_j\cap \mathbb {H}\subset \widehat{U}^\rho _{\sigma _j}$.

The second property implies that $\tau _{j-1}< \sigma _j< \tau _j$. Now we define two events. Let $F_<=\{\sigma _j<\widehat{\tau }_j\}$ and $F_\ge =\{\widehat{\tau }_j\le \sigma _j<\tau _j\}$. Then $A_{(j,j)}\subset F_<\cup F_\ge$.

Case 4.1 Suppose $F_\ge$ occurs. Let $N=\lceil \log (R_j/r_j)\rceil \in \mathbb {N}$. Let $\zeta _k=\{z\in \mathbb {H}:|z-z_j|=(R_j^{N-k} r_j^k)^{1/N}\}$, $0\le k\le N$. Note that $\zeta _0=\widehat{\xi }_j$ and $\zeta _N=\xi _j$. Then $F_{\ge }\subset \bigcup _{k=1}^N F_k$, where

$$\begin{aligned} F_k:=\{\tau _{\zeta _{k-1}}\le \sigma _j<\tau _{\zeta _k}<\infty \},\quad 1\le k\le N. \end{aligned}$$

See Fig. 4 for an illustration of $F_k$. If $F_k$ occurs, then $\zeta _k\subset \widehat{U}^\rho _{\sigma _j}$. Since $\zeta _{k-1}\cap H_{\sigma _j}$ has a connected component $\zeta _{k-1}^{\sigma _j}$, which disconnects $\zeta _k$ from $\rho _{\sigma _j}$ in $H_{\sigma _j}$, by the boundary estimate, we get

$$\begin{aligned} \mathbb {P}\big [\tau _{\zeta _k}<\infty |{\mathcal {F}}_{\sigma _j},F_k\big ]\le C e^{-\alpha \pi d_{H_{\sigma _j}}(\rho _{\sigma _j},\zeta _{k-1}^{\sigma _j})}. \end{aligned}$$

According to whether $\zeta _k$ belongs to $D_0$ or $\widehat{D}_0$, we may find a connected component $\eta _{\sigma _j}$ of $J_0\cap H_{\sigma _j}$ or $\widehat{J}_0\cap H_{\sigma _j}$ that disconnects $\zeta _{k-1}^{\sigma _j}$ from $\rho _{\sigma _j}$ and $\infty$ in $H_{\sigma _j}$. Moreover, we may find a connected component $\zeta _0^{\sigma _j}$ of $\zeta _0\cap H_{\sigma _j}$ that disconnects $\eta _{\sigma _j}$ from $\zeta _{k-1}^{\sigma _j}$. From the composition law of extremal length and (A.2) we get

$$\begin{aligned} d_{H_{\sigma _j}}\big (\rho _{\sigma _j},\zeta _{k-1}^{\sigma _j}\big )\ge & {} d_{H_{\sigma _j}}(\rho _{\sigma _j},\eta _{\sigma _j}) +d_{H_{\sigma _j}}\big (\zeta _0^{\sigma _j},\zeta _{k-1}^{\sigma _j}\big ) \ge \frac{1}{2} d_{{\mathbb {C}}}(J_0,\widehat{J}_0)\\&+\frac{k-1}{2\pi N} \log \Big (\frac{R_j}{r_j}\Big )\Big . \end{aligned}$$

Thus, we get

$$\begin{aligned} \mathbb {P}\big [\tau _{\zeta _k}<\infty |{\mathcal {F}}_{\sigma _j},F_k\big ] \le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2}\Big (\frac{r_j}{R_j}\Big )^{\frac{\alpha }{2} \frac{k-1}{N}}. \end{aligned}$$

From the one-point estimate, we get

$$\begin{aligned}&\mathbb {P}\big [F_k|{\mathcal {F}}_{\tau _{j-1}}, \tau _{j-1}<\widehat{\tau }_j\big ]\le C \frac{P_{y_j}\big (\big (R_j^{N-k+1} r_j^{k-1}\big )^{1/N}\big )}{P_{y_j}(R_j)};\\&\quad \mathbb {P}\big [\tau _j<\infty |{\mathcal {F}}_{\tau _{\zeta _k}},F_k\big ]\le C \frac{P_{y_j}(r_j)}{P_{y_j}\big (\big (R_j^{N-k} r_j^{k}\big )^{1/N}\big )}. \end{aligned}$$

The above three displayed formulas together imply that

$$\begin{aligned} \mathbb {P}\big [\tau _j< \infty ,F_k|{\mathcal {F}}_{\tau _{j-1}},\tau _{j-1}<\widehat{\tau }_j\big ]\le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2} \Big (\frac{r_j}{R_j}\Big )^{\frac{\alpha }{2} \frac{k-1}{N}}\Big (\frac{r_j}{R_j}\Big )^{-\alpha /N}\frac{P_{y_j}(r_j)}{P_{y_j}(R_j)} . \end{aligned}$$

Since $F_{\ge }\subset \bigcup _{k=1}^N F_k$, by summing up the above inequality over k, we get

$$\begin{aligned}&\mathbb {P}\big [\tau _j<\infty ,F_{\ge } |{\mathcal {F}}_{\tau _{j-1}},\tau _{j-1}<\widehat{\tau }_j\big ]\nonumber \\&\quad \le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2} \frac{P_{y_j}(r_j)}{P_{y_j}(R_j)}\left[ \Big (\frac{r_j}{R_j}\Big )^{-\alpha /N} \frac{1-\big (\frac{r_j}{R_j}\big )^{\alpha /2}}{1-\big (\frac{r_j}{R_j}\big )^{\alpha /(2N)}}\right] \nonumber \\&\quad \le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2} \frac{P_{y_j}(r_j)}{P_{y_j}(R_j)}, \end{aligned}$$

(A.4)

where the second inequality holds because the quantity inside the square bracket is bounded above by $\frac{e^\alpha }{1-e^{-\alpha /4}}$. To see this, consider the cases $R_j/r_j\le e$ and $R_j/r_j>e$ separately.

Case 4.2 Suppose $F_{<}$ occurs. Then $\widehat{\xi }_j\subset \widehat{U}^\rho _{\sigma _j}$. According to whether $\widehat{\xi }_j$ belongs to $D_0$ or $\widehat{D}_0$, we may find a connected component $\eta _{\sigma _j}$ of $J_0\cap H_{\sigma _j}$ or $\widehat{J}_0\cap H_{\sigma _j}$ that disconnects $\widehat{\xi }_j$ from $\rho _{\sigma _j}$ and $\infty$ in $H_{\sigma _j}$. By the boundary estimate, we get

$$\begin{aligned} \mathbb {P}\big [\widehat{\tau }_j<\infty |{\mathcal {F}}_{\sigma _j},F_{<}\big ]\le C e^{-\alpha \pi d_{H_{\sigma _j}}(\rho _{\sigma _j}, \eta _{\sigma _j})}\le Ce^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2}, \end{aligned}$$

which together with (A.1) implies that

$$\begin{aligned} \mathbb {P}\big [\tau _j<\infty , F_{<}|{\mathcal {F}}_{\tau _{j-1}}\big ]\le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2} \frac{P_{y_j}(r_j)}{P_{y_j}(R_j)}. \end{aligned}$$

(A.5)

Combining (A.4) and (A.5), we get

$$\begin{aligned} \mathbb {P}\big [\tau _j<\infty , A_{(j,j)}|{\mathcal {F}}_{\tau _{j-1}},\tau _{j-1}<\widehat{\tau }_j\big ]\le C e^{-\alpha \pi d_{{\mathbb {C}}}(J_0,\widehat{J}_0)/2} \frac{P_{y_j}(r_j)}{P_{y_j}(R_j)}, \end{aligned}$$

which together with (A.1) implies that (A.3) holds for $\iota =(j,j)$, $n_0\le j\le m$.

Case 5 Suppose for some $1\le j\le n_0-1$, $A_{(j,j)}$ occurs. Define a stopping time

$$\begin{aligned} \sigma _j=\inf \big \{t\ge \tau _{j-1}:D_j\cap \mathbb {H}\subset U^\rho _{t}\big \}. \end{aligned}$$

To derive properties of $\sigma _j$, we claim that the following are true.

• (i) If $D_j\cap \mathbb {H}\subset H_{t_0}{\setminus }U^\rho _{t_0}$, then there is $\varepsilon >0$ such that $D_j\cap \mathbb {H}\subset H_t{\setminus }U^\rho _{t}$ for $t_0\le t<t_0+\varepsilon$;

• (ii) If $D_j\cap \mathbb {H}\subset U^\rho _{t_0}$, and if $\gamma (t_0)$ is not an endpoint of a connected component of $\rho \cap H_{t_0}$ that disconnects $D_j\cap \mathbb {H}$ from $\infty$ in $H_{t_0}$, then there is $\varepsilon >0$ such that $D_j\cap \mathbb {H}\subset U^\rho _{t }$ for $t_0-\varepsilon <t\le t_0$.

To see that (i) holds, we consider two cases. Case 1. $D_j\cap \mathbb {H}\subset H_{t_0}{\setminus }\widehat{U}^\rho _{t_0}$. From [17, Lemma 2.2], there is $\varepsilon >0$ such that for $t_0\le t<t_0+\varepsilon$, $D_j\cap \mathbb {H}\subset H_t{\setminus }\widehat{U}^\rho _{t}$, which implies that $D_j\cap \mathbb {H}\subset H_t{\setminus }U^\rho _{t}$. Case 2. $D_j\cap \mathbb {H}\subset \widehat{U}^\rho _{t_0}{\setminus }U^\rho _{t_0}$. Then there is a curve $\zeta$ in $H_{t_0}$, which connects $\xi _0'$ with $D_j$, and does not intersect $\rho$. In this case, there is $\varepsilon >0$ such that for $t_0\le t<t_0+\varepsilon$, $\zeta \subset H_t$ and $D_j\cap \mathbb {H}\subset H_t$, which imply that $D_j\cap \mathbb {H}\subset H_t{\setminus }U^\rho _{t}$.

Now we consider (ii). Since $D_j\cap \mathbb {H}\subset U^\rho _{t_0}$, there is a connected component $\zeta$ of $\rho \cap H_{t_0}$, which is contained in $\widehat{U}^\rho _{t_0}$, and disconnects $D_j\cap \mathbb {H}$ from $\xi _0'$ and $\infty$ in $H_{t_0}$. From the assumption, $\gamma (t_0)$ is not an end point of $\zeta$. By the continuity of $\gamma$, there is $\varepsilon _1>0$ such that $\gamma [t_0-\varepsilon _1,t_0]\cap \overline{\zeta }=\emptyset$. This implies that, for $t_0-\varepsilon _1<t\le t_0$, $\zeta$ is also a crosscut of $H_t$. Since $H_t$ is simply connected, $\zeta$ also disconnects $D_j\cap \mathbb {H}$ from $\xi _0'$ and $\infty$ in $H_{t}$. Since $\rho _{t_0}$ is a connected component of $\rho \cap H_{t_0}$ that disconnects $\widehat{U}^\rho _{t_0}\supset U^\rho _{t_0}\supset D_j\cap \mathbb {H}$ from $\infty$, $\gamma (t_0)$ is also not an endpoint of $\rho _{t_0}$. Since $\zeta \subset \widehat{U}^\rho _{t_0}$, from [17, Lemma 2.2], there is $\varepsilon \in (0,\varepsilon _1)$ such that for $t_0-\varepsilon < t\le t_0$, $\zeta \subset \widehat{U}^\rho _{t }$, which implies that $D_j\cap \mathbb {H}\subset U^\rho _t$.

From (i) and (ii) we conclude that

• $\gamma (\sigma _j)$ is an endpoint of a connected component of $\rho \cap H_{\sigma _j}$ that disconnects $D_j\cap \mathbb {H}$ from $\infty$ in $H_{\sigma _j}$. Let this crosscut be denoted by $\widetilde{\rho }_{\sigma _j}$.

• $D(z_j,r_j)\cap \mathbb {H}\subset U^\rho _{\sigma _j}$.

Following the proof of Case 4 with $\widetilde{\rho }_{\sigma _j}$ and $U^\rho _{\sigma _j}$ in place of $\rho _{\sigma _j}$ and $\widehat{U}^\rho _{\sigma _j}$, respectively, we conclude that (A.3) holds for $\iota =(j,j)$, $1\le j\le n_0-1$. See Fig. 4 for an illustration of the subcase $F_<$ of Case 5. The proof is now complete. $\square$

Fig. 4

The two pictures above illustrate the subcase $F_k\subset F_{\ge }$ of Case 4 (left) and the subcase $F_<$ of Case 5 (right), respectively. In both pictures, the zigzag curve is $\gamma$ up to $\sigma _j$, and the three big arcs are $\widehat{J}_0$, $\rho$ and $J_0$ restricted to $\mathbb {H}$. The acs that contribute the factors from the boundary estimate ($\widetilde{\rho }_{\tau _j}$, $\eta _{\tau _j}$, $\zeta ^{\sigma _j}_{0}$ and $\zeta ^{\sigma _j}_{k-1}$ on the left, $\rho _{\tau _j}$ and $\eta _{\tau _j}$ on the right) are labeled and colored red

Let $\Xi$ be a family of mutually disjoint circles with centers in $\overline{\mathbb {H}}$, each of which does not pass through or enclose 0. Define a partial order on $\Xi$ such that $\xi _1<\xi _2$ if $\xi _2$ is enclosed by $\xi _1$. One should keep in mind that a smaller element in $\Xi$ has bigger radius, but will be visited earlier (if it happens) by a curve started from 0.

Suppose that $\Xi$ has a partition $\{\Xi _e\}_{e\in {\mathcal {E}}}$ with the following properties:

• For each $e\in {\mathcal {E}}$, the elements in $\Xi _e$ are concentric circles with radii forming a geometric sequence with common ratio 1 / 4. We denote the common center $z_e$, the biggest radius $R_e$, and the smallest radius $r_e$, and let $y_e={{\mathrm{Im}}}z_e$.

• Let $A_e=\{r_e\le |z-z_0|\le R_e\}$ be the closed annulus associated with $\Xi _e$, which is a single circle if $R_e=r_e$, i.e., $|\Xi _e|=1$. Then the annuli $A_e$, $e\in {\mathcal {E}}$, are mutually disjoint.

Note that every $\Xi _e$ is a totally ordered set w.r.t. the partial order on $\Xi$.

Lemma A.4

Suppose that $J_1$ and $J_2$ are disjoint Jordan curves in ${\mathbb {C}}$, which are disjoint from all $\xi \in \Xi$. Suppose that 0 is not contained in or enclosed by $J_1$, $J_1$ is enclosed by $J_2$, and that every $\xi \in \Xi$ that lies in the doubly connected domain bounded by $J_1$ and $J_2$ disconnects $J_1$ from $J_2$. Suppose $\xi _a<\xi _b\in \Xi$ are both enclosed by $J_1$, and $\xi _c\in \Xi$ neither encloses $J_2$, or is enclosed by $J_2$. Let E denote the event that $\tau _\xi <\infty$ for all $\xi \in \Xi$, and $\tau _{\xi _a}<\tau _{\xi _c}<\tau _{\xi _b}$. Then

$$\begin{aligned} \mathbb {P}[E]\le C_{|{\mathcal {E}}|} e^{-\frac{\alpha }{4|{\mathcal {E}}|}\pi d_{{\mathbb {C}}}(J_1,J_2)} \prod _{e\in {\mathcal {E}}} \frac{P_{y_e}(r_e)}{P_{y_e}(R_e)}, \end{aligned}$$

where $C_{|{\mathcal {E}}|}\in (0,\infty )$ depends only on $\kappa$ and $|{\mathcal {E}}|$.

Discussion From [17, Theorem 3.2], we know that $\mathbb {P}[\tau _\xi <\infty ,\xi \in \Xi ]\le C_{|{\mathcal {E}}|} \prod _{e\in {\mathcal {E}}} \frac{P_{y_e}(r_e)}{P_{y_e}(R_e)}$. Now we need to derive the additional factor $e^{-\frac{\alpha }{4|{\mathcal {E}}|}\pi d_{{\mathbb {C}}}(J_1,J_2)}$ using the condition $\tau _{\xi _a}<\tau _{\xi _c}<\tau _{\xi _b}$.

Proof

We write $\mathbb {N}_n$ for $\{k\in \mathbb {N}:k\le n\}$. Let S denote the set of bijections $\sigma :\mathbb {N}_{|\Xi |}\rightarrow \Xi$ such that $\xi _1<\xi _2$ implies that $\sigma ^{-1}(\xi _1)<\sigma ^{-1}(\xi _2)$, and $\sigma ^{-1}(\xi _a)<\sigma ^{-1}(\xi _c)<\sigma ^{-1}(\xi _b)$. Let

$$\begin{aligned} E^\sigma =\{\tau _{\sigma (1)}<\tau _{\sigma (2)}<\cdots<\tau _{\sigma (|\Xi |)}<\infty \},\quad \sigma \in S. \end{aligned}$$

Then we have

$$\begin{aligned} E=\bigcup _{\sigma \in S} E^\sigma . \end{aligned}$$

(A.6)

We will derive an upper bound of $\mathbb {P}[E^\sigma ]$ in (A.11).

Fix $\sigma \in S$. For $e\in {\mathcal {E}}$, if there is no $\xi \in \Xi$ such that $\xi >\max \Xi _e$, then we say that e is a maximal element in E. In this case, we define $\widehat{\Xi }_e=\Xi _e$ and $\xi _e^*=\max \Xi _e$. If e is not a maximal element in E, let $\xi _e^*$ denote the first $\xi >\max \Xi _e$ that is visited by $\gamma$ on the event $E^\sigma$, and define $\widehat{\Xi }_e=\Xi _e\cup \xi _e^*$. This definition certainly depends on $\sigma$. Label the elements of $\widehat{\Xi }_e$ by $\xi ^e_0<\cdots <\xi ^e_{N_e}=\xi _e^*$, where $N_e=|\widehat{\Xi }_e|-1$.

For $e\in E$, define

$$\begin{aligned} J_e=\big \{1\le n\le N_e:\sigma ^{-1}\big (\xi ^e_n\big )> \sigma ^{-1}(\xi ^e_{n-1})+1\big \}. \end{aligned}$$

Roughly speaking, $n\in J_e$ means that between $\tau _{\xi ^e_{n-1}}$ and $\tau _{\xi ^e_n}$, $\gamma$ visits other element in $\Xi$ that it has not visited before on the event $E_\sigma$.

Order the elements of $J_e\cup \{0\}$ by $0=s_e(0)<\cdots <s_e(M_e)$, where $M_e=|J_e|$. Set $s_e(M_e+1)=N_e+1$. Every $\widehat{\Xi }_e$ can be partitioned into $M_e+1$ subsets:

$$\begin{aligned} \widehat{\Xi }_{(e,j)}=\big \{\xi ^e_n:s_e(j)\le n\le s_e(j+1)-1\big \},\quad 0\le j\le M_e. \end{aligned}$$

The meaning of the partition is that, after $\gamma$ visits the first element in $\widehat{\Xi }_{(e,j)}$, which must be $\xi ^e_{s_e(j)}$, it then visits all elements in $\widehat{\Xi }_{(e,j)}$ without visiting any other circles in $\Xi$ that it has not visited before. Let $I=\{(e,j):e\in {{\mathcal {E}}}, 0\le j\le M_e\}$. Then $\{\widehat{\Xi }_{\iota }:\iota \in I\}$ is a cover of $\Xi$. Note that every $\sigma ^{-1}(\widehat{\Xi }_\iota )$, $\iota \in I$, is a connected subset of $\mathbb {Z}$.

For $\iota \in I$, let $e_\iota$ denote the first coordinate of $\iota$, $z_\iota =z_{e_\iota }$ and $y_\iota ={{\mathrm{Im}}}z_\iota$. Define $P_\iota$ for each $\iota \in I$. If $\max \widehat{\Xi }_\iota \in \Xi _{e_\iota }$, define $P_\iota = \frac{P_{y_{\iota }}(R_{\max \widehat{\Xi }_\iota })}{P_{y_{\iota }}(R_{\min \widehat{\Xi }_\iota })}$, where we use $R_\xi$ to denote the radius of $\xi$. If $\max \widehat{\Xi }_\iota \not \in \Xi _{e_\iota }$, which means $\max \widehat{\Xi }_\iota =\xi ^*_{e_\iota }> \max \Xi _{e_\iota }$, then we consider two subcases. If $\widehat{\Xi }_\iota$ contains only one element (i.e., $\xi ^*_{e_\iota }$) or two elements (i.e., $\xi ^*_{e_\iota }$ and $\max \Xi _{e_\iota }$), then let $P_\iota =1$; otherwise let $P_\iota = \frac{P_{y_{\iota }}(R_{\max \Xi _{e_\iota }})}{P_{y_{\iota }}(R_{\min \widehat{\Xi }_\iota })}$. From the one-point estimate, we get

$$\begin{aligned} \mathbb {P}\big [\tau _{\max \widehat{\Xi }_\iota }<\infty |{\mathcal {F}}_{\min \widehat{\Xi }_\iota }\big ]\le C P_\iota ,\quad \iota \in I. \end{aligned}$$

(A.7)

Let $P_e= \frac{P_{y_e}(r_e)}{P_{y_e}(R_e)}$, $e\in {\mathcal {E}}$. From Lemma 2.1 we get

$$\begin{aligned} \prod _{j=0}^{M_e} P_{(e,j)}\le 4^{\alpha M_e} P_e,\quad e\in {\mathcal {E}}. \end{aligned}$$

(A.8)

We have $|I|=\sum _{e\in {\mathcal {E}}}(M_e+1)$. Considering the order that $\gamma$ visits $\widehat{\Xi }_\iota$, $\iota \in I$, we get a bijection map $\sigma _I:\mathbb {N}_{|I|}\rightarrow I$ such that $n_1<n_2$ implies that $\max \sigma ^{-1}(\widehat{\Xi }_{\sigma _I(n_1)})\le \min \sigma ^{-1}(\widehat{\Xi }_{\sigma _I(n_2)})$, and $n_1=n_2-1$ implies that $\min \sigma ^{-1}(\widehat{\Xi }_{\sigma _I(n_2)})-\max \sigma ^{-1}(\widehat{\Xi }_{\sigma _I(n_1)})\in \{0,1\}$. The difference may take value 0 if $\max \widehat{\Xi }_{\sigma _I(n_1)}=\xi _e^*\not \in \Xi _e$ for $e=e_{\sigma _I(n_1)}$. We may express $E^\sigma$ as

$$\begin{aligned} E^\sigma =\big \{\tau _{\min \widehat{\Xi }_{\sigma _I(1)}}\le \tau _{\max \widehat{\Xi }_{\sigma _I(1)}} \le \tau _{\min \widehat{\Xi }_{\sigma _I(2)}} \le \cdots \le \tau _{\min \widehat{\Xi }_{\sigma _I(|I|)}}<\tau _{\max \widehat{\Xi }_{\sigma _I(|I|)}}<\infty \big \}. \end{aligned}$$

Fix $e_0\in {\mathcal {E}}$. Let $n_j=\sigma _I^{-1}((e_0,j))$, $0\le j\le M_{e_0}$. Then $n_{j+1}\ge n_{j}+2$, $0\le j\le M_{e_0}-1$. Fix $0\le j\le M_{e_0}-1$. Let $m=n_{j+1}-n_{j}-1$. If $\max \widehat{\Xi }_{\sigma _I(n_j+k)}$ and $\min \widehat{\Xi }_{\sigma _I(n_j+k)}$ are concentric for $1\le k\le m$, applying Lemma A.3 with $\widehat{J}_0=\min \Xi _{e_0}$, $J_0=\max \widehat{\Xi }_{({e_0},j)}=\max \widehat{\Xi }_{\sigma _I(n_{j})}$, $J'_0= \min \widehat{\Xi }_{({e_0},j+1)}=\min \widehat{\Xi }_{\sigma _I(n_{j+1})}$, $\{|z-z_k|=R_k\}=\min \widehat{\Xi }_{\sigma _I(n_{j}+k)}$ and $\{|z-z_k|=r_k\}=\max \widehat{\Xi }_{\sigma _I(n_j+k)}$, $1\le k\le m$, we get

$$\begin{aligned} \mathbb {P}\big [E^\sigma _{\big [\max \widehat{\Xi }_{\sigma _I(n_{j})}, \min \widehat{\Xi }_{\sigma _I(n_{j+1})}\big ]} |{\mathcal {F}}_{\tau _{\max \widehat{\Xi }_{\sigma _I(n_{j})}}}\big ]\le C^m 4^{-\alpha /4(s_{e_0}(j+1)-1)} \prod _{n=n_{j}+1}^{n_{j+1}-1}P_{\sigma _I(n)},\nonumber \\ \end{aligned}$$

(A.9)

where $E^\sigma _{[\max \widehat{\Xi }_{\sigma _I(n_{j})},\min \widehat{\Xi }_{\sigma _I(n_{j+1})}]}$ is the event

$$\begin{aligned} \big \{\tau _{\max \widehat{\Xi }_{\sigma _I(n_{j})}}\le \tau _{\min \widehat{\Xi }_{\sigma _I(n_{j}+1)}}\le \tau _{\max \widehat{\Xi }_{\sigma _I(n_{j}+1)}}\le \cdots \le \tau _{\max \widehat{\Xi }_{\sigma _I(n_{j}+m)}}\le \tau _{\min \widehat{\Xi }_{\sigma _I(n_{j+1})}}<\infty \big \}. \end{aligned}$$

Because of the definition of $P_\iota$, $\iota \in I$, the above estimate still holds in the general case, i.e., there may be some $1\le k\le n$ such that $\max \widehat{\Xi }_{\sigma _I(n_j+k)}=\xi _e^*\not \in \Xi _e$, where $e=e_{\sigma _I(n_j+k)}$.

We say that $\gamma$ makes a $(J_1,J_2)$ jump during $[\max \widehat{\Xi }_{\sigma _I(n_{j})},\min \widehat{\Xi }_{\sigma _I(n_{j+1})}]$ if $\min \Xi _{e_0}$ is enclosed by $J_1$, and there is at least one $k_0\in \mathbb {N}_m$ such that $\min \widehat{\Xi }_{\sigma _I(n_{j}+k_0)}$ is not enclosed by $J_2$. In this case, applying Lemma A.3 with $J_0=J_1$ and $\widehat{J}_0=J_2$, we get

$$\begin{aligned} \mathbb {P}\big [E^\sigma _{\big [\max \widehat{\Xi }_{\sigma _I(n_{j})}, \min \widehat{\Xi }_{\sigma _I(n_{j+1})}\big ]} |{\mathcal {F}}_{\tau _{\max \widehat{\Xi }_{\sigma _I(n_{j})}}}\big ]\le C^m e^{-\alpha \pi d_{{\mathbb {C}}}(J_1,\widehat{J}_2)/2} \prod _{n=n_{j}+1}^{n_{j+1}-1}P_{\sigma _I(n)}. \end{aligned}$$

Combining this with (A.9), we get

$$\begin{aligned}&\mathbb {P}\big [E^\sigma _{\big [\max \widehat{\Xi }_{\sigma _I(n_{j})}, \min \widehat{\Xi }_{\sigma _I(n_{j+1})}\big ]} |{\mathcal {F}}_{\tau _{\max \widehat{\Xi }_{\sigma _I(n_{j})}}}\big ]\nonumber \\&\quad \le C^m e^{-\frac{\alpha }{4}\pi d_{{\mathbb {C}}}(J_1,\widehat{J}_2)} 4^{-\frac{\alpha }{8}(s_{e_0}(j+1)-1)} \prod _{n=n_{j}+1}^{n_{j+1}-1}P_{\sigma _I(n)}. \end{aligned}$$

(A.10)

Letting j vary between 0 and $M_{e_0}-1$ and using (A.7) and (A.9), we get

$$\begin{aligned} \mathbb {P}[E^\sigma ]\le C^{|I|} 4^{-\alpha /4 \sum _{j=1}^{M_{e_0}} (s_{e_0}(j)-1)} \prod _{\iota \in I} P_\iota . \end{aligned}$$

Using (A.8) and $|I|=\sum _e (M_e+1)$, we find that

$$\begin{aligned} \mathbb {P}[E^\sigma ]\le C^{|{{\mathcal {E}}}|} C^{\sum _{e\in {\mathcal {E}}}M_e}4^{-\frac{\alpha }{4} \sum _{j=1}^{M_{e_0}} s_{e_0}(j)}\prod _{e\in {\mathcal {E}}} P_e. \end{aligned}$$

Since $\sigma ^{-1}(\xi _a)<\sigma ^{-1}(\xi _c)<\sigma ^{-1}(\xi _b)$, $\xi _a<\xi _b$ are enclosed by $J_1$, and $\xi _c$ is not enclosed by $J_2$, there must exist some $e_0\in {\mathcal {E}}$ and some $j\in [0,M_{e_0}-1]$ such that $\gamma$ makes a $(J_1,J_2)$ jump during $[\max \widehat{\Xi }_{\sigma _I(n_{j})},\min \widehat{\Xi }_{\sigma _I(n_{j+1})}]$. In that case, using (A.7), (A.9), and (A.10), we get

$$\begin{aligned} \mathbb {P}[E^\sigma ]\le C^{|{{\mathcal {E}}}|} C^{\sum _{e\in {\mathcal {E}}}M_e}e^{-\frac{\alpha }{4}\pi d_{{\mathbb {C}}}(J_1,\widehat{J}_2)} 4^{-\frac{\alpha }{8} \sum _{j=1}^{M_{e_0}} s_{e_0}(j)}\prod _{e\in {\mathcal {E}}} P_e. \end{aligned}$$

Taking a geometric average of the above upper bounds for $\mathbb {P}[E^\sigma ]$ over $e_0\in {\mathcal {E}}$, we get

$$\begin{aligned} \mathbb {P}[E^\sigma ]\le C^{|{{\mathcal {E}}}|} C^{\sum _{e\in {\mathcal {E}}}M_e}e^{-\frac{\alpha }{4|{\mathcal {E}}|}\pi d_{{\mathbb {C}}}(J_1,\widehat{J}_2)} 4^{-\frac{\alpha }{8|{{\mathcal {E}}}|}\sum _{e\in {\mathcal {E}}} \sum _{j=1}^{M_{e}} s_{e}(j)}\prod _{e\in {\mathcal {E}}} P_e. \end{aligned}$$

(A.11)

So far we have omitted the $\sigma$ on I, $M_e$, $s_e(j)$ and etc; we will put $\sigma$ on the superscript if we want to emphasize the dependence on $\sigma$. From (A.6) and (A.11), we get

$$\begin{aligned}&\mathbb {P}[E]\le C^{|{{\mathcal {E}}}|} \sum _{\big (M_e;(s_e(j))_{j=0}^{M_e}\big )_{e\in {\mathcal {E}}}}|S_{(M_e,(s_e(j)))}| C^{\sum _{e\in {\mathcal {E}}}M_e}e^{-\frac{\alpha }{4|{\mathcal {E}}|}\pi d_{{\mathbb {C}}}(J_1,\widehat{J}_2)}\nonumber \\&\quad 4^{-\frac{\alpha }{8|{{\mathcal {E}}}|}\sum _{e\in {\mathcal {E}}} \sum _{j=1}^{M_{e}} s_{e}(j)}\prod _{e\in {\mathcal {E}}} P_e, \end{aligned}$$

(A.12)

where

$$\begin{aligned} S_{(M_e,(s_e(j)))}:=\big \{\sigma \in S:M^\sigma _e=M_e, s^\sigma _e(j)=s_e(j), 0\le j\le M_e,e\in {\mathcal {M}}\big \}, \end{aligned}$$

and the first summation in (A.12) is over all possible $(M_e;(s_e(j))_{j=0}^{M_e})_{e\in {\mathcal {E}}}$, namely, $M_e\ge 0$ and $0=s_e(0)<s_e(1)<\cdots s_e(M_e)\le N_e$ for every $e\in {\mathcal {E}}$. It now suffices to show that

$$\begin{aligned} \sum _{(M_e;(s_e(j))_{j=1}^{M_e})_{e\in {\mathcal {E}}}}|S_{(M_e,(s_e(j)))}| C^{\sum _{e\in {\mathcal {E}}}M_e}4^{-\frac{\alpha }{8|{{\mathcal {E}}}|} \sum _{e\in {\mathcal {E}}} \sum _{j=1}^{M_{e}} s_{e}(j)} \le C_{|{\mathcal {E}}|}, \end{aligned}$$

(A.13)

for some $C_{|{\mathcal {E}}|}<\infty$ depending only on $|{\mathcal {E}}|$ and $\kappa$.

We now bound the size of $S_{(M_e,(s_e(j)))}$. Note that when $M^\sigma _e$ and $s^\sigma _e(j)$, $0\le j\le M^\sigma _e$, $e\in {\mathcal {E}}$, are given, $\sigma$ is then determined by $\sigma _I:\mathbb {N}_{|I^\sigma |}\rightarrow I^\sigma$, which is in turn determined by $e_{\sigma _I(n)}$, $1\le n\le |I^\sigma |=\sum _{e\in {\mathcal {E}}}(M^\sigma _e+1)$. Since each $e_{\sigma _I(n)}$ has at most $|{\mathcal {E}}|$ possibilities, we have $|S_{(M_e,(s_e(j)))}|\le |{{\mathcal {E}}}|^{\sum _{e\in {\mathcal {E}}}(M_e+1)}$. Thus, the left-hand side of (A.13) is bounded by

$$\begin{aligned}&|{{\mathcal {E}}}|^{|{{\mathcal {E}}}|} \sum _{(M_e;(s_e(j))_{j=0}^{M_e})_{e\in {\mathcal {E}}}} \prod _{e\in {\mathcal {E}}} (C|{{\mathcal {E}}}|)^{ M_e}4^{-\frac{\alpha }{8|{{\mathcal {E}}}|} \sum _{j=1}^{M_{e}} s_{e}(j)}\\&\quad =|{{\mathcal {E}}}|^{|{{\mathcal {E}}}|} \prod _{e\in {\mathcal {E}}} \sum _{M_e=0}^{N_e} (C|{{\mathcal {E}}}|)^{ M_e} \sum _{0=s_e(0)<\cdots <s_e(M_e)\le N_e}4^{-\frac{\alpha }{8|{{\mathcal {E}}}|} \sum _{j=1}^{M_{e}} s_{e}(j)}\\&\quad \le |{{\mathcal {E}}}|^{|{{\mathcal {E}}}|} \prod _{e\in {\mathcal {E}}} \sum _{M=0}^{\infty } (C|{{\mathcal {E}}}|)^{ M} \sum _{s(1)=1}^\infty \cdots \sum _{s(M)=M}^\infty 4^{-\frac{\alpha }{8|{{\mathcal {E}}}|} \sum _{j=1}^{M} s(j)}\\&\quad \le |{{\mathcal {E}}}|^{|{{\mathcal {E}}}|} \prod _{e\in {\mathcal {E}}} \sum _{M=0}^{\infty } (C|{{\mathcal {E}}}|)^{ M}\prod _{j=1}^M \sum _{s(j)=j}^\infty 4^{-\frac{\alpha }{8|{{\mathcal {E}}}|} s(j)}\\&\quad = \left[ |{{\mathcal {E}}}| \sum _{M=0}^{\infty } \left( \frac{C|{{\mathcal {E}}}|}{1-4^{-\frac{\alpha }{8|{{\mathcal {E}}}|} }}\right) ^{ M} 4^{-\frac{\alpha }{16|{{\mathcal {E}}}|} M(M+1)} \right] ^{|{{\mathcal {E}}}|}. \end{aligned}$$

The conclusion now follows since the summation inside the square bracket equals to a finite number depending only on $\kappa$ and $|{\mathcal {E}}|$. $\square$

Proof of Theorem 3.1

By (2.7), we may change the order of the points $z_1,\dots ,z_n$. Thus, it suffices to show that

$$\begin{aligned} \mathbb {P}\big [\tau ^{z_j}_{r_j}<\infty ,1\le j\le n; \tau ^{z_{1}}_{s_{1}}<\tau ^{z_{2}}_{r_{2}}<\tau ^{z_1}_{r_1}\big ] \le C_n \prod _{j=1}^n \frac{P_{y_j}(r_j)}{P_{y_j}(l_j)}\cdot \Big (\frac{s_{1}}{|z_{1}-z_{2}|\wedge |z_{1}|}\Big )^{\frac{\alpha }{32n^2}},\nonumber \\ \end{aligned}$$

(A.14)

for any distinct points $z_1,\dots ,z_n\in \overline{\mathbb {H}}{\setminus }\{0\}$, $r_j\in (0,d_j)$, $1\le j\le n$, and $s_1\ge 0$, where $y_j,l_j,d_j$ are defined by (2.3). If $s_1\le r_1$, the event on the LHS is empty, and the formula trivially holds; if $s\ge |z_{1}-z_{2}|\wedge |z_{1}|$, the formula follows from [17, Theorem 1.1]. For the rest of the proof, we assume that $s_1\in (r_1,|z_{1}-z_{2}|\wedge |z_{1}|)$.

We want to deduce the theorem from Lemma A.4, so we want to construct a family $\Xi$ of mutually disjoint circles and Jordan curves $J_1,J_2$.

Suppose $4^{h_j} r_j\le l_j\le 4^{h_j+1} r_j$ for some $h_j\in \mathbb {N}$, $1\le j\le n$. By increasing the value of $s_1$, we may assume that $s_1=4^{\widetilde{h}_1} r_1$, where $\widetilde{h}_1\in \mathbb {N}$ and $\widetilde{h}_1>h_1$. Define

$$\begin{aligned} \xi _j^s=\big \{|z-z_j|=4^{h_j-s} r_j\big \}, \quad 1\le j\le n,\quad 1\le s\le h_j. \end{aligned}$$

The family $\{\xi _j^s:1\le j\le n,\quad 1\le s\le h_j\}$ may not be mutually disjoint. So we can not define $\Xi$ to be this family. To solve this issue, we will remove some circles as follows. For $1\le j<k\le n$, let $D_k=\{|z-z_k|\le l_k/4\}$, which contains every $\xi _k^r$, $1\le r\le h_k$, and

$$\begin{aligned} I_{j,k}=\big \{\xi _j^s: 1\le s\le h_j, \xi _j^s\cap D_k\ne \emptyset \big \}. \end{aligned}$$

(A.15)

Then $\Xi :=\{\xi _j^s:1\le j\le n, 1\le s\le h_j\}{\setminus }\bigcup _{1\le j<k\le n} I_{j,k}$ is mutually disjoint. If ${{\mathrm{dist}}}(\gamma ,z_j)\le r_j$, then $\gamma$ intersects every $\xi _j^s$, $1\le s\le h_j$. So we get

$$\begin{aligned} \mathbb {P}\big [{{\mathrm{dist}}}(\gamma ,z_j)\le r_j,1\le j\le n\big ]\le \mathbb {P}\Big [\bigcap _{j=1}^n \bigcap _{s=1}^{h_j}\{\gamma \cap \xi _j^s\ne \emptyset \}\Big ] \le \mathbb {P}\Big [\bigcap _{\xi \in \Xi } \{\gamma \cap \xi \ne \emptyset \}\Big ].\nonumber \\ \end{aligned}$$

(A.16)

Next, we construct a partition $\{\Xi _e:e\in {\mathcal {E}}\}$ of $\Xi$. We introduce some notation: if e is a family of circles centered at $z_0\in \overline{\mathbb {H}}$ with biggest radius R and smallest radius r, then we define $A_e=\{r\le |z-z_0|\le R\}$ and $P_e=\frac{P_{{{\mathrm{Im}}}z_0}(r)}{P_{{{\mathrm{Im}}}z_0}(R)}$.

First, $\Xi$ has a natural partition $\Xi _j$, $1\le j\le n$, such that $\Xi _j$ is composed of circles centered at $z_j$. For each j, we construct a graph $G_j$, whose vertex set is $\Xi _j$, and $\xi _1\ne \xi _2\in \Xi _j$ are connected by an edge iff the bigger radius is 4 times the smaller one, and the open annulus between them does not contain any other circle in $\Xi$. Let ${{\mathcal {E}}}_j$ denote the set of connected components of $G_j$. Then we partition $\Xi _j$ into $\Xi _e$, $e\in {{\mathcal {E}}}_j$, such that every $\Xi _e$ is the vertex set of $e\in {{\mathcal {E}}}_j$. Then the circles in every $\Xi _e$ are concentric circles with radii forming a geometric sequence with common ratio 1 / 4, and the closed annuli $A_e$ associated with $\Xi _e$, $e\in {{\mathcal {E}}}_j$, are mutually disjoint. From the construction we also see that for any $j<k$, and $e\in {{\mathcal {E}}}_j$, $A_e$ does not intersect $D_k$, which contains every $A_e$ with $e\in {{\mathcal {E}}}_k$. Let ${{\mathcal {E}}}=\bigcup _{j=1}^n {{\mathcal {E}}}_j$. Then $A_e$, $e\in {\mathcal {E}}$, are mutually disjoint. Thus, $\{\Xi _e:e\in {\mathcal {E}}\}$ is a partition of $\Xi$ that satisfies the properties before Lemma A.4.

We observe that for $j<k$, $\bigcup _{\xi \in \Xi _k}\xi \subset D_k$ can be covered by an annulus centered at $z_j$ with ratio less than 4 because

$$\begin{aligned} \frac{\max _{z\in D_k}\{|z-z_j|\}}{\min _{z\in D_k}\{|z-z_j|\}}\le \frac{|z_j-z_k|+l_k/4}{|z_j-z_k|-l_k/4}\le \frac{l_k+l_k/4}{l_k-l_k/4}<4. \end{aligned}$$

Thus, every $I_{j,k}$ defined in (A.15) contains at most one element. We also see that, for $j<k$, $\bigcup _{\xi \in \Xi _k}\xi \subset D_k$ intersects at most 2 annuli from $\{4^{h_j-s} r_j\le |z-z_j|\le 4^{h_j-s+1} r_j\}$, $2\le s\le h_j$. If $j>k$, by construction, $\bigcup _{\xi \in \Xi _k}\xi$ is disjoint from the annuli $\{4^{h_j-s} r_j\le |z-z_j|\le 4^{h_j-s+1} r_j\}$, $2\le s\le h_j$, which are contained in $D_j$.

From [17, Theorem 1.1], we have $\mathbb {P}[\tau ^{z_j}_{r_j}<\infty ,1\le j\le n]\le C_n \prod _{j=1}^n \frac{P_{y_j}(r_j)}{P_{y_j}(l_j)}$. So we may assume that $|z_2-z_1|\wedge |z_1|>4^{4n+1} s_1$. Since for $k\ge 2$, $\bigcup _{\xi \in \Xi _k}\xi \subset D_k$ can be covered by an annulus centered at $z_1$ with ratio less than 4, by pigeon hole principle, we can find a closed annulus centered at $z_1$ with two radii $r<R$ satisfying $s_1\le r<R\le |z_2-z_1|\wedge |z_1|$ and $R/r\le (\frac{|z_2-z_1|\wedge |z_1|}{s_1})^{1/2n}$ that is disjoint from all $\bigcup _{\xi \in \Xi _k}\xi \subset D_k$, $k\ge 2$. Moreover, we may choose R and r such that the boundary circles are disjoint from every $\xi \in \Xi$. Applying Lemma A.4 with $J_1=\{|z-z_1|=r\}$, $J_2=\{|z-z_1|=R\}$, $\xi _a=\{|z-z_1|=s_1\}$, $\xi _b=\{|z-z_1|=r_1\}$, $\xi _c=\{|z-z_2|=r_2\}$, and $\{\Xi _e:e\in {\mathcal {E}}\}$, we find that

$$\begin{aligned} \mathbb {P}\big [\tau ^{z_j}_{r_j}<\infty ,1\le j\le n; \tau ^{z_{1}}_{s_{1}}<\tau ^{z_{2}}_{r_{2}}<\tau ^{z_1}_{r_1}\big ] \le C_{|{\mathcal {E}}|}\Big (\frac{s_1}{|z_1-z_2|\wedge |z_1|}\Big )^{\frac{\alpha }{16n|{\mathcal {E}}|}} \prod _{j=1}^n \prod _{e\in {{\mathcal {E}}}_j} P_e.\nonumber \\ \end{aligned}$$

(A.17)

Here we set $\prod _{e\in {{\mathcal {E}}}_j}P_e=1$ if ${{\mathcal {E}}}_j=\emptyset$. We will finish the proof by proving that $|{{\mathcal {E}}}|\le 2n$ and $\prod _{e\in {{\mathcal {E}}}} P_e\le C_n\frac{P_{y_j}(r_j)}{P_{y_j}(l_j)}$.

We now bound $|{{\mathcal {E}}}|=\sum _{j=1}^n |{{\mathcal {E}}}_j|$. For $1\le m\le n$, we use ${{\mathcal {E}}}^{(m)}_j$, $1\le j\le m$, to denote the set of connected components of the graph $G^{(m)}_j$ obtained by removing the circles in $I_{j,k}$, $j<k\le m$, from $\Xi _j$. Let ${{\mathcal {E}}}^{(m)}=\bigcup _{j=1}^m {{\mathcal {E}}}^{(m)}_j$. Then ${{\mathcal {E}}}={{\mathcal {E}}}^{(n)}$. For $2\le m\le n$, and $1\le j\le m-1$, we may define a map $f_{m}:\bigcup _{j=1}^{m-1}{{\mathcal {E}}}^{(m)}_j \rightarrow {{\mathcal {E}}}^{(m-1)}$ such that for every $e\in {{\mathcal {E}}}^{(m)}_j$, $1\le j\le m-1$, $f_{m}(e)$ is the unique element in ${{\mathcal {E}}}^{(m-1)}_j$ that contains e. Then each $e\in {{\mathcal {E}}}^{(m-1)}$ has at most 2 preimages, and $e\in {{\mathcal {E}}}^{(m-1)}$ has exactly 2 preimages iff $D_m$ is contained in the interior of $A_e$. Since the annuli $A_e$, $e\in {{\mathcal {E}}}^{(m-1)}$, are mutually disjoint, at most one of them has two preimages. Since ${{\mathcal {E}}}^{(m)}_m$ contains only one element, we find that $|{{\mathcal {E}}}^{(m)}|\le |{{\mathcal {E}}}^{(m-1)}|+2$. From $|{{\mathcal {E}}}^{(1)}|=1$ and ${{\mathcal {E}}}={{\mathcal {E}}}^{(n)}$, we get $|{{\mathcal {E}}}|\le 2n-1$.

To estimate $\prod _{e\in {{\mathcal {E}}}} P_e$, we introduce $S_j$ to be the family of pairs of circles $\{\{|z-z_j|=4^s r_j\},\{|z-z_j|=4^{s-1} r_j\}\}$, $s\in \mathbb {N}$. Let $S^{(m)}_j$ denote the set of $e'\in S_j$ such that $A_{e'}\subset \bigcup _{e\in {{\mathcal {E}}}^{(m)}_j} A_e$. Then $\prod _{e\in {{\mathcal {E}}}^{(m)}_j} P_e=\prod _{e'\in S^{(m)}_j} P_{e'}$. Note that, for $m>j$, $A_{e'}$, $e'\in S^{(m)}_j$ can be obtained from $A_{e'}$, $e'\in S^{(m-1)}_j$, by removing the annuli in the latter group that intersects $D_m$. Since $D_m$ can be covered by an annulus centered at $z_j$ with ratio less than 4, it can intersect at most two of $A_{e'}$, $e'\in S_j$. Using Lemma 2.1, we find that $\prod _{e\in {{\mathcal {E}}}^{(m)}_j} P_e\le 4^{2\alpha } \prod _{e\in {{\mathcal {E}}}^{(m-1)}_j} P_e$. Since $l_j\le 4^{h_j+1} r_j$, we get $\prod _{e\in {{\mathcal {E}}}^{(j)}_j} P_e=\frac{P_{y_j}(r_j)}{P_{y_j}(4^{h_j}r_j)}\le 4^\alpha \frac{P_{y_j}(r_j)}{P_{y_j}(l_j)}$. Thus, $\prod _{e\in {{\mathcal {E}}}^{(n)}_j} P_e\le 4^{\alpha (2n-2j+1)} \frac{P_{y_j}(r_j)}{P_{y_j}(l_j)}$, which implies that

$$\begin{aligned} \prod _{e\in {{\mathcal {E}}}^{(n)}} P_e=\prod _{j=1}^n \prod _{e\in {{\mathcal {E}}}^{(n)}_j} P_e \le \prod _{j=1}^n 4^{\alpha (2n-2j+1)}\frac{P_{y_j}(r_j)}{P_{y_j}(l_j)}=4^{\alpha n^2} \prod _{j=1}^n \frac{P_{y_j}(r_j)}{P_{y_j}(l_j)}. \end{aligned}$$

The proof is now complete. $\square$