Regularity and Convergence of Random Curves

Abstract

In this chapter, we study regularity properties curves in the capacity parametrization and their convergence with respect to the uniform norm on compact time intervals.

In this chapter, we study regularity properties curves in the capacity parametrization and their convergence with respect to the uniform norm on compact time intervals. Sometimes the term strong convergence is used for the convergence under the uniform norm and either given or unspecified parametrization.

When we consider random curves, we are interested, in probability theoretic sense, in so called weak convergence of probability measures or equivalently convergence of random curves in distribution with respect to the above topology. Before that subject, we aim to prove Theorem 5.2 in Sect. 6.2. Section 6.1 can be seen as an introduction to Sects. 6.2 and 6.3.

6.1 Continuity Properties of the Loewner Chains

In this section we will switch to the following notation for Loewner chains.

$$\begin{aligned} W(t), \quad \gamma (t), \quad g(t,z), \text { etc.}\quad \text {and} \quad W_n(t), \quad \gamma _n(t), \quad g_n(t,z), \text { etc.} \end{aligned}$$

(6.1)

This allows us to denote, for instance, a sequence of driving terms by $(W_n(t))_{t \in [0,T_n)}$. Let us also use the notation

$$\begin{aligned} F(t,y) = f(t,W(t)+{\mathrm {i}}y) . \end{aligned}$$

(6.2)

The path $y \mapsto F(t,y)$, $y > 0$ is the shortest path in a conformal sense between the point $\infty$ and the “tip” of $K_t$, that is, the points $\lim _{y \rightarrow \infty } F(t,y)$ and $\lim _{y \rightarrow 0} F(t,y)$.

The variable t takes values in [0, T) for $W(t), \gamma (t), g(t,z)$, etc. and in $[0,T_n)$ for $W_n(t), \gamma _n(t), g_n(t,z)$, etc. The variables T and $T_n$ can be finite or infinite. Since we often consider uniform convergence in compact sets of the time variable we often restrict to $t \in [0,T']$ and consider any $T'$ which is finite and less than T or $T_n$.

We study in this section the dependencies of modes of convergence for Loewner chains, Loewner curves and Loewner driving terms. The most important motivation for this is to clarify the relations of different aspects of Loewner chains. In a later section, that insight is used when studying an example case of a random curve converging to SLE$(\kappa )$.

6.1.1 Carathéodory Kernel Convergence

Let us first define what is meant by the convergence of sequences of Loewner chains or simply connected domains.

For a given point $z_0$ and a sequence of simply connected domains $(\varOmega _n)_{n \in \mathbb {Z}_{>0}}$ such that $z_0 \in \varOmega _n \ne \mathbb {C}$, define conformal and onto maps $\phi _n : \mathbb {D}\rightarrow \varOmega _n$ such that $\phi _n(0)=z_0$ and $\phi _n'(0)>0$.

Definition 6.1

We say that $(\varOmega _n)_{n \in \mathbb {Z}_{>0}}$ converges in the Carathéodory sense if the sequence $(\phi _n)_{n \in \mathbb {Z}_{>0}}$ converges uniformly on compact subsets of $\mathbb {D}$.

Remark 6.1

Either $\lim _n \phi _n = const. = w_0$ or $\lim _n \phi _n$ is a conformal map.

Definition 6.2

We say that $( (K_n(t))_{t \in [0,T']} )_{n \in \mathbb {Z}_{>0}}$ converges in the Carathéodory sense if for each compact $G \subset \mathbb {H}$, the sequence $(f_n)_{n \in \mathbb {Z}_{>0}}$ converges uniformly on $[0,T'] \times G$.

The Carathéodory convergence is convenient since it depends on conformal maps, which we can estimate in many ways. The next theorem makes the definition more concrete by giving an equivalent geometric description. See [9] for the proof.

Theorem 6.1

(Carathéodory kernel theorem) The locally uniform convergence of $(\phi _n)_{n \in \mathbb {Z}_{>0}}$ is equivalent to the kernel convergence of $U_n \rightarrow U$ as $n \rightarrow \infty$ with respect to $w_0$ in the sense that

• either $U=\{w_0\}$ or U is a domain $\ne \mathbb {C}$ with $w_0 \in U$ such that every $w \in U$ has a neighborhood that lies in $U_n$ for large enough n,

• for each $w \in \partial U$ there exists a sequence $w_n \in \partial U_n$ such that $w_n \rightarrow w$ as $n \rightarrow \infty$.

6.1.2 Continuity Properties of the Mappings $W \mapsto F$ and $W \mapsto g$

We first present auxiliary lemmas. The following estimate appears for instance in [5].

Lemma 6.1

For each $\delta >0$ and $T>0$ there exists a constant $C(T,\delta )$ such that the following holds. Let $h_k(t,z)$, $k=1,2$ be the solutions of (4.19) with the continuous driving terms $(W_k(t))_{t \in [0,T]}$, $k=1,2$, respectively. Then they satisfy

$$\begin{aligned} |h_1 (T,z_1) - h_2 (T,z_2)| \le C(T,\delta ) ( \Vert W_1 - W_2 \Vert _{\infty ,[0,T]} + |z_1 - z_2| ) \end{aligned}$$

(6.3)

for any $z_1, z_2$ such that $\mathrm{{Im}}z_k> \delta > 0$.

Proof

Fix $\delta >0$, $T>0$ and $z_k \in \mathbb {H}$, $k=1,2$, such that $\mathrm{{Im}}z_k > \delta$, $k=1,2$. Let $h_k(t,z)$, $k=1,2$ be the solutions of (4.19) with the continuous driving terms $(W_k(t))_{t \in [0,T]}$, which we also consider to be fixed. Write $\psi (t)= h_1 (t,z_1) - h_2 (t,z_2)$. Then

$$\begin{aligned} \partial _t \psi (t) = \zeta (t) \, ( \psi (t) - D(t) ) \end{aligned}$$

where $\zeta (t) = 2/( (h_1 (t,z_1)- W_1(t))(h_2 (t,z_2)- W_2(t)))$ and $D(t) = W_1(t) - W_2(t)$.

We can write $\partial _t \left( e^{ - \int _0^t \zeta (s) \, {\mathrm {d}}s }\, \psi (t) \right) = - \zeta (t) e^{ - \int _0^t \zeta (s) \, {\mathrm {d}}s } \, D(t)$ using an integrating factor. Hence

$$\begin{aligned} \psi (t) = e^{ \int _0^t \zeta (s) \, {\mathrm {d}}s }\, \psi (0) - \int _0^t \zeta (u) e^{ \int _u^t \zeta (s) \, {\mathrm {d}}s } \, D(u) {\mathrm {d}}u . \end{aligned}$$

We find using $| e^{ \int _0^t \zeta (s) \, {\mathrm {d}}s } | \le e^{ \int _0^t \left| \zeta (s)\right| \, {\mathrm {d}}s }$ that

$$\begin{aligned} \left| \int _0^t \zeta (u) e^{ \int _u^t \zeta (s) {\mathrm {d}}s } \, D(u) {\mathrm {d}}u \right|&\le \Vert D \Vert _{\infty ,[0,T]} \int _0^t |\zeta (u)| e^{ \int _0^u \left| \zeta (s)\right| {\mathrm {d}}s } {\mathrm {d}}u \nonumber \\&= \Vert D \Vert _{\infty ,[0,T]} \left( e^{ \int _0^t \left| \zeta (s)\right| \, {\mathrm {d}}s } - 1\right) . \end{aligned}$$

By the Cauchy–Schwarz inequality

$$\begin{aligned} \int _0^t \left| \zeta (s)\right| {\mathrm {d}}s \le \sqrt{I_1 I_2} \end{aligned}$$

where $I_k = \int _0^t \frac{2 {\mathrm {d}}s}{|h_k(t,z_k) - W_k(t)|^2}$. On the other hand

$$\begin{aligned} \partial _t \log \mathrm{{Im}}h_k(t,z_k) = \frac{2 {\mathrm {d}}s}{|h_k(t,z_k) - W_k(t)|^2} \end{aligned}$$

and therefore

$$\begin{aligned} I_k = \log \frac{ \mathrm{{Im}}h_k(t,z_k) }{y} \le \log \frac{\sqrt{y^2 + 4 t}}{y}. \end{aligned}$$

Here we used the upper bound $\partial _t \mathrm{{Im}}h_k(t,z_k) \le 2 ( \mathrm{{Im}}h_k(t,z_k))^{-1}$ to derive an upper bound for $\mathrm{{Im}}h_k(t,z_k)$.

Thus easily from the bounds above, we establish the bound

$$\begin{aligned} |\psi (t)| \le \frac{\sqrt{\delta ^2 + 4 t}}{\delta }\, |\psi (0)| + \left( \frac{\sqrt{\delta ^2 + 4 t}}{\delta } - 1\right) \, \Vert D \Vert _{\infty ,[0,T]} . \end{aligned}$$

This gives the claim. $\square$

A similar result for the forward Loewner equation is the following.

Lemma 6.2

For each $\delta >0$ and $T>0$ there exists a constant $C(T,\delta )$ such that the following holds. Let $g_k(t,z)$, $k=1,2$ be the solutions of the Loewner equation (4.14) with the continuous driving terms $(W_k(t))_{t \in [0,T]}$, $k=1,2$, respectively. Then they satisfy

$$\begin{aligned} |g_1 (T,z_1) - g_2 (T,z_2)| \le C(T,\delta ) ( \Vert W_1 - W_2 \Vert _{\infty ,[0,T]} + |z_1 - z_2| ) \end{aligned}$$

(6.4)

for any $z_1, z_2$ such that $\mathrm{{Im}}g_k(T,z_k)> \delta > 0$.

Proof

The proof is similar to the proof of Lemma 6.1. The only difference is that we replace $\psi (t)$ by $\psi (t)=g_1(t,z_1) -g_2(t,z_1)$ and $\zeta (t)$ by $\zeta (t)=-2/( (g_1 (t,z_1)- W_1(t))(g_2 (t,z_2)- W_2(t)))$. Then $I_k$ is given as and bounded by $I_k = \int _0^t 2 |g_k (s,z_k)- W_k(s)|^{-2} {\mathrm {d}}s \le \log \frac{\mathrm{{Im}}z_k}{\mathrm{{Im}}g_k(t,z_k)} \le \log \frac{\mathrm{{Im}}z_k}{\max \{\delta , \sqrt{ ( (\mathrm{{Im}}z_k)^2 - 4t)^+ } \} }$ where $a^+ = \max \{ a, 0\}$. $\square$

The following results establish continuous dependency of solutions of Loewner equations on the driving term (i.e. continuity of the mappings $W \mapsto f$ and $W \mapsto g$).

Proposition 6.1

The mapping $W \mapsto f$ is continuous with respect to the convergence in the Carathéodory sense. More specifically, for any compact $G \subset \mathbb {H}$, there exists a constant $C>0$ such that if $f_k$, $k = 1,2$, are two (inverse) Loewner chains, then

$$\begin{aligned} \Vert f_1 - f_2 \Vert _{\infty , [0,T] \times G} \le C \Vert W_1 - W_2 \Vert _{\infty ,[0,T]} . \end{aligned}$$

(6.5)

Proof

The claim follows directly from Lemma 6.1 and from the fact that $h_k$ and $f_k$ are related a time reversal, see Lemma 4.10. $\square$

A similar result for the (direct) Loewner maps is the following.

Proposition 6.2

Let $K_0$ be a hull and $G \subset \mathbb {H}\setminus K_0$ be a compact set. Then there exists a constant $C>0$ such that if if $g_k$, $k = 1,2$, are two Loewner chains such that $K_k(T) \subset K_0$ for $k=1,2$, then

$$\begin{aligned} \Vert g_1 - g_2 \Vert _{\infty , [0,T] \times G} \le C \Vert W_1 - W_2 \Vert _{\infty ,[0,T]} . \end{aligned}$$

(6.6)

Proof

The claim follows directly from Lemma 6.2. $\square$

6.1.3 Continuity Properties of the Mapping $\gamma \mapsto G$

The mapping from $(\gamma (t))_{t \in [0,T']}$ to $g(T',\cdot )$ is continuous by Theorem 6.1 with respect to the convergence in the Carathéodory sense. It is easy to extend this to continuity of the mapping $(\gamma (t))_{t \in [0,T']}$ to $(g(t,\cdot ))_{t \in [0,T']}$ the convergence in the Carathéodory sense. For example, this can be using an compactness argument in the following way.

Make a counter assumption that there are $\gamma$, $\gamma _n$, $t_n \in [0,T']$ and $z_n \in G$, where G is a compact set as in Definition 6.2, such that $\gamma _n$ converges to $\gamma$ uniformly on $[0,T']$ but $\liminf _n |g_n(t_n,z_n) - g(t_n,z_n)| =\mathrel {\mathop :}\varepsilon >0$. Using compactness we can suppose that $t_n$ and $z_n$ converge to $t \in [0,T']$ and z, respectively, as $n \rightarrow \infty$ and then again by Theorem 6.1 $g_n(t_n,\cdot )$ converges to $g(t,\cdot )$ and $g(t_n,\cdot )$ converges to $g(t,\cdot )$ as $n \rightarrow \infty$. Therefore $\lim _n |g_n(t_n,z_n) - g(t_n,z_n)| = 0$ (along a subsequence), which leads to a contradiction.

Proposition 6.3

The mapping $\gamma \mapsto g$ is continuous with respect to the convergence in the Carathéodory sense.¹

The next corollary follows when this proposition is combined with the continuous dependency of the solution of the Loewner equation on the driving term.

Corollary 6.1

Suppose that $\gamma _n$ is a sequence of curves and $W_n$ are their driving terms. If $\gamma _n$ tends to $\gamma$ and $W_n$ tends to W uniformly on [0, T] as n tends to $\infty$, then $\gamma$ is driven by W, that is, the Loewner chain g of $\gamma$ satisfies the Loewner equation with the driving term W.

6.1.4 Continuity Properties of the Mapping $\gamma \mapsto W$

Below we will study the continuity properties of the mapping $(\gamma (t))_{t \in [0,T)} \mapsto (W_t)_{t \in [0,T)}$.

Theorem 6.2

Let $X\subset C([0,T),\mathbb {C})$ be the set of curves in the upper half-plane that generate a Loewner chain. The mapping $(\gamma (t))_{t \in [0,T)} \mapsto (W_t)_{t \in [0,T)}$ from X to $C([0,T),\mathbb {R})$ is continuous. However, it is not uniformly continuous.

Proof

Both the domain and the range of the map $(\gamma (t))_{t \in [0,T)} \mapsto (W_t)_{t \in [0,T)}$ are metrizable topological spaces. Hence it is sufficient to consider $T' \in (0,T)$ and a sequence $\gamma _n$ that converges to $\gamma$ in the uniform norm on $[0,T']$ and to establish that the corresponding driving terms $W_n$ of $\gamma _n$ converge to a limit W in the uniform norm on $[0,T']$ and that $\gamma$ is driven by W.

Since $\gamma _n$ tends to $\gamma$, the family $\gamma _n$ is equicontinuous on $[0,T']$. By the same argument as in the proof of Theorem 4.2 (specifically the proof that the first statement implies the second one) the family $W_n$ is equicontinuous on $[0,T']$. It is also uniformly bounded and we can apply Arzelà–Ascoli theorem to find a converging subsequence $W_{n_i}$. Denote its limit by W. By taking a sequence $T'_n$ that increases to T, we can suppose that $W_{n_i}$ converges to W uniformly on any $[0,T'] \subset [0,T)$. By Corollary 6.1, $\gamma$ is driven by W.

To prove the first claim, we need to show that $W_n$ converges to W. Assume the contrary and suppose that there is a subsequence of $W_n$ that stays at a positive distance away from W in the uniform norm on $[0,T']$, for some $T' \in (0,T)$. By the same argument as above we can find a subsequence of that sequence such that it converges to some $W_1$. By Corollary 6.1, $\gamma$ is driven by $W_1$ which leads to a contradiction since $W_1 \ne W$ by the assumption we made. Thus we have shown the first claim.

To prove the second claim let $\gamma _n$ be the broken line 0, ${\mathrm {i}}$, ${\mathrm {i}}+1/n$, $(1/2) \, {\mathrm {i}}+ 1/n$ parametrized with the half-plane capacity and let $\tilde{\gamma }_n(t) = -\overline{\gamma _n(t)}$, i.e., the reflection with respect to the y-axis. Let the corresponding driving terms be $W_n$ and $\tilde{W}_n$. Let then by a simple argument using, say, harmonic measure there exists $c>0$ and $T'$ such that the capacity of $\gamma _n$ and $\tilde{\gamma }_n$ is greater than $T'$ and $W_n(T')>c$ and $\tilde{W}_n(T')<-c$. Hence $\Vert \gamma _n - \tilde{\gamma }_n \Vert _{\infty ,[0,T']} = 2/n$ and $\Vert W_n - \tilde{W}_n \Vert _{\infty ,[0,T']} > 2c$. This shows that the map $(\gamma (t))_{t \in [0,T)} \mapsto (W_t)_{t \in [0,T)}$ is not uniformly continuous. $\square$

Remark 6.2

To get uniform continuity, we need to improve the topology of curves, say, by keeping track of the harmonic measures of the left-hand and right-hand sides of $\mathbb {R}\cup \gamma [0,t]$. We don’t try to formulate the topology here, but we wanted to mention that it can be done in principle.

6.1.5 Continuity Properties of the Mapping $W \mapsto \gamma$

Lemma 6.3

The mapping $(W_t)_{t \in [0,T)} \mapsto (\gamma (t))_{t \in [0,T)}$ is not continuous.

For the proof see Fig. 6.1 (See also [7], Example 4.49 where the example is originally from.) where an example of a sequence of curves is given such that their driving terms converge uniformly to a constant,² while the sequence of curves doesn’t have any subsequence that would converge. A similar example of an obstruction to the convergence of $\gamma _n$ is related to the tip being hidden from infinity during a non-trivial interval. Consider a path $\gamma _n$ of Fig. 6.2. Then it is quite standard using, say, harmonic measure, to notice that $W_n$ converges to some W while $\gamma _n$ doesn’t converge uniformly in the capacity parametrization.

Fig. 6.1

The proof of Lemma 6.3 by a picture: consider a sequence of curves $\gamma _n$ which all are broken lines. The corners of $\gamma _n$ lie on three lines $\mathrm{{Re}}z = -\frac{1}{n}, 0 , \frac{1}{n}$ which are the three dashed lines in the picture. On $\mathrm{{Re}}z = \pm \frac{1}{n}$, the corners are at height $\mathrm{{Im}} z = \frac{1}{n}, \frac{2}{n}, \frac{3}{n}, \ldots$ and on $\mathrm{{Re}}z =0$, they are at height $\mathrm{{Im}} z = 0,\frac{1}{2n}, \frac{3}{4n}, \frac{1}{n}, \frac{5}{4n}, \frac{3}{2n} \ldots$ These points are connected by line segments in the order indicated in the picture ($\mathrm{{Re}}z= 0 , -\frac{1}{n}, 0 , \frac{1}{n}, 0 , -\frac{1}{n},\ldots$ and the height is ordered on each line). The tip of the curve at time t is indicated by the circle in the picture. The value of $W_n(t)$ is between the images of the triangles $z_\text {left}$, $z_\text {right}$ under $g(t,\cdot )$. Concentrate on the rightmost triangle $z_\text {right}$. Let L be the broken line $-\frac{1}{n}, z_\text {left}, z_\text {right}$. The value of $g(t,z_\text {right}) =g_{K_t}(z_\text {right})$ increases if we add to $K_t$ the path L and all the points disconnected from infinity as a consequence. In a similar manner, removing “matter” from the right-hand side of L increases $g_{K}(z_\text {right})$. Therefore $W_t$ is at most $g_{L}(z_\text {right})$, which can be estimated using the explicit Loewner map of the vertical slit to be at most $C \max \{t^{1/4} n^{-1/2}, n^{-1}\}$ where $C>0$ is some constant. By symmetry we have shown that $|W_n(t)| \le C \max \{t^{1/4} n^{-1/2}, n^{-1}\}$ and hence $W_n$ converges uniformly to zero. We leave as an exercise to verify the fact that $\gamma _n$ doesn’t contain any subsequence that would converge uniformly on any [0, T], $T>0$

Fig. 6.2

The solid horizontal line in the bottom of the picture is the real axis and the solid broken line is $\gamma _n$. Suppose that the vertices are $z_0,z_1,\ldots ,z_8$ and that the other vertices are fixed except that $z_4$ and $z_7$ converge towards $z_1$ as $n \rightarrow \infty$ (along the dashed lines)

Based on the observation of the previous lemma, we need to consider more restrictive class of curves. Denote by F the function

$$\begin{aligned} F(t,y)= f(t, W_t + y{\mathrm {i}}) . \end{aligned}$$

(6.7)

For any $T' \in \mathbb {R}_{>0}$, $\delta _0>0$ and any function $\lambda : (0,\delta _0] \rightarrow \mathbb {R}_{\ge 0}$ such that $\lim _{y \rightarrow 0}\lambda (y)=0$, define

$$\begin{aligned} \mathscr {E}_{\lambda ,T',\delta _0} = \left\{ W \in C([0,T]) \,:\, \begin{array}{c} W \text {drives a curve }\gamma \text { and } \\ |F(t,y)- \gamma (t)|\le \lambda (y) \text { for all } (t,y)\in [0,T']\times (0,\delta _0] \end{array} \right\} . \end{aligned}$$

By Theorem 6.4 below, the uniform convergence of the mappings $t \mapsto F_{t}(y)$ as y tends to zero is sufficient and necessary for the Loewner chain to be driven by a curve.

Lemma 6.4

For each $T',\delta$ there exists $C(T',\delta )$ such that the following holds. If $W_1,W_2 \in \mathscr {E}_{\lambda ,T',\delta _0}$ for some $\lambda : [0,\delta _0] \rightarrow \mathbb {R}_{\ge 0}$, then

$$\begin{aligned} \Vert \gamma _1 - \gamma _2 \Vert _{\infty , [0,T']} \le C(T',\delta ) \Vert W_1 - W_2 \Vert _{\infty , [0,T']} + 2 \lambda (\delta ) . \end{aligned}$$

(6.8)

Proof

Write using the triangle inequality that

$$\begin{aligned} | \gamma _1(t) - \gamma _2(t) | \le |F_1(t,\delta ) -F_2(t,\delta )| + |F_1(t,\delta ) - \gamma _1(t)| + |F_2(t,\delta )- \gamma _2(t)| . \end{aligned}$$

(6.9)

By the assumption, when $t \in [0,T']$, the second and third terms on the right-hand side are bounded by $\lambda (\delta )$. Notice that $F_k(t,\delta )=f_k(t, z_k(t))$ where $z_k(t) = W_k(t) + {\mathrm {i}}\delta$. Hence by Lemma 6.1 there exists an explicit constant $C(T',\delta )$ such that the first term on the right-hand side of the above inequality is bounded by $C(T',\delta ) \Vert W_1 - W_2 \Vert _{\infty , [0,T']}$. $\square$

Proposition 6.4

Let $\mathscr {E} \subset C([0,T))$ be a collection of driving terms of curves such that for each $T \in (0,T')$ there exists $\delta _0$ and $\lambda$ as above such that any $W \in \mathscr {E}$ when restricted to the time interval $[0,T')$ belongs to $\mathscr {E}_{\lambda ,T',\delta _0}$. Then the mapping $(W(t))_{t \in [0,T)} \mapsto (\gamma (t))_{t \in [0,T)}$ is uniformly continuous in $\mathscr {E}$ with respect to the topology of local uniform convergence for $(W(t))_{t \in [0,T)}$ and $(\gamma (t))_{t \in [0,T)}$.

Proof

Let $T' \in (0,T)$ and let $\lambda$ and $\delta _0$ be such that any $W \in \mathscr {E}$ belongs to $\mathscr {E}_{\lambda ,T',\delta _0}$.

Let $\varepsilon >0$. Take $\delta _1 \in (0,\delta _0]$ such that $4\lambda (\delta _1) \le \varepsilon$. Then take $\delta = \varepsilon /(2 C(T',\delta _1))$ where $C(T',\delta _1)$ is as in Lemma 6.4. Now by Lemma 6.4, $\Vert \gamma _1 - \gamma _2 \Vert _{\infty ,[0,T']} \le \varepsilon$ for any $W_1,W_2 \in \mathscr {E}$ such that $\Vert W_1 - W_2 \Vert _{\infty ,[0,T']} \le \delta$. $\square$

6.1.5.1 The Modulus of Continuity for $\gamma$ Driven by $W \in \mathscr {E}_{\lambda ,T',\delta _0}$

Using and extending the idea of Lemma 6.4, it is possible to prove the following result on the modulus of continuity of the curve driving a Loewner chain. The proof is omitted here.

Theorem 6.3

For any $T', \delta _0, \lambda$ as above and any function $\psi :(0,1] \rightarrow (0,\infty )$ such that $\lim _{\delta \rightarrow 0} \psi (\delta ) = 0$, there exists $\phi$ such that $\lim _{\delta \rightarrow 0} \phi (\delta ) = 0$ and the following holds. If $W \in \mathscr {E}_{\lambda ,T',\delta _0}$ and $|W(t) - W(s)| \le \psi (|t-s|)$, then $\gamma$ satisfies

$$\begin{aligned} |\gamma (t) - \gamma (s)| \le \phi (|t-s|) . \end{aligned}$$

(6.10)

6.2 Continuity of SLE$(\kappa )$

6.2.1 Existence of the Trace for Loewner Chains

The following lemma shows that a Loewner chain is tame at certain time instances, in the sense that $\lim _{y \rightarrow 0} F_t ( y)$ exists.

Lemma 6.5

Let $z_0 \in \overline{\mathbb {H}} \setminus \{ W_0 \}$, $0< r < |z_0 - W_0|$ and $B = B(z_0,r) \cap \mathbb {H}$. Suppose $t>0$ is such that $K_t \cap \overline{B}$ is non-empty and $K_s \cap \overline{B}$ is empty for each $s \in [0,t)$. Suppose also that $\overline{B} \setminus K_t$ is non-empty, that is, $\overline{B}$ is hit, but not swallowed by $K_t$. Then there exists $z_1 \in \partial B$ such that $K_t \cap \overline{B} = \{z_1\}$ and moreover, $z_1 = \lim _{y \rightarrow 0} F_t ( y)$.

Proof

Let $z_0,r,B$ and t be as stated above. Then by the assumptions $(\partial K_t) \cap \overline{B} \ne \emptyset$. Let $z_1 \in (\partial K_t) \cap \overline{B}$. By the Carathéodory kernel convergence theorem (Theorem 6.1) there exists $w_n \in \partial K_{s_n}$ such that $s_n$ increases to t and $w_n$ tends to $z_1$ as $n \rightarrow \infty$. Since $w_n \in \mathbb {C}\setminus \overline{B}$ for all n, it follows that $z_1 \in \partial B$. Hence $(K_t \cap \overline{B}) \subset \partial B$.

Since the line segment from $z_1$ to $z_0$ lies in $H_t$ except the endpoint $z_1$, the end point $z_1$ is accessible,³ see [9] Sect. 2.5 Exercise 5, and thus there exists $x \in \mathbb {R}$ such that $z_1 = \lim _{y \rightarrow 0} f_t(x+{\mathrm {i}}y)$. From the facts that $\sup \{ |z - W_{t-\delta }| \,:\, z\in g_{t-\delta }( K_t \setminus K_{t-\delta })\} = o(1)$ as $\delta \rightarrow 0$, see Theorem 4.2, and that $s \mapsto W_s$ is continuous, it follows that $x=W_s$. $\square$

Let $H \subset \mathbb {H}$. Denote by $\partial _+ H$ the set points, which have the property that every neighborhood (in $\mathbb {C}$) of the point intersects both H and $\mathbb {H}\setminus H$. In other words, $\partial _+ H$ is the boundary of H in $\mathbb {H}$.

Remember that $(K_t)_{t \in [0,T)}$ is generated by a curve $\gamma :[0,T) \rightarrow \mathbb {C}$, if $H_t$ is the unbounded component of $\mathbb {H}\setminus \gamma [0,t]$. The next result is a basic tool to verify that a Loewner chain is generated by a curve.

Theorem 6.4

If $t \mapsto F_{t}(y)$ converges to some $\gamma$ uniformly on compact subsets of [0, T) as $y>0$ tends to 0, then $\gamma$ is a continuous curve and $(K_t)_{t \in [0,T)}$ is generated by $\gamma$. Furthermore for each $t \in [0,T)$, the map $z \mapsto f_t(z)$ extends continuously to $\overline{\mathbb {H}}$.

Proof

For each fixed $y>0$, the map $t \mapsto F_{t}(y)$ is continuous by Lemma 6.1. Hence the uniform convergence of $F_{t}(y) \rightarrow \gamma (t)$ as $y \rightarrow 0$ on compact subsets of [0, T) implies that $\gamma : [0,T) \rightarrow \mathbb {C}$ is continuous.

It remains to show that for each $t \in [0,T)$, $H_t$ is the unbounded component of $\mathbb {H}\setminus (\gamma [0,t])$. Since $\gamma (s) \in \partial _+ H_s$, it follows that $\gamma [0,t] \subset \bigcup _{s \in [0,t]} \partial _+ H_s$. Hence it is sufficient to show that $\partial _+ H_t \subset \gamma [0,t]$.

Let $z_0 \in \partial _+ H_t$. If $z_0=W_0$, then clearly $z_0 \in \gamma [0,t]$. Suppose then that $z_0 \ne W_0$ and take any $\varepsilon \in (0,|z_0 - W_0|)$. Let

$$\begin{aligned} t_\varepsilon = \inf \left\{ s \in \mathbb {R}_{\ge 0}\,:\, K_s \cap \overline{B(z_0,\varepsilon )} \ne \emptyset \right\} . \end{aligned}$$

Then $0< t_\varepsilon < t$. Since $z_0 \in \partial _+ H_t$, the set $\overline{\mathbb {H}\cap B(z_0,\varepsilon )} \setminus K_{t_\varepsilon }$ is non-empty. By Lemma 6.5, $|z_0 - \gamma (t_\varepsilon )|=\varepsilon$. Therefore $z_0 = \lim _{\varepsilon \rightarrow 0} \gamma (t_\varepsilon )$ and consequently $z_0 \in \overline{\gamma [0,t)} = \gamma [0,t]$. Thus $\partial _+ H_t \subset \gamma [0,t]$ as was claimed. The set $H_t$ is therefore the unbounded component of $\mathbb {H}\setminus \gamma [0,t]$ and since $\partial H_t$ is locally connected $f_t$ extends continuously to $\overline{\mathbb {H}}$ as claimed. $\square$

Let’s formulate the following corollary to stress the importance of the result.

Corollary 6.2

A Loewner chain $(K_t)_{t \in [0,T)}$ is generated by a curve if and only if for each $T' \in [0,T)$, there exists a function $\lambda : (0,1] \rightarrow \mathbb {R}_{\ge 0}$ such that $\lim _{y \rightarrow 0} \lambda (y) = 0$ and

$$\begin{aligned} | F_t(y_1) - F_t(y_2) | \le \lambda (y) \end{aligned}$$

(6.11)

for all $y \in (0,1]$, $y_1,y_2 \in (0,y]$ and $t \in [0,T']$.

6.2.2 Auxiliary Results on Conformal Maps

The next result is a version of Koebe distortion theorem in $\mathbb {H}$. The proof, which is straightforward, is given in Appendix D. See also [8, 12].

Lemma 6.6

(Koebe distortion in $\mathbb {H}$) There exists a constant C such that for any $y>0$, $s \in [\frac{1}{2},2]$, $x \in \mathbb {R}$ and any conformal map $f: \mathbb {H}\rightarrow \mathbb {C}$,

$$\begin{aligned}&C^{-1} |f'({\mathrm {i}}y)| \le |f'({\mathrm {i}}s y)| \le C |f'({\mathrm {i}}y)| \end{aligned}$$

(6.12)

$$\begin{aligned} C^{-1} (1+x^2)^{-3} |&f'({\mathrm {i}}y)| \le |f'(y(x + {\mathrm {i}}))| \le C (1+x^2)^3|f'({\mathrm {i}}y)|. \end{aligned}$$

(6.13)

The next result is based on the Loewner equation and thus the proof is given here.

Lemma 6.7

There exists a constant C such that for any solution $f_t$ of the Loewner equation for the inverse Loewner map and for any $x + {\mathrm {i}}y \in \mathbb {H}$, ${t \in \mathbb {R}_{\ge 0}}$ and $s \in [0,y^2]$

$$\begin{aligned} C^{-1} \, |f_t'(x+ {\mathrm {i}}y)|&\ \le |f_{t+s}'(x + {\mathrm {i}}y) | \le C \, |f_t'(x+{\mathrm {i}}y)| \end{aligned}$$

(6.14)

$$\begin{aligned} |f_{t+s}(x + {\mathrm {i}}y)&- f_{t} ( x + {\mathrm {i}}y)| \le C \, y \, |f_t'(x+{\mathrm {i}}y)| . \end{aligned}$$

(6.15)

Proof

By differentiating the Loewner equation and using the triangle inequality and the inequality $|x + {\mathrm {i}}y- W_t| \ge y$, it follows that

$$\begin{aligned} |\partial _t f_t'(x + {\mathrm {i}}y) | \le \frac{2|f_t''(x+{\mathrm {i}}y)|}{y} + \frac{2|f_t'(x+{\mathrm {i}}y)|}{y^2} . \end{aligned}$$

To estimate $|f_t''(z)|$, for fixed $z = x +{\mathrm {i}}y \in \mathbb {H}$, let $\phi (\zeta ) = x + {\mathrm {i}}y \frac{1-\zeta }{1+\zeta }$. Then $\phi$ is a Möbius map from $\mathbb {D}$ onto $\mathbb {H}$ and it has expansion $\phi (\zeta ) = x + {\mathrm {i}}y ( 1 + 2 \sum _{n=1}^\infty (-1)^n \zeta ^n )$. Thus $\phi (0)=z$, $\phi '(0)=-2{\mathrm {i}}y$ and $\phi ''(0)=4{\mathrm {i}}y$.

The function $(f_t \circ \phi (\zeta ) - f_t(z))/(f_t'(z) \phi '(0))$ has expansion

$$\begin{aligned} \zeta + \frac{f_t''(z) (\phi '(0))^2 + f_t'(z) \phi ''(0)}{2 f_t'(z) \phi '(0)} \zeta ^2 + \cdots . \end{aligned}$$

around $\zeta =0$. Using (3.9), it follows that $|f_t''(z) | |\phi '(0)|^2 \le | f_t'(z)| (|\phi ''(0)| + 4 |\phi '(0)|)$ and thus $|f_t''(z) | \le 6 \,| f_t'(z)|\, y^{-1}$. Combining this with the above estimate gives $|\partial _t f_t'(x + {\mathrm {i}}y) | \le 14|f_t'(x+{\mathrm {i}}y)| \, y^{-2}$. Thus $|\partial _t \log f_t'(x + {\mathrm {i}}y) | \le 14\, y^{-2}$ and hence

$$\begin{aligned} - \frac{14}{y^2} \le \partial _t \log |f_t'(x + {\mathrm {i}}y)| \le \frac{14}{y^2} \end{aligned}$$

where we used the inequality $-|z| \le \mathrm{{Re}}z \le |z|$.

By integrating this inequality with respect to t, we get the first claim easily. The second claim is derived from the first one by plugging it in to the Loewner equation, which is then integrated with respect to t. This gives an upper bound which is proportional to $|f'_t(x+{\mathrm {i}}y)| \frac{s}{y} \le |f'_t(x+{\mathrm {i}}y)| y$. $\square$

6.2.3 Proof of Theorem 5.2

Definition 6.3

An increasing, continuous function $\psi : [0,\infty ) \rightarrow (0,\infty )$ is said to be a subpower function if $\lim _{x \rightarrow \infty } \frac{\log \psi (x)}{\log x} = 0$ or equivalently if $\lim _{x \rightarrow \infty } x^{-\mu } \psi (x) = 0$ for all $\mu > 0$.

Remark 6.3

One way to write this is $\psi (x) = e^{o(\log x)}$. If $\psi _1$ and $\psi _2$ are subpower functions also $\psi _1 \psi _2$, $\psi _1 + \psi _2$ and $\psi (x)=\psi _1(x^p)$, $p>0$, are subpower functions.

Proof

(Proof of Theorem 5.2) By Theorem 6.4, it is enough to prove that the functions $t \mapsto f_t(W_t + {\mathrm {i}}y)$ converges uniformly as y tends to 0.

Our goal is to prove this based on the following bounds: As we saw above for each $\kappa \ne 8$, there exist a constant $\theta >0$ and a random variable C which is almost surely finite such that

$$\begin{aligned} |\tilde{f}_t'(i 2^{-n})| \le C \, 2^{n(1-\theta )} \end{aligned}$$

(6.16)

for all $t \in \mathscr {D}_{2n}$ and for any $n \in \mathbb {N}$. Remember also that since $(W_t)_{t \in \mathbb {R}_{\ge 0}}$ is a Brownian motion, there is an almost surely finite random variable $\tilde{C}$ such that

$$\begin{aligned} |W_{t+s} - W_t| \le \tilde{C} \sqrt{s \, \log (1/s)} \end{aligned}$$

(6.17)

for any $t,s \in [0,1]$. Fix a realization of the driving process and the Loewner chain such that the bounds (6.16) and (6.17) hold for some finite C and $\tilde{C}$.

Let $t \in [0,1],y \in (0,1)$. Take $n \in \mathbb {N}$ and $t_0 \in \mathscr {D}_{2n}$ such that

$$\begin{aligned} 2^{-n} \le y< 2^{-n+1}, \qquad t_0 \le t < t_0 + 2^{-2n} , \end{aligned}$$

that is, $n = \lceil \log _2 (1/y) \rceil$ and $t_0 = \lfloor t 2^{2n} \rfloor 2^{-2n}$. By (6.16), (6.17) and Lemmas 6.6 and 6.7, it follows that

$$\begin{aligned} |\tilde{f}_t'(i y)|&= |f_t'(W_t + i y)| \le c|f_{t_0}'(W_t + i y)| \nonumber \\&\le c|f_{t_0}'(W_t + i y_0)| \le c \left( 1 + \frac{|W_t - W_{t_0}|^2}{y_0^2} \right) ^3 |f_{t_0}'(W_{t_0} + i y)| \nonumber \\&\le c n^r 2^{n(1-\theta )} \le y^{\theta -1} \psi (1/y) \end{aligned}$$

(6.18)

for some subpower function $\psi$. Here c is a generic constant that might change from line to line .

Let’s integrate the bound (6.18). For any $0< y_1< y_2 \le y < 1$, by the triangle inequality and a change of integration variable

$$\begin{aligned} |\tilde{f}_t ({\mathrm {i}}y_2) - \tilde{f}_t ({\mathrm {i}}y_1)| \le \int _{y_1}^{y_2} | \tilde{f}_t'({\mathrm {i}}u)| \, {\mathrm {d}}u \le \int _0^y u^{\theta -1} \psi (1/u) \, {\mathrm {d}}u = y^{\theta } \tilde{\psi }(1/y) \end{aligned}$$

where $\tilde{\psi } (x) = \int _0^1 u^{\theta -1} \psi (x/u) \, {\mathrm {d}}u$. It is not difficult to check that $\tilde{\psi }$ is a subpower function. Hence $\gamma (t) = \lim _{y \searrow 0} \tilde{f}_t ({\mathrm {i}}y)$ exists and satisfies

$$\begin{aligned} |\gamma (t) - \tilde{f}_t ({\mathrm {i}}y)| \le y^\theta \tilde{\psi }(1/y) . \end{aligned}$$

(6.19)

Now by Theorem 6.4, $\gamma$ continuous and generates $(K_t)_{t \in \mathbb {R}_{\ge 0}}$. $\square$

It turns out that the following more quantitative result holds.

Proposition 6.5

For each $\kappa \ne 8$, there exist a constant $\alpha _0>0$ such that $t \mapsto \gamma (t)$ is Hölder continuous for any exponent $\alpha < \alpha _0$

Remark 6.4

We can choose $\alpha _0 = \theta _0/2$ where $\theta _0$ is as in Remark 5.12.

6.3 Convergence of Interfaces in the Site Percolation Model

In this section, we apply the ideas of Sect. 6.1 to the convergence of random curves to SLEs in a specific example case of the site percolation model on triangular lattice. Most of the proofs are based on estimates established for this discrete model. We spend large portion of the time establishing the set of sufficient estimates of [6] for the convergence of capacity parameterized curves in the present example case.

6.3.1 Definition of the Site Percolation Model

A graph $\ G$ is a pair $G=(V,E)$ where V is a finite set and $E \subset \{ \{v,w\} \subset V \,:\, v\ne w\}$. An element $v \in V$ is called a vertex or a site and an element $e \in E$ is called an edge or a link . Two vertices connected by an edge are said to be neighbors on the graph, and the number of neighbors is called the degree of the vertex. All the graphs we are going to consider are planar , that is, they come together with an embedding to the plane so that the vertices are distinct points and edges are represented by simple paths which connect the vertices and which are pair-wise disjoint.

The complement (as a planar point set) of the embedded graph consist of a finite number of components, that we can interpret as polygons whose edges and vertices are edges and vertices of the graph. (The face of) Such a polygon is said to be a face of the graph. The centers (or otherwise chosen points, one for each face) of the faces form a finite set $V^*$. The set $E^*$ is defined to be edges that connect two centers v, w of faces if and only if the faces are adjacent, i.e., the distinct faces share at least one edge. The graph $G^* = (V^*, E^*)$ is called the dual graph of G . We can arrange so that there is one-to-one correspondence between edges E and dual edges $E^*$ by saying that $e \in E$ and $f \in E^*$ are dual to each other if and only they cross.⁴ In this case we define $e^* \mathrel {\mathop :}= f$.

Let us extend the notion of graphs so that we allow V to be an infinite set. An approach is to take $G=(V,E)$ to be a pair (together with an embedding to the plane) such that the restriction of V to any bounded region is finite and all vertices have finite degree. An important class of infinite graphs are the lattices . We define a lattice to be an infinite graph such that its faces are translates of each other and they cover the plane, or more generally, its faces tile the plane and the tiling is formed from translates of a finite pattern of polygons. Examples are the square $\mathbb {Z}^2$, triangular $\mathbb {L}_ tri$ and hexagonal $\mathbb {L}_ hex$ lattices which are formed from regular squares, triangles and hexagons, respectively.

Let $G=(V,E)$ be a finite of infinite graph. We always think that G is either a lattice or a subgraph of lattice. Let $p \in [0,1]$ be a parameter and consider a family of random variables, one for each site of G, taking values in the set $\{ open , closed \}$. We say that a site $v \in V$ is open or closed depending on the value that random variable takes. To define the law precisely, it is assumed that

$$\begin{aligned} \mathsf {P}[ v\text { is open} ] = p, \qquad \mathsf {P}[ v \text { is closed} ] = 1-p \end{aligned}$$

(6.20)

for each $v \in V$ and that the random variables are independent. A probability space with the percolation model can be constructed as a product space $\{\text {open},\text {closed}\}^V$ with a product measure of laws of Bernoulli random variables.

The model is called the site percolation on G . The closed sites model a random media and the open sites form cavities through which a fluid can flow. Fundamental questions in the percolation model are therefore connectivity properties of the subgraph of open sites. For small values of p, we expect that it is rare to see large clusters of open sites and that for large p, in an infinite graph, almost surely there exists an open infinite cluster.

If we want to stress the dependency of the model from the parameter, we denote the probability measure as $\mathsf {P}_p$.

6.3.1.1 The Percolation Exploration Process

We wish to define a process that explores a part of a percolation configuration. This process will be a simple path on the hexagonal lattice that keeps open sites of the triangular lattice on its left, say, and closed sites on its right. The exploration process is the blue curve in Fig. 1.8

Let us first define a type of domain on which we will define the exploration. For later purposes, let’s introduce a lattice mesh $\delta >0$. Suppose that $\varOmega _\delta$ is a simply connected domain such that $\partial \varOmega _\delta \ne \emptyset$ is a path on the lattice $\delta \mathbb {L}_ hex$. Let $\tilde{V}$ be the set of sites on $\delta \mathbb {L}_ tri$ that lie inside $\varOmega _\delta$ and let $V_1$ be the set of those $v \in \tilde{V}$ such that the hexagon corresponding to v has at least one common edge with $\partial \varOmega _\delta$. Let $V=\tilde{V} \setminus V_1$. The set V is now the one where we are going to put the percolation configuration. So if we want that V is a given shape, say, a rhombic domain $V=\delta R(v,\lceil a \delta ^{-1} \rceil ,\lceil b \delta ^{-1} \rceil )$, then for $\varOmega _\delta$ we need to add a layer of hexagons around V.

Suppose now that we have defined a percolation configuration on V. The set $V_1$ is connected and we can interpret it as a unique non-self-crossing closed path $\pi$ (with counterclockwise orientation) on the triangular lattice. Let a and b be two distinct points on the hexagonal lattice such that at a and b, exactly one of the three edges of $\mathbb {L}_ hex$ belongs to $\partial \varOmega _\delta$ or that property holds and the property that $\varOmega _\delta$ is a simply connected domain remains valid if a suitable edge is added to a and b or one of them. Then at a and b, of the three neighboring hexagons, two have centers in $V_1$ (and correspond to the above mentioned edge) and third in either $V_1$ or V. We say that a and b are points on the boundary of V.

To define the exploration process, pick for both a and b one of the neighboring edges that cross $\pi$ and denote the two halves of $\pi$ as ab and ba.⁵ The exploration process is the unique simple path from a to b on the hexagonal lattice such that all the hexagons on its right are closed and all on its left are open in the way that ab is considered to be closed and ba open when defining the process.

6.3.1.2 Percolation Interface and Its Scaling Limit

Definition 6.4

For a domain $\varOmega$ and its distinct boundary points a and b and an approximating sequence $(\varOmega _\delta ,a_\delta ,b_\delta )_\delta$, where $\delta >0$ runs over a set of points accumulating to 0, we define $\mu ^{(\varOmega ,a,b)}_\delta$ to be the probability law of the percolation interface $\gamma$ in $\varOmega _\delta$ starting at $a_\delta$ and ending at $b_\delta$. The scaling limit of the percolation interface is $\mu ^{(\varOmega ,a,b)} = \lim _{\delta \rightarrow 0} \mu ^{(\varOmega ,a,b)}_\delta$ where the mode of convergence is specified soon.

Remark 6.5

Let us collect the approximating sequence $(\varOmega _\delta ,a_\delta ,b_\delta )_\delta$ to a family $\mathscr {D}$ of triplets (U, a, b) and let’s assume that the parameter $\delta$ is implicitly given (it is the length of any edge (line segment) in $\partial \varOmega _\delta$). Let’s also collect $\mu ^{(\varOmega ,a,b)}_\delta$ to a family $\mathscr {M}$ of probability measures.

In what follows, we mostly don’t explicitly refer $\mathscr {D}$ or $\mathscr {M}$ before Theorem 6.8. In practice, the statements hold for any collection of domain triplets $(\varOmega ,a,b)$ and they don’t need to form an approximating sequence. Hence we drop the notation for the lattice mesh $\delta >0$ and think that any $(U,a,b) \in \mathscr {D}$ is a discrete domain. We also use notation $\mu \in \mathscr {M}$ without superscripts or even just the standard notation for probability $\mathsf {P}$.

Remark 6.6

We will use the topology of a uniform convergence of continuous functions for curves parametrized with the capacity. The mode of convergence of (generalized, curve-valued) random variables is specified in the next section.

6.3.2 A Probability Bound on Crossings by Multiple Open Percolation Paths

In this section, we establish bounds on multiple crossings of open paths in percolation. These basic estimates are in general needed when we show that the exploration process is sufficiently regular as a curve.

6.3.2.1 Increasing Events and the FKG Inequality

There is a natural ordering among percolation configurations, namely, a configuration $(\omega _x)_{x \in V}$ is greater that a configuration $(\omega _x')_{x \in V}$ if and only if for each $x \in V$, $\omega _x \ge \omega _x'$. This order relation is denoted by $\succ$. A random variable that respects the order $\succ$ is said to be increasing , that is, for an increasing random variable $X: \varOmega \rightarrow \mathbb {R}$, it holds that $X( (\omega _x)_{x \in V} ) \ge X( (\omega _x')_{x \in V} )$ for any configurations $(\omega _x)_{x \in V} \succ (\omega _x')_{x \in V}$. An event said to be increasing if and only if its indicator function is an increasing random variable.

The following result shows that increasing events are positively correlated. One way to formulate this is that if A, B are increasing events in a percolation model, then $\mathsf {P}[A \,|\, B] \ge \mathsf {P}[A]$. For the proof of the theorem, see for instance [3] or [4].

Theorem 6.5

(Fortuin-Kasteleyn-Ginibre (FKG) inequality ) For increasing, non-negative random variables X, Y in a percolation model, it holds that $\mathsf {E}[X\,Y] \ge \mathsf {E}[X] \, \mathsf {E}[Y]$. In particular, this holds for indicator functions of increasing events.

6.3.2.2 Critical Point and RSW Estimates

There are many (equivalent) ways to characterize the critical parameter $p=p_c$ of the site percolation model on a given lattice. We will choose the following definition which is suitable for our needs.

Let $e_1 =1$ and $e_2 = e^{{\mathrm {i}}\pi /3}$, which are unit vectors that generate the triangular lattice $\mathbb {L}_ tri$. Consider rhombi

$$\begin{aligned} R(v,a,b)= \{ v + s\,e_1 + t\, e_2 \,:\, s \in \llbracket 0,a\rrbracket , t \in \llbracket 0,b\rrbracket \}, \end{aligned}$$

(6.21)

where $v \in V(\mathbb {L}_ tri )$ is the lower left corner and $a,b \in \mathbb {Z}_{>0}$ are the side lengths of the rhombus. In any percolation configuration on R(v, a, b), either there is an open path from left to right in R(v, a, b) or a closed path from top to bottom. We set $p_c=1/2$, since satisfies the following crossing property. At $p=p_c$, by symmetry

$$\begin{aligned} \mathsf {P}[ \text {left to right crossing in } R(v,a,a)] = \frac{1}{2} . \end{aligned}$$

(6.22)

Notice that the equality (6.22) holds for all a. In particular, we see that the crossing probability remains bounded away from 0 and 1 as a tends to $\infty$. For subcritical p, that is $p<p_c$, the crossing probability would tend to zero and for supercritical p, that is $p>p_c$, it would tend to one.

Define also a triangle $T(v,a) = \{ v + s\,e_1 + t\, e_2 \,:\, s,t \in \mathbb {Z}_{\ge 0}, s+t \le a\}$ and a trapezoid $\tilde{R}(v,a,b)= R(v,a,b) \cup T(v+a \,e_1, b)$. Let’s call the bottom of $T(v+a \,e_1, b)$ the bottom-right side of the trapezoid $\tilde{R}(v,a,b)$. Denote by

$$\begin{aligned} \mathscr {S}_{ L-R } (R) \qquad \text {or} \qquad \mathscr {S}_{ L-R } (\tilde{R}) \end{aligned}$$

the event of left–right crossing of a rhombus R or trapezoid $\tilde{R}$ and by

$$\begin{aligned} \mathscr {S}_{ L-BR }(\tilde{R}) \end{aligned}$$

the event of a crossing from the left side to the bottom-right of the trapezoid $\tilde{R}$.

The Russo–Seymour–Welsh estimates (RSW) allow us to extend the Eq. (6.22) as inequalities for other rhombi R(v, a, b), $a \ne b$, and other shapes.

Lemma 6.8

For $a,b \in \mathbb {Z}_{>0}$ such that $b \le a$, the following inequalities hold

• When $p \ge p_c$, $\mathsf {P}_{p} [\mathscr {S}_{ L-BR } (\tilde{R}(v,a,b))] \ge \frac{1}{2} \mathsf {P}_{p} [ \mathscr {S}_{ L-BR } (R(v,a,b))]$

• For all p, $\mathsf {P}_{p} [\mathscr {S}_{ L-R } (\tilde{R}(v,a,b))] \ge \mathsf {P}_{p} [ \mathscr {S}_{ L-BR } (\tilde{R}(v,a,b))]^2$

• When $p \ge p_c$, $\mathsf {P}_{p} [\mathscr {S}_{ L-R } ( R(v,2 a,b))] \ge \frac{1}{2} \mathsf {P}_{p} [ \mathscr {S}_{ L-R } (\tilde{R}(v,a,b))]^2$

Consequently, $\mathsf {P}_{p} [\mathscr {S}_{ L-R } ( R(v,2 a,b))] \ge \frac{1}{32} \mathsf {P}_{p} [ \mathscr {S}_{ L-R } (R(v,a,b))]^4$ when $p \ge p_c$.

Proof

Notice first that the probabilities of the crossing events we consider are independent of v and their values remain invariant under lattice rotations.

The argument for the first claim is similar to one given in [13]. It is the main observation of the present proof. Namely, if the event $\mathscr {S}_{ L-R } (R(v,a,b))$ occurs, we can find the topmost simple open path $\pi$ in the percolation configuration crossing the rhombus R(v, a, b). It is illustrated as the dark blue path in Fig. 6.3a. It turns out that such path can be found so that we have to “reveal” only the site above and on the path, for details see [13]. Consequently, conditionally on $\pi$, the configuration restricted to the sites below $\pi$ has still the percolation distribution. Consider the mirror image $R'$ of R(v, a, b) with respect to the right side of R(v, a, b) and the mirror image $\pi '$ of $\pi$. See Fig. 6.3a. In the kite shaped domain formed by the union of R(v, a, b) and $R'$ on the sites below the path concatenated from $\pi$ and $\pi '$, by symmetry, the probability of an open path from sites next to $\pi$ to the lower left side of $R'$ (the solid line in Fig. 6.3a) is at least $\frac{1}{2}$. The occurrence of this event together with existence of $\pi$ implies that $\mathscr {S}_{ L-BR } (\tilde{R}(v,a,b))$ occurs. Summing over the paths $\pi$ gives the claim.

The second claim follows from the FKG inequality, when we consider the events $\mathscr {S}_{ L-BR } (\tilde{R}(v,a,b))$ and the similar crossing event from bottom left to right side of $\tilde{R}(v,a,b)$. See also Fig. 6.3b.

The third claim follows also from the FKG inequality. Namely, if we put two trapezoid so that they partially overlap as in Fig. 6.3b and form an equilateral rhombus in the middle, then by FKG inequality the joint occurrence of open crossings from left to right in both trapezoids and from bottom to top in the rhombus has probability at least the product of the probabilities of the three events. Thus the claim follows.

The last claim follows by combining all the other inequalities. $\square$

Fig. 6.3

The proof of Lemma 6.8 is based on ideas presented in these figures

The previous bounds will give us easily the following bounds.

Corollary 6.3

(Crossing probability of long rhombi or rectangles) For any $\rho \ge 1$, there exists $\varepsilon \in (0,1)$ such that for every $n \in \mathbb {Z}_{>0}$

$$\begin{aligned} \varepsilon \le \mathsf {P}_{p_c}[ \mathscr {S}_{ L-R } (R(v, \lceil \rho n \rceil ,n)) ] \le 1-\varepsilon . \end{aligned}$$

Similar bound holds for rectangles.

Proof

The upper bound follows from the fact that $\mathsf {P}_{p_c}[ \mathscr {S}_{ L-R } (R(v, \lceil \rho n \rceil ,n)) ]$ is bounded from above by $\mathsf {P}_{p_c}[ \mathscr {S}_{ L-R } (R(v, n,n)) ] = 1/2$.

The lower bound follow from $\mathsf {P}_{p_c} [\mathscr {S}_{ L-R } ( R(v,2^k n,n))] \ge 32^{-k} \mathsf {P}_{p_c} [ \mathscr {S}_{ L-R } (R(v,n,n))]^{4 k}$ and $\mathsf {P}_{p_c}[ \mathscr {S}_{ L-R } (R(v, n,n)) ] = 1/2$. $\square$

Together with the FKG inequality this implies the following bound. Consider an annulus $A=A(z_0,r,R)$. We say that a path $\pi$ crosses ⁶ A if $\pi$ intersects both of the connected components of $\mathbb {C}\setminus A$. Denote the crossing event of $A(z_0,r,R)$ by an open percolation path as $\mathscr {S}_{}(A(z_0,r,R))$. Similarly define a discrete annulus $\tilde{A}(v,n,m)$ as the set of sites whose distance on the lattice to v is not less than n or greater than m. Notice that $\tilde{A}(v,n,m)$ is the set between two concentric hexagonal boundary arcs of side lengths n and m lattice steps. Let $\mathscr {S}_{}(\tilde{A}(v,n,m))$ be the crossing event of $\tilde{A}(v,n,m)$, that is, the event that there is an open path connecting the two boundary arcs.

Corollary 6.4

(Crossing of discrete annuli) For any $m \in \mathbb {Z}_{>0}$, there exists $\varepsilon \in (0,1)$ such that for every $n \in \mathbb {Z}_{>0}$

$$\begin{aligned} \varepsilon \le \mathsf {P}_{p_c}[ \mathscr {S}_{}(\tilde{A}(v,n, m\,n)) ] \le 1-\varepsilon . \end{aligned}$$

Proof

For the lower bound, superimpose the annulus with a (long) rhombus, whose ends lie fully in the complement of the annulus, one inside and one outside. The bound follows from Corollary 6.3.

For the upper bound, notice that at $p_c$ we can flip the state of each site (“open” between “closed”). Notice also that either there is a closed crossing of the annulus or an open path, which forms a loop that is non-contractible in the annulus. Next we construct such a non-trivial loop from open crossings of six rhombi by arranging them as in Fig. 6.4.⁷ The joint occurrence of those crossing events is bounded from below by the product of their probabilities by the FKG inequality. Thus the upper bound of the claim follows.$\square$

Fig. 6.4

Construction of a non-trial cycle in an annulus from six “left-to-right” crossing in rhombi

Let’s call $A(z_1,r_1,R_1)$ a subannulus of $A(z_0,r,R)$ if $A(z_1,r_1,R_1) \subset A(z_0,r,R)$ and $A(z_1,r_1,R_1)$ separates the connected components of $\mathbb {C}\setminus A(z_0,r,R)$.

Corollary 6.5

(Crossing of annuli) There exist constant $K_1$ and $\varDelta _1>0$ such that for any $z_0 \in \mathbb {C}$ and $1<r<R$,

$$\begin{aligned} \mathsf {P}_{p_c}[ \mathscr {S}_{}(A(z_0,r,R)) ] \le K_1 \left( \frac{r}{R}\right) ^{\varDelta _1} . \end{aligned}$$

Proof

We will establish the bound when $R/r>2$. For the complementary range the bound follows easily by choosing $K_1 > 2^{\varDelta _1}$ which we can always do.

Since $r>1$, there exists a lattice site v within the distance r from $z_0$ thus A(v, 2r, R / 2) is a subannulus of $A(z_0,r,R)$. Here we used $R/r>2$. Denote by $\tilde{B}(v,n)$ the discrete ball of radius r, i.e., the filled hexagon centered at v with sides on the lattices with length n. Let $n_0 = \lceil 4r/\sqrt{3} \rceil$ and k be the maximal integer such that $2^k \, (n_0 + 1) -2 \le R/2$, that is, $k =\lfloor \log _2 ((R+4)/(n_0+1)) \rfloor -1$. Then since $\tilde{B}(v,n) \subset \overline{B(v,n)} \le \tilde{B}(v, \lceil \frac{2n}{\sqrt{3}} \rceil )$, the discrete annuli $\tilde{A}_j \mathrel {\mathop :}= \tilde{A}(v, 2^j \, (n_0 + 1) -2, 2^j \, (n_0 + 1) -1)$, $j=0,1,\ldots ,k$, are subannuli of $A(z_0,r,R)$ and they are disjoint. By independence of percolation in disjoint sets,

$$\begin{aligned} \mathsf {P}_{p_c}[ \mathscr {S}_{}(A(z_0,r,R)) ]&\le \mathsf {P}_{p_c} [ \bigcap _{j=0}^k \mathscr {S}_{}(\tilde{A}_j) ] = \prod _{j=0}^k \mathsf {P}_{p_c} [ \mathscr {S}_{}(\tilde{A}_j) ] \le (1-\varepsilon )^{k+1} . \end{aligned}$$

(6.23)

Notice $n_0 +1 \le 4r/\sqrt{3} +2 < 5 r$ and $k+1 \ge \log _2 ((R+4)/(n_0+1)) \ge \log _2 (R/(5r))$.

$$\begin{aligned} (1-\varepsilon )^{k+1} \le e^{ \frac{\log \frac{1}{1-\varepsilon }}{\log 2} \, (\log 5 + \log \frac{r}{R} )} = K_1 \left( \frac{r}{R} \right) ^{\varDelta _1} \end{aligned}$$

where $\varDelta _1 =\log _2 \frac{1}{1-\varepsilon }$ and $K_1 = \exp ( (\log _2 \frac{1}{1-\varepsilon })( \log 5))$ $\square$

6.3.2.3 BK Inequality and Multiple Crossings of Quadrilaterals and Annuli

Let A and B two events in a percolation model. We will denote by $A \square B$ the event that A and B both occur and they occur disjointly . We define it so that $\omega \in A \square B$ if and only if there exist disjoint sets F and G of sites (which might depend on $\omega$) such that knowledge of $\omega$ restricted to F implies that $\omega \in A$ and knowledge of $\omega$ restricted to G implies that $\omega \in B$. For more details see [4], Sect. 4.3.

We will use below the following inequality. We omit the proof which can be found in [3] or [4].

Theorem 6.6

(van den Berg–Kesten (BK) inequality ) For increasing events A, B in a percolation model, it holds that $\mathsf {P}[A \square B] \le \mathsf {P}[A ] \mathsf {P}[B]$.

Remark 6.7

For all events A, B, it holds that $A \square B \subset A \cap B$. It is natural to interpret the BK inequality as a counterpart for the FKG inequality.

Proposition 6.6

(Disjoint open crossings of annuli) For each $C>1$ and $\varepsilon >0$ there exists $n_0 \in \mathbb {Z}_{>0}$ such that

$$\begin{aligned} \mathsf {P}\left[ \mathscr {S}_{n} (A(z_0,r,R)) \right] \le \varepsilon \end{aligned}$$

(6.24)

for all $n \ge n_0$, $z_0\in \mathbb {C}$, $r,R \in \mathbb {R}_{>0}$ such that $R/r \ge C$. Consequently, there exist non-negative constants $\varDelta _n$ and $K_n$ such that $\varDelta _n$ tends to infinity as n tends to infinity and for every n, $z_0 \in \mathbb {C}$ and $0<r<R$,

$$\begin{aligned} \mathsf {P}\left[ \mathscr {S}_{n} (A(z_0,r,R)) \right] \le K_n \, \left( \frac{r}{R}\right) ^{\varDelta _n} . \end{aligned}$$

Remark 6.8

We can choose $\varDelta _n$ to be non-decreasing and by this result $\varDelta _n > 0$ for all $n \ge n_0$ for some $n_n \in \mathbb {Z}_{>0}$.

Proof

The claim follows easily from Theorem 6.6 and Corollary 6.5 when $r>1$. For $r \le 1$, we need to notice that for $n \ge 10$, say, $\mathsf {P}[ \mathscr {S}_{n} (A(z_0,r, R)) ]=0$. $\square$

6.3.3 Distortion Estimates of Annuli

Throughout this subsection U is a simply connected domain and $\phi : \mathbb {D}\rightarrow U$ is a conformal map. We study how annuli are distorted under conformal maps. We claim that the conformal image of an annulus has a subannulus (here we naturally extend the concept of a subannulus slightly) such that their conformal moduli, i.e., logarithms of ratios of the radii, are proportional. First we study annuli fully contained in the domain.

Lemma 6.9

(Distortion of annuli contained in $\mathbb {D}$) For any $\rho '>1$ sufficiently large, there exists $\rho >1$ such that the following holds. Suppose U and $\phi$ are as above and that $A=A(z_0,r,R) \subset \mathbb {D}$ with $R/r > \rho$ and $R\le \frac{1}{2} (1-|z_0|)$. Then there exists an annulus $A'=A(z_0',r',R')$ with $R'/r' > \rho '$ such that $A' \subset \phi (A)$ and $A'$ separates the boundary components of $\phi (A)$ in $\mathbb {C}$. Furthermore the dependency of $\rho$ and $\rho '$ can be made linear.

Proof

Let $A=A(z_0,r,R)$ be such that $|z_0|<1$ and $0<\rho ^{-1} R< r<R<\frac{1}{2} (1-|z_0)$. Let $\psi$ be a Möbius selfmap of $\mathbb {D}$ that sends $z_0$ to 0. As usual, the map $\psi$ can be explicitly written and one can verify that $|\psi (z_0 + \tilde{r} e^{{\mathrm {i}}\theta })| = \tilde{r}/ |1-|z_0|^2 - \overline{z}_0 \, \tilde{r} e^{{\mathrm {i}}\theta }|$ and thus

$$\begin{aligned} \frac{1}{3} \frac{\tilde{r}}{1-|z_0|} \le |\psi (z_0 + \tilde{r} e^{{\mathrm {i}}\theta })| \le \frac{\tilde{r}}{1-|z_0|} \end{aligned}$$

(6.25)

for all $\tilde{r} \in (0, 1-|z_0|)$. This shows that there exists an annulus $A_1 = A_1(0,r_1,R_1)$ such that $A_1 \subset \psi (A)$ and $R_1/r_1 = (1/3)(R/r)$.

When $R \le (1-|z_0)/2$, by the right-hand side of (6.25), we can choose $R_1 \le \frac{1}{2}$. A similar argument to the above one using the Koebe distortion theorem, Theorem 3.9, shows that $\phi \circ \psi ^{-1}$ distorts the two boundary components of $A_1$ proportionally. Consequently we can find $A' = A'(\phi (0),r',R')$ such that $A' \subset \phi (A)$ separates the boundary components of $\phi (A)$ and $r'/R' = \text {const.} (R_1/r_1)$ where the universal constant comes from the Koebe distortion theorem. $\square$

Next we will apply extremal length (see also Sect. 3.4.2) to show a distortion estimate for annuli intersecting the boundary.

Lemma 6.10

(Distortion of annuli not fully contained in $\mathbb {D}$) For any $\rho '>1$ sufficiently large, there exists $\rho >1$ such that the following holds. Suppose U and $\phi$ are as above and that $A=A(z_0,r,R)$ with $R/r > \rho$ is such that $1-|z_0|<r$ and $R<1$ (that is, $\partial \mathbb {D}$ crosses A). Then there exists an annulus $A'=A(z_0',r',R')$ with $R'/r' > \rho '$ such that there exists a connected component O of $U \cap A'$ such that $O \subset \phi (A \cap \mathbb {D})$ and O separates the components of $\phi ((\partial A) \cap \mathbb {D})$ in U.

Proof

The proof is based on the extremal length. The extremal length of the curve family connecting the components of $(\partial A) \cap \mathbb {D}$ in $A \cap \mathbb {D}$ is at least the extremal length of the curve family connecting the boundary components of A in A. The latter one is equal to $\frac{1}{2\pi } \log (R/r)$.

Let $Q=\phi (A \cap \mathbb {D})$. Then Q is a topological quadrilateral which then has four “marked” sides $S_1,S_2,S_3,S_4$. Suppose that $S_1 \cup S_3 = \phi ( (\partial A) \cap \mathbb {D})$. Let $d_1$ be the Euclidian distance inside Q from $S_1$ to $S_3$.

Let $\gamma _1$ be a path connecting $S_2$ and $S_4$ in Q whose length is (strictly) less than $2 d_1$. Let $z_0'$ to be its mid point (with respect to its length). Then $\gamma _1 \subset B(z_0,d_1)$.

Let $\gamma$ be a path connecting $S_1$ and $S_3$ in Q whose diameter is d. Calculate a lower bound for the extremal length of the curve family connecting $S_1$ to $S_3$ in Q by using a metric $\rho$ equal to 1 in a $d_1$-neighborhood of $\gamma$. Then the $\rho$-length of any curve connecting $S_1$ to $S_3$ in Q is at least $d_1$ and the $\rho$-area of Q is at most $(d+2 d_1)^2$. Consequently the extremal length of the curve family is at least $d_1^2/(d+2 d_1)^2$. By the reciprocity of the complementary extremal lengths in a topological quadrilateral, it holds now that

$$\begin{aligned} \frac{d_1^2}{(d+2 d_1)^2} \le \frac{2 \pi }{\log (R/r)} =\mathrel {\mathop :}m^{-2}. \end{aligned}$$

(6.26)

Thus the diameter of $\gamma$ satisfies $d \ge (m-2)d_1$. In particular, either for $S_1$ or $S_3$, it holds that any path from $\gamma _1$ to that arc has to have diameter at least $(d - d_1)/2 \ge ((m-3)/2) d_1$. Otherwise we could construct a path connecting $S_1$ or $S_3$ that has diameter less than d. Therefore $\gamma$ has to intersect the complement of $B(z_0,((m-3)/2) d_1)$.

Thus we have shown that any path connecting $S_1$ to $S_3$ in Q has to make at least one crossing of $A'=A'(z_0',d_1, ((m-3)/2) d_1)$. Let $\gamma$ now be the path that makes the minimal number of such crossings.⁸ Then each connected component of $U \cap A'$, whose closure contains a minimal subcrossing of $\gamma$, separates $S_1$ and $S_3$ in U. Choose any one them and denote it by O. Then O has the claimed properties. $\square$

The same proof can be used for the following result. Let $\tilde{\phi }: \mathbb {H}\rightarrow U$ be a conformal, onto map.

Lemma 6.11

(Distortion of annuli not fully contained in $\mathbb {H}$) For any $\rho '>1$ sufficiently large, there exists $\rho >1$ such that the following holds. Suppose U and $\phi$ are as above and that $A=A(z_0,r,R)$ with $R/r > \rho$ is such that $|\mathrm{{Im}}z_0| <r$ and $R>0$ (that is, $\partial \mathbb {H}$ crosses A). Then there exists an annulus $A'=A(z_0',r',R')$ with $R'/r' > \rho '$ such that there exists a connected component O of $U \cap A'$ such that $O \subset \tilde{\phi }(A \cap \mathbb {H})$ and O separates the components of $\tilde{\phi }((\partial A) \cap \mathbb {H})$ in U.

Let us now use Lemmas 6.9 and 6.10 to estimate the probability of the n-arms event in an annulus $A=A(z_0,r,R)$ by open paths of the percolation configuration transformed to $\mathbb {D}$ conformally. We call a path $x_k \in \mathbb {D}$, $k\in \llbracket 0,n\rrbracket$, an open crossing of A in $\mathbb {D}$ if $\phi (x_k) \in \varOmega$ are open sites of the triangular lattice of the percolation configuration in $\varOmega$, they form a lattice path (that is, $|\phi (x_{k+1}) - \phi (x_k)|=\delta$ where $\delta$ is the lattice mesh) and $|x_0 - z_0|\le r$ and $|x_n - z_0|\ge R$. If there are n disjoint open crossings of A we say that (monochromatic) n-arms event occurs in A. Let $C>1$ and define $m = \lfloor (\log (R/r))/(\log C) \rfloor$ and annuli

$$\begin{aligned} A_k = A\left( z_0,C^{k-\frac{2}{3}} r, C^{k-\frac{1}{3}} r\right) , \qquad \hat{A}_j = A\left( z_0,C^{\frac{j-1}{3}} r, C^{\frac{j}{3}} r\right) , \end{aligned}$$

for $k \in \llbracket 1,m\rrbracket$ and $j \in \llbracket 1,3m\rrbracket$.

Let $D_j$ be the infimum of $\mathrm{diam}(\phi (\gamma ))$ where $\gamma \subset \mathbb {D}$ is a simple curve (which is open or close) that separates the components of $\mathbb {C}\setminus \hat{A}_j$. Similarly let $L_j$ be the infimum of $\mathrm{diam}(\gamma )$ where $\gamma \subset \mathbb {D}$ is a simple curve that connects the components of $\mathbb {C}\setminus \hat{A}_j$. Then by a simple argument shows that $D_j \ge \delta$; otherwise there couldn’t be $n \ge 2$ open crossings of $\hat{A}_j$. An argument using the extremal length similar to the proof of Lemma 6.10, shows that for any $M>0$, there is $C_0$ such that if $C>C_0$, then $L_j/D_j \ge M$. Choose $M=10$, say, and notice that this implies that $\phi (\mathbb {D}\cap \hat{A}_j)$ contains a path of neighboring hexagons that separate the components of $\phi (\mathbb {D}\setminus \hat{A}_j)$. Consequently, the crossing events in $A_k$ for different $k \in \llbracket 1,m\rrbracket$ are independent. Thus $\mathsf {P}[ \mathscr {S}^\mathbb {D}_{n}(A) ] \le \prod _k \mathsf {P}[ \mathscr {S}^\mathbb {D}_{n}(A_k) ]$.

By Lemmas 6.9 and 6.10, there exists a constant $C'>0$ such that we can select annuli $A'_k =A(z_k',r_k', C' r_k')$ such that $A'_k$ is a subannulus of $\phi (A_k)$. Let $\varepsilon >0$. Apply Proposition 6.6 and in particular the inequality (6.24) to select $n_0$ such that $\mathsf {P}[ \mathscr {S}^\mathbb {D}_{n} (A'_k)] \le \varepsilon$ for all $n \ge n_0$. Then $\mathsf {P}[ \mathscr {S}^\mathbb {D}_{n}(A)] \le \varepsilon ^m$. Using the definition of m and some algebra gives then the following probability bound.

Proposition 6.7

(Probability bound on multiple open crossings in $\mathbb {D}$) There exist non-negative constants $\varDelta _n$ and $K_n$ such that $\varDelta _n$ tends to infinity as n tends to infinity and for every n, $z_0 \in \mathbb {C}$ and $0<r<R$,

$$\begin{aligned} \mathsf {P}\left[ \mathscr {S}^\mathbb {D}_{n} (A(z_0,r,R)) \right] \le K_n \, \left( \frac{r}{R}\right) ^{\varDelta _n} . \end{aligned}$$

6.3.4 Analysis of Tortuosity

6.3.4.1 Tortuosity and the Connection to Hölder Continuity

We follow here Aizenman’s and Burchard’s seminal paper [1]. Let’s make the following definition for a curve $\gamma :[0,1] \rightarrow \mathbb {C}$. Define $[\gamma ]$ to be the equivalence class of all reparameterizations of $\gamma$.⁹ Define $M(\gamma ,l)$ to be the minimum number of segments of $\gamma$ with diameters less or equal to l that are needed to cover $\gamma$, which is a reparameterization invariant. A bound of the form

$$\begin{aligned} M(\gamma ,l) \le \frac{1}{\eta (l)}, \end{aligned}$$

(6.27)

where $\eta : (0,1] \rightarrow (0,1]$ is a non-degreasing function, a tortuosity bound.

The following lemma establishes in one direction the connection between tortuosity bounds and Hölder continuity. For the other direction, see [1]. The proof of this lemma is given in Appendix D.

Lemma 6.12

If $\gamma$ satisfies (6.27), then it can be parametrized so that the parametrization $\tilde{\gamma }\in [\gamma ]$ satisfies $\tilde{\eta }(| \tilde{\gamma }(t) - \tilde{\gamma }(s) | ) \le |t-s|$ where $\tilde{\eta }(y) =\displaystyle \frac{\eta (y/4)}{2 \log (8/(y))^2 }$.

Remark 6.9

If $\eta (y) = C y^{\alpha }$, then for each $\varepsilon >0$ there exists a constant $\tilde{C}_\varepsilon$ such that $| \tilde{\gamma }(t) - \tilde{\gamma }(s) | \le \tilde{C}_\varepsilon |t-s|^{\frac{1}{\alpha }-\varepsilon }$ by Lemma 6.12.

Definition 6.5

Suppose that $\varepsilon >0$, $r_0>0$ and $k \in \mathbb {Z}_{>0}$. We say that $\gamma$ has $(\varepsilon ,r_0,k)$-tempered crossing property if for all $r \in (0,r_0)$ it holds that $\gamma$ doesn’t cross k or more times any annulus of the form $A=A(z_0, r^{1+\varepsilon },r)$.

Define $N(\gamma ,l)$ to be the minimal number of sets of diameter less or equal to l needed to cover $\gamma$. Then obviously $N(\gamma ,l) \le M(\gamma ,l)$. The following result gives a complementary inequality.

Lemma 6.13

If $\gamma$ has the $(\varepsilon ,r_0,k)$-tempered crossing property, then for any $l \in (0,r_0)$, $M(\gamma ,2l) \le k N(\gamma ,l^{1+\varepsilon })$.

Proof

Cover $\gamma$ by segments of diameter less than 2l recursively by choosing points $x_0,x_1,\ldots ,x_n$ so that $x_{k+1}$ is the first point after $x_k$ that lies on the boundary of $B(x_k,l)$ and $x_0$ and $x_n$ are the endpoints of $\gamma$. Then $M(\gamma ,2l) \le n$.

Cover also $\gamma$ with $N(\gamma ,l^{1+\varepsilon })$ balls of diameter $l^{1+\varepsilon }$. Let $B=\overline{B(z_0,l^{1+\varepsilon }/2)}$ be one of those balls. Then since $\gamma$ doesn’t make k or more crossings of $A(z_0, (l/2)^{1+\varepsilon }, l/2)$, it holds that at most k of the points $x_0,x_1,\ldots ,x_n$ can be contained in B. Therefore the claim follows. $\square$

If the diameter of $\gamma$ is at most R, then it can be covered with $\lceil R\sqrt{2}/l \rceil ^2$ squares of diameter l. Therefore

$$\begin{aligned} N(\gamma ,l) \le 3 R^2 l^{-2} \end{aligned}$$

(6.28)

for all $l \in (0,R)$. Therefore the following result holds by Lemmas 6.12 and 6.13 and the remark after Lemma 6.12.

Proposition 6.8

Let $\varepsilon >0$. If $\gamma$ is bounded, explicitly, $\gamma \subset B(z_0,R)$, and has $(\varepsilon ,r_0,k)$-tempered crossing property for some $r_0>0$ and $k \in \mathbb {Z}_{>0}$, then for all $l \in (0,r_0)$ it holds that

$$\begin{aligned} M(\gamma ,l) \le \tilde{C}\, k\, R^2\, l^{-2(1+\varepsilon )} . \end{aligned}$$

(6.29)

Here $\tilde{C}$ is an absolute constant.

Here is an interesting corollary, which we don’t directly use, but which clarifies the role of tortuosity bounds.

Corollary 6.6

Let $\varepsilon >0$. If $\gamma$ is bounded, explicitly, $\gamma \subset B(z_0,R)$, and has $(\varepsilon ,r_0,k)$-tempered crossing property for some $r_0>0$ and $k \in \mathbb {Z}_{>0}$, then for each $\alpha \in (0, 1/(2+2 \varepsilon ))$, $\gamma$ can be parametrized as $\gamma :[0,1] \rightarrow \mathbb {C}$ such that for all $s,t \in [0,1]$

$$\begin{aligned} | \tilde{\gamma }(t) - \tilde{\gamma }(s) | \le \tilde{C} |t-s|^{\alpha } . \end{aligned}$$

(6.30)

Here $\tilde{C}$ doesn’t depend directly on $\gamma$, but can depend on $\varepsilon ,r_0,k,R$ and $\alpha$.

6.3.4.2 The Percolation Interface Satisfies a Tortuosity Bound

Define $\gamma$ to be the percolation interface in a discrete domain (U, a, b) at criticality, that is, when $p=p_c$. Let $\phi : \mathbb {D}\rightarrow U$ be a conformal, onto map such that $\phi (-1)= a$ and $\phi (+1) = b$. Define $\hat{\gamma } = \phi ^{-1} \circ \gamma$. That is, $\gamma$ is the interface on the original domain and $\hat{\gamma }$ is its conformal image on the unit disc.

Proposition 6.9

There exists universal constants $\tilde{K}_n$ and $\tilde{\varDelta }_n$ for each n such that $\tilde{\varDelta }_n \rightarrow \infty$ as $n \rightarrow \infty$ and the following holds. Let $\hat{\gamma }$ be as above and $z_0 \in \mathbb {D}$ and $0< r< R <1$. Then

$$\begin{aligned} \mathsf {P}[ \hat{\gamma } \text { makes }n\text { crossings of } A(z_0,r,R) ] \le \tilde{K}_n \left( \frac{r}{R} \right) ^{\tilde{\varDelta }_n} . \end{aligned}$$

(6.31)

Proof

We may assume that R / r is sufficiently large that we can apply Lemmas 6.9 and 6.10 below. Namely, for $R/r \le M$, we can arrange so that the inequality (6.31) holds by choosing $\tilde{K}_n$ larger than $M^{\tilde{\varDelta }_n}$.

Let $d = 1 -|z_0|$. One of the following occurs: (i) $d < r$, (ii) $d \in [r,R]$ or (iii) $d>R$. In the cases (i) and (iii), set $r_1=r$ and $R_1 =R$. In the case (ii), set $r_1 = \sqrt{r \, R}$ and $R_1=R$, if $d < \sqrt{r \, R}$, and $r_1 = r$ and $R_1 = \sqrt{r \, R}$, otherwise. Then $r \le r_1 < R_1 \le R$, $r_1/R_1 \le \sqrt{ r/R}$ and either $d \le r_1$ or $d \ge R_1$.

Let us then consider the crossing events of $\hat{\gamma }$. Suppose first that $d \le r_1$ and $\hat{\gamma }$ makes exactly n crossings of $A_1 \mathrel {\mathop :}= A(z_0,r_1,R_1)$. Then there are at least either $\lfloor n/2 \rfloor$ disjoint open crossing or $\lfloor n/2 \rfloor$ disjoint closed crossing, which are in addition disjoint from the boundary. We can suppose that they are open. Since the crossing are disjoint from the boundary we can change the state of the boundary sites, and we may as well suppose that all the boundary sites of U are closed. Then the crossings are automatically disjoint from the boundary. This leaves the probability unchanged. Thus

$$\begin{aligned} \mathsf {P}[ \hat{\gamma } \text { makes }n\text { crossings of } A(z_0,r,R) ] \le \mathsf {P}[ \lfloor n/2 \rfloor \text { open crossings of } A_1 \cap \mathbb {D}] . \end{aligned}$$

The same upper bound holds easily also when $d \ge R_1$; we don’t have to worry about the boundary in that case.

Next we apply the conformal transformation $\phi$. When $d \le r_1$, by Lemma 6.10, we can find $z_2$, $r_2$ and $R_2$ such that $R_2/r_2 \ge const. (R_1/r_1)$ and $A_2\mathrel {\mathop :}=A(z_2,r_2,R_2)$ is such that there exists a connected component O of $U \cap A_2$ such that $O \subset \phi (A \cap \mathbb {D})$ and O separates the components of $\phi ((\partial A) \cap \mathbb {D})$ in U. It follows that if there are at least $\lfloor n/2 \rfloor$ open crossings of $A_1 \cap \mathbb {D}$, then there are at least $\lfloor n/2 \rfloor$ open crossings of $A_2$. We can remove all the closed boundary sites in $A_2$ and we get an upper bound by using the probability of at least $\lfloor n/2 \rfloor$ open crossings in the whole $A_2$ (without any boundary sites).

On the other hand, if $d \ge R_1$ apply Lemma 6.9, to show that there exists $z_3$, $r_3$ and $R_3$ such that $R_3/r_3 \ge const. (R_1/r_1)$ and $A_3 \mathrel {\mathop :}= A(z_3,r_3,R_3)$ is a subannulus of $\phi ( A(z_0,r_1,R_1))$. It follows that if there are at least $\lfloor n/2 \rfloor$ open crossings of $A_1 \cap \mathbb {D}$, then there are at least $\lfloor n/2 \rfloor$ open crossings of $A_3$.

The claim follows now from Proposition 6.6. $\square$

Proposition 6.10

Let $\varepsilon >0$ and $k \in \mathbb {Z}_{>0}$ such that $\frac{\varepsilon }{1+\varepsilon } \varDelta _k > 2$. Then there exists a random variable $r_0>0$ such that $\hat{\gamma }$ has $(\varepsilon ,r_0,k)$-tempered crossing property. Furthermore, $\mathsf {P}[ r_0 < r] \le C r^{\varepsilon \varDelta _k - 2(1+\varepsilon )}$ with some constant C.

Proof

Let $\varepsilon >0$ and $k \in \mathbb {Z}_{>0}$ be as in the statement of the proposition and let $I_n = ([-1,1] \cap 2^{-n} (\mathbb {Z}+ \frac{1}{2}) )^2$ for any $n \in \mathbb {Z}_{>0}$.

For any $r >0$, let

$$\begin{aligned} n_r = \left\lfloor \log _2 \frac{1}{r^{1+\varepsilon }} \right\rfloor -2 . \end{aligned}$$

Notice that holds that $2^{-n_r-3} < r^{1+\varepsilon } \le 2^{-n_r-2}$. For any $z_0 \in \overline{\mathbb {D}}$, we can choose $z_1 \in I_{n_r}$ such that $|z_0-z_1|<2^{-n_r - 1/2}$. Thus for any z such that $|z-z_1| \ge 2^{-n_r}$, it holds that $|z - z_0| > r^{1+\varepsilon }$. Similarly, $|z-z_0| \le r$ for any z such that $|z-z_1| \le \frac{1}{16}2^{-\frac{n_r}{1+\varepsilon }}$ and for small enough r. Consequently, $A(z_1,2^{-n_r} ,\frac{1}{16}2^{-\frac{n_r}{1+\varepsilon }})$ is a subannulus of $A(z_0,r^{1+\varepsilon },r)$ for small enough r.

Therefore if we set

$$\begin{aligned} \eta = \sup \left\{ n \in \mathbb {Z}_{>0}\,:\, \begin{array}{l} A(z_1, 2^{-n}, (1/16) 2^{-n/(1+\varepsilon )}) \text { contains} \\ k\text {-fold crossing for some } z_1 \in I_n \end{array} \right\} \end{aligned}$$

then for any r such that $r^{1+\varepsilon } \le 2^{-\eta -3}$ it holds that $A(z_0,r^{1+\varepsilon },r)$ doesn’t contain k-fold crossings for any $z_0 \in \mathbb {C}$. Then $\eta$ is almost surely finite and has exponential tails, since

$$\begin{aligned} \mathsf {P}[ \eta \ge n ]&\le \sum _{l=n}^\infty \sum _{z_1 \in I_l} \mathsf {P}[A(z_1, 2^{-l}, (1/16) 2^{-\frac{l}{1+ \varepsilon }}) \text { contains }k-\text {fold crossing} ] \nonumber \\&\le C_1 \sum _{l=n}^\infty 2^{ (2- \frac{\varepsilon }{1+\varepsilon } \varDelta _k)l} = C_2 2^{ (2- \frac{\varepsilon }{1+\varepsilon } \varDelta _k)n} \end{aligned}$$

when $\frac{\varepsilon }{1+\varepsilon } \varDelta _k > 2$. This implies the claim. $\square$

6.3.5 Regularity of the Percolation Interface in the Capacity Parametrization and Existence of Subsequent Scaling Limits

6.3.5.1 The Speed of Approach to the Tip

Consider the simple curves in $\mathbb {D}$; more specifically, consider the collection

$$\begin{aligned} \left\{ \gamma \in C(\mathbb {R}_{\ge 0},\mathbb {C}) \,:\, \begin{array}{l} \gamma (0)=-1, \gamma \text { is simple}, \\ \gamma (t) \in \mathbb {D}\text { for all } t>0 \text { and } \lim _{t \rightarrow \infty } \gamma (t) = +1 \end{array} \right\} . \end{aligned}$$

(6.32)

Remember that for a curve $\gamma$ in the set (6.32), the standard way to transform $\gamma$ to $\mathbb {H}$ is to define $\psi (z)= {\mathrm {i}}(z+1)/(1-z)$ and $\gamma _\mathbb {H}= \psi \circ \gamma$.

In this subsection, we study the following event.

Definition 6.6

Fix a (small) constant $\rho > 0$ and set $\tilde{B}_\rho = \psi ^{-1}(\mathbb {H}\setminus B(0,\frac{1}{\rho })$. Define a subset E(r, R) of the set (6.32) such that there exists $s,t \in \mathbb {R}_{\ge 0}$ such that $s<t$ and the following statements are satisfied

• $\mathrm{diam}( \gamma [s,t]) \ge R$ and

• there exists a crosscut C in the domain $\mathbb {D}\setminus \gamma (0,s]$ such that $\mathrm{diam}(C) \le r$ and C separates $\gamma (s,t]$ from $\tilde{B}_\rho$ in $\mathbb {D}\setminus \gamma (0,s]$.

Remark 6.10

The set $\tilde{B}_\rho$ is the intersection of the unit disc and a closed ball of radius $2\rho /(1-\rho ^2)$. It is easy to verify that $B(1,\frac{2\rho }{1+\rho }) \subset \tilde{B}_\rho \subset B(1,\frac{2\rho }{1-\rho ^2})$. We skip the details of this Möbius function calculation.

Remark 6.11

In fact, if $\gamma \in E(r,R)$, we can choose a pair (s, t) such that $\gamma (s)$ is one of the endpoints of C and $|\gamma (t) - \gamma (s)| \ge R/2$. This follows from the next lemma, by which $\gamma (u) \in \overline{C}$ for some $u \in [0,t]$ and then we can choose s to be maximal such u.

Lemma 6.14

Let $R \ge 2r$ and $r < \min \{2,\rho \}$. Then $\overline{C} \cap \gamma [0,t] \ne \emptyset$.

Proof

Assume the opposite, that is, that $\{x_1,x_2\} \mathrel {\mathop :}= \overline{C} \setminus C$ is a subset of $\partial \mathbb {D}$.

Write $\mathbb {D}\setminus C = D_1 \cup D_2$ where $D_k$ are the connected components. such that ${\text {Length}}(\partial D_1 \cap \partial \mathbb {D}) < \pi$ and ${\text {Length}}(\partial D_2 \cap \partial \mathbb {D}) = 2\pi -{\text {Length}}(\partial D_1 \cap \partial \mathbb {D}) > \pi$. Notice then that $C \subset \overline{B(x_1,r)}$ and thus $D_1 \subset B(x_1,r)$.

Since $B(1,\rho ) \cap C = \emptyset$, it follows that $B(1,\rho ) \cap \mathbb {D}$ is a subset of either $D_1$ or $D_2$. If $(B(1,\rho ) \cap \mathbb {D}) \subset D_1$, then $\rho \le r$, which is a contradiction. Therefore $(B(1,\rho ) \cap \mathbb {D}) \subset D_2$. Since $\gamma [0,t] \cap C = \emptyset$, $\gamma [0,t]$ is in a similar manner a subset of either $D_1$ or $D_2$. Since C separates $\gamma [0,t]$ and $B(1,\rho )$ in $\mathbb {D}$, $\gamma [0,t] \subset D_1$.

Therefore $\mathrm{diam}( \gamma [0,t] ) < 2r$. This is a contradiction and the claim follows. $\square$

The following result connects the above annulus-crossing-type event E(r, R) (we will clarify this later) to the speed of convergence of radial limit of a conformal map towards the tip of $\gamma _\mathbb {H}$.

Proposition 6.11

There exists a constant $K>0$ and an increasing function $\mu :[0,1] \rightarrow \mathbb {R}_{\ge 0}$ such that $\lim _{r \rightarrow 0} \mu (r) = 0$ and that the following holds. Let $r < \min \{2,\rho \}$, $R \ge 2r$ and $\gamma$ be in (6.32). Assume that $\gamma (0,t] \subset \mathbb {D}\setminus \tilde{B}_{2 \rho }$. If $\gamma$ is not in E(r, R), then

$$\begin{aligned} \sup _{y \in (0,\mu (r)]} |\gamma _\mathbb {H}(t) - F(t,y)| \le K \, \rho ^{-2} \, R . \end{aligned}$$

(6.33)

Remark 6.12

To apply the result, let $r=R^{1+\varepsilon }$ for some $\varepsilon >0$ and let $y_0 = \mu (r)$. Then $K \, \rho ^{-2} \,R= K \, \rho ^{-2} \,r^{\frac{1}{1+\varepsilon }} = K \, \rho ^{-2} \,(\mu ^{-1}(y_0))^{\frac{1}{1+\varepsilon }} =\mathrel {\mathop :}\lambda (y_0)$. Thus we can write (6.33) in the form

$$\begin{aligned} \sup _{y \in (0,y_0]} |\gamma _\mathbb {H}(t) - F(t,y)| \le \lambda (y_0) . \end{aligned}$$

(6.34)

Proof

Let $\mu (r) = \exp \left( - \frac{2 \pi ^2}{r^2} \right)$. Fix $t \in \mathbb {R}_{\ge 0}$ and let $C_y = \{\psi ^{-1} \circ f_t(W_t + y e^{{\mathrm {i}}\theta }) \,:\, \theta \in (0,\pi ) \}$ and $z_y = \psi ^{-1} \circ f_t(W_t + {\mathrm {i}}y)$.

By Lemma 4.6, for each $r>0$, there exists $y_r \in [\mu (r), \sqrt{\mu (r)}]$ such that $C_{y_r}$ has diameter less than r. Then by the assumption that $\gamma \notin E(r,R)$, the path with least diameter from $z_{y_r}$ to $\gamma (t)$ has diameter at most $r + R < 2R$. By the Gehring–Hayman theorem, see [9], the diameter of $J \mathrel {\mathop :}= \psi ^{-1} \circ f_t( \{W_t + {\mathrm {i}}y \,:\, y \in (0,y_0]\})$ is less than KR, where K is an absolute constant.

Next notice that ${{\mathrm{dist}}}(\mathbb {D}\setminus \tilde{B}_{2 \rho } , \tilde{B}_{\rho }) > \rho$ and thus $J \subset \mathbb {D}\setminus \tilde{B}_{\rho }$. It is easy to verify that $|\psi '(z)| = 2/|1-z|^2$ and thus $\frac{1}{2}< |\psi '(z)|< \frac{1}{4}(1+\frac{1}{\rho })^2<\frac{1}{\rho ^2}$. Consequently, the diameter of the set $\psi (J)= f_t( \{W_t + {\mathrm {i}}y \,:\, y \in (0,y_0]\})$ is at most $K \, \rho ^{-2} \, R$. Hence (6.33) holds. $\square$

The following definition enables the use of crossing probability bounds to establish a bound for the speed of convergence of the radial limit to the tip.

Definition 6.7

For $\tilde{\varepsilon }>0$, $r_0$, $\rho >0$, we say that the path $\hat{\gamma }$ has the $(\tilde{\varepsilon }, r_0, \rho )$-tempered 6-fold crossing property if for any $R< r_0$, we cannot find for any pair $(r,R)=(R^{1+\tilde{\varepsilon }},R)$, parameters s, t so that the property of Definition 6.6 would hold.

Theorem 6.7

For the site percolation interface $\hat{\gamma }$ (transformed to $\mathbb {D}$) for any $\tilde{\varepsilon } >0$, there exists a tight random variable $r_0$ such that $\hat{\gamma }$ has the $(\tilde{\varepsilon }, r_0, \rho )$-tempered 6-fold crossing property.

Proof

Let $r>0$ and $R>12 r$. Let $\sigma _k$ be defined by $\sigma _0=0$ and recursively by $\sigma _{k+1} = \sup \{ t \ge \sigma _k \,:\, \mathrm{diam}( \sigma [\sigma _k,t] ) < R/4 \}$. Then there exists finite random N such that $\sigma _{N-1}< \infty$, but $\sigma _N=\infty$. Let $J_k = \gamma [\sigma _{k-1},\sigma _k]$, $k=1,2,3,\ldots ,N$ and let $J_k$, $k>N$, be a partition of $\partial \mathbb {D}$ into arcs of diameter at most R / 4—the number of such arcs can be chosen to be at most a constant times 1 / R. Observe that if the curve is divided into pieces that have diameter at most $R/4-\varepsilon$, $\varepsilon >0$, then none of these pieces can contain more than one of the $\gamma (\sigma _k)$. Therefore $N \le \inf _{\varepsilon >0} M(\gamma ,R/4 - \varepsilon ) \le M(\gamma ,R/8)$ where M is as in Sect. 6.3.4.1. By Propositions 6.8 and 6.10, N is a tight random variable, which we will use below.

Define also stopping times $\tau _{j,k} = \inf \{ \, t \in [\sigma _{k-1}, \sigma _k] \;:\; {{\mathrm{dist}}}_t ( \gamma (t), J_j ) \le r \} .$ for $k\in \llbracket 1,N\rrbracket$ and $j \in \llbracket 1,k-1\rrbracket$ or $j>N$. Here ${{\mathrm{dist}}}_t(\gamma (t),A)$ is the infimum of the numbers l such that $\gamma (t)$ can be connected to the set A by a path of diameter less than l in $\mathbb {D}\setminus \gamma (0,t]$.

Suppose that the event E(r, R) occurs. Take any C, s, t as in the definition of E(r, R).

Let j, k be such that the end points of C are on $J_j$ and $J_k$. Notice that $j \ne k$ and j, k can’t both be larger than N and hence we can suppose that $k\in \llbracket 1,N\rrbracket$ and $j \in \llbracket 1,k-1\rrbracket$ or $j>N$. Also notice that ${{\mathrm{dist}}}(J_j,J_k) \le r$ and hence $\tau _{j,k}$ is finite.

Let $\tilde{C}$ be a path of diameter less than 2r in $\mathbb {D}\setminus \gamma (0,\tau _{j,k}]$ connecting $\gamma (\tau _{j,k})$ to $J_j$. We claim that $\tilde{C}$ disconnects $\gamma (t)$ from $+1$ in $\mathbb {D}\setminus \gamma (0,\tau _{j,k}]$ and that $| \gamma (t) - \gamma (\tau _{j,k}) | > R/2$.

Let $\tilde{J}_j$ and $\tilde{J}_k$ be the subpaths of $J_j$ and $J_k$, respectively, that connect an endpoint of C to an endpoint of $\tilde{C}$. Let $\Gamma$ be the concatenation of C, $\tilde{J}_j$, $\tilde{C}$ and $\tilde{J}_k$ which closes to a loop. Then the points disconnected by C from $+1$ in $\mathbb {D}\setminus \gamma (0,s]$ but not by $\tilde{C}$ in $\mathbb {D}\setminus \gamma (0,\tau _{j,k}]$, form a subset of (closure of) interior of $\Gamma$.

Since $\Gamma$ is contained in $\overline{B(\gamma (s),R/4+3r)}$ and $|\gamma (t) - \gamma (s)| \ge R$, $\gamma (t)$ is not in the closure of the interior of $\Gamma$. Consequently $\gamma (t)$ is disconnected by $\tilde{C}$ from $+1$ in $\mathbb {D}\setminus \gamma (0,\tau _{j,k}]$. In addition, by triangle inequality, $|\gamma (t) - \gamma (\tau _{j,k})|> R - (R/4+3r) > R/2$.

Let $A_{j,k}=A(\gamma (\tau _{j,k}),2r,R/r)$, $\tilde{A}_{j,k} = (A_{j,k} \cap (\mathbb {D}\setminus \gamma (0,\tau _{j,k}]))$ and

$$\begin{aligned} V_{j,k} = \left\{ z \in \tilde{A}_{j,k} \,:\, \begin{array}{l} \text {connected component of }z \text {in }\tilde{A}_{j,k}\text { is disconnected} \\ \text {from }+1\text { by }\tilde{C}\text { in }\mathbb {D}\setminus \gamma (0,\tau _{j,k}] \end{array} \right\} . \end{aligned}$$

Notice that each component of $V_{j,k}$ has percolation boundary sites of only one type, either they are all open or all closed. Then on the event E(r, R), when $C,\tilde{C},j,k$ are as above, then $\hat{\gamma }(u)$, $u > \tau _{j,k}$, crosses the annulus $A_{j,k}$ using the set $V_{j,k}$. As usual, this implies an open or closed percolation path crossing of $A_{j,k}$ in a component of $V_{j,k}$, whose boundary sites have the opposite state. By the RSW estimate of Corollary 6.5, using similarly Lemmas 6.9 and  6.10 as in the proof of Proposition 6.9. we can show that

$$\begin{aligned} \mathsf {P}[ \exists t \ge \tau _{j,k} \text { s.t. } \gamma (\tau _{j,k},t] \text { crosses }A_{j,k} in V_{j,k} ] \le K_1 \left( \frac{r}{R} \right) ^{\varDelta _1} . \end{aligned}$$

By Propositions 6.8 and 6.10, summing over pairs (j, k) we find that

$$\begin{aligned} \mathsf {P}[E(r,R) ] \le const.\left( R^{\varepsilon \varDelta _k - 2(1+\varepsilon )} + R^{-4(1+\varepsilon )} \left( \frac{r}{R} \right) ^{\varDelta _1} \right) \end{aligned}$$

If we choose $r =c R^{1+\tilde{\varepsilon }}$, $\tilde{\varepsilon }\in (0,\frac{4(1+\varepsilon )}{\varDelta _1})$, then $\mathsf {P}[E(c R^{1+\tilde{\varepsilon }},R) ] \le C R^{\alpha }$ for some $\alpha >0$. If we set $R= 2^{-n}$ and sum over n, we see that by the Borel-Cantelli lemma, there exists a random variable $r_0>0$ such that $\hat{\gamma } \notin E(R^{1+\tilde{\varepsilon }},R)$ for $R \in (0,r_0)$. $\square$

6.3.5.2 Regularity of the Driving Process

Let’s start by recalling some facts from the proof of Proposition 4.1. Notice firs that all $\gamma$ in the set (6.32) once mapped to $\mathbb {H}$ conformally such that $+1$ is mapped on $\infty$ are eligible for description as Loewner chains. Therefore the conclusions in the proof of Proposition 4.1 apply to them.

Consider a simple curve $\gamma$ of $\mathbb {H}$ of the Loewner type parametrized by the capacity. Let $0 \le s \le t$ and define

$$\begin{aligned} \tilde{\gamma }_s (t) = g_s(\gamma (t)) \end{aligned}$$

Then by the proof of Proposition 4.1,

$$\begin{aligned} \max _{u \in [s,t]} \mathrm{{Im}}\tilde{\gamma }_s(u) \le 2 \sqrt{|t-s|} , \qquad \max _{u \in [s,t]} | \mathrm{{Re}}\tilde{\gamma }_s(u) - W_s| \le \max _{u \in [s,t]} | W_u -W_s| . \end{aligned}$$

Therefore if $\gamma _s(u)$ exits the rectangle $[W_s -L, W_s+L] \times [0,2\sqrt{|t-s|}]$ from the sides $\mathrm{{Re}}z = W_s \pm L$, then $\max _{u \in [s,t]} | W_u -W_s| \ge L$. Also a kind of a converse is true as shown next.

Lemma 6.15

$\frac{1}{2} \sup _{u \in [s,t]} |W(u)-W(s)| - 2 \sqrt{|s-t|} \le \sup _{u \in [s,t]} |\mathrm{{Re}}\tilde{\gamma }s(u)- W(s)| \le \sup _{u \in [s,t]} |W(u)-W(s)|$

Proof

The upper bound follows from the above considerations.

For the lower bound, assume without loss of generality that $s=0$ and $W(0)=0$. Then if $M= \Vert \mathrm{{Re}}\gamma \Vert _{\infty , [0,t]}$ and $R= \sqrt{M^2 + (2 \sqrt{t})^2}$, then $\gamma [0,t] \subset \overline{B(0,R)}$. Consequently $W_u \in [g_u(-R),g_u(R)]$ by monotonicity of Loewner maps on the boundary. Next notice that $g_u(R) \le \phi (R)$ and $g_u(-R) \ge \phi (-R)$ where $\phi (z) = z + R^2 z^{-1}$. Thus $-2R< g_u(-R)< W_u< g_u(R) < 2R$ and hence $\frac{1}{2} |W_u| \le R \le M + 2\sqrt{t}$. $\square$

Proposition 6.12

For percolation interface $\hat{\gamma }$, it holds that for each $\alpha < \frac{1}{2}$, there exists a random variable $C_\alpha$ such that $|W(t) - W(s)| \le C_\alpha \, |t-s|^\alpha$ for $t,s \in [0,T]$. Furthermore, $C_\alpha$ is a tight random variable.

Proof

Let us first show that there are constants K and $\varepsilon$ such that

$$\begin{aligned} \mathsf {P}\left[ \sup _{u \in [s,t]} |W(u) - W(s)| \ge L \right] \le K \exp \left( -\varepsilon \frac{L}{ \sqrt{|s-t|} } \right) . \end{aligned}$$

(6.35)

If $\sup _{u \in [s,t]} |W(u) - W(s)| \ge L$, then by Lemma 6.15 $\sup _{u \in [s,t]} |\mathrm{{Re}}\tilde{\gamma }_s(u)- W(s)| \ge \frac{L}{2} - 2 \sqrt{|s-t|}$. On the other hand, $\sup _{u \in [s,t]} \mathrm{{Im}}\gamma _s (u) \le 2 \sqrt{|s-t|}$ always by (4.18).

Choose $\rho '>0$ such that when $R/r > \rho '$ then the right-hand side of the inequality in Corollary 6.5 is less than $e^{-1}$. Then choose $\rho >0$ such that $\rho$ and $\rho '$ are as in Lemma 6.11. Let $C_k = [x_k, x_k + {\mathrm {i}}2 \sqrt{|s-t|}]$ where $x_k = 2\rho k \sqrt{|s-t|}$. By above if $\sup _{u \in [s,t]} |W(u) - W(s)| \ge L$, then $\gamma _s (u)$, $u \in [s,t]$, hits either all $C_1,C_2,\ldots ,C_m$ or all $C_{-1},C_{-2},\ldots ,C_{-m}$ where $m = \lfloor L/ (2\rho \sqrt{|s-t|}) \rfloor$. When $L/ \sqrt{|s-t|}$ is large enough, then $m \ge L/ (4\rho \sqrt{|s-t|})$. Thus $\mathsf {P}[ \sup _{u \in [s,t]} |W(u) - W(s)| \ge L ] \le 2 e^{-m}$ by Corollary 6.5 and Lemma 6.11, and (6.35) follows for some constant $\varepsilon >0$.

Choose any $\alpha \in (0,1/2)$ and set $\delta _n = T 2^{-n}$ and $L_n = 2^{-\alpha n}$ for $n \in \mathbb {Z}_{>0}$. Then

$$\begin{aligned}&\mathsf {P}\left[ \sup _{u \in [(k-1) \delta _n,k \delta _n]} |W( u) - W( (k-1) \delta _n)| \ge L_n \text { for some } k \in \llbracket 1,2^n\rrbracket \right] \nonumber \\ \le&2^n K \exp \left( -\frac{\varepsilon }{ \sqrt{T} } 2^{ (\frac{1}{2} - \alpha ) n} \right) . \end{aligned}$$

(6.36)

Since the upper bound is summable over n, by the Borel–Cantelli lemma, there exists a random variable N such that for all $n \ge N$ and for all $k \in \llbracket 1,2^n\rrbracket$ it holds that

$$\begin{aligned} \sup _{u \in [(k-1) \delta _n,k \delta _n]} |W( u) - W( (k-1) \delta _n)| < L_n . \end{aligned}$$

(6.37)

Thus if $\delta _{n+1} < |u-v| \le \delta _n$, then by the triangle inequality, $|W(u) - W(v)| \le 3 L_n = 3 \, (2^{-n})^\alpha < 2^{2+\alpha } T^{-\alpha } |u-v|^\alpha$. Thus $|W(u) - W(v)| \le 2^{2+\alpha } T^{-\alpha } |u-v|^\alpha$ for all $u,v \in [0,T]$ such that $|u-v| \le \delta _N$.

Notice also that $\sup _{u \in [0,T]} |W(u)| \le 2^N L_N$ which is finite. Thus the first claim of the proposition follows.

For the second claim, use (6.36) to give uniform estimates for $\mathsf {P}[ N>n]$ as $n \rightarrow \infty$. $\square$

6.3.5.3 Tightness of the Capacity Parametrized Random Curves

Remember that $\gamma _\mathbb {D}$ was the percolation interface transformed to $\mathbb {D}$. Let $\gamma _\mathbb {H}$ be $\phi ( \gamma _\mathbb {D})$, where $\phi (z) = {\mathrm {i}}\frac{1 + z}{1- z}$, reparameterized with capacity.

Remember also that a metric on $C(\mathbb {R}_{\ge 0},\mathbb {C})$ is given by

$$\begin{aligned} {\mathrm {d}}(\gamma _1,\gamma _2 ) = \sum _{n=1}^\infty 2^{-n}( 1 \wedge \Vert \gamma _1 - \gamma _2 \Vert _{[0,n],\infty } ) \end{aligned}$$

Theorem 6.8

Let $\mathscr {D}$ be a collection of quadruplets $(U,a,b,\delta )$ and let $\mathscr {M}$ be the collection of all probability laws $\mu _{(U,a,b,\delta )}$ of $\gamma _\mathbb {H}$, where $(U,a,b,\delta )$ runs over all quadruplets in $\mathscr {D}$ and $\gamma$ is the percolation interface in (U, a, b) with lattice mesh $\delta >0$. Then the collection $\mathscr {M}$ is tight.

Proof

Let $\varepsilon >0$. We will find a relatively compact set E such that $\mu _{(U,a,b,\delta )}[E] \ge 1 - \varepsilon$.

Let $n \in \mathbb {Z}_{>0}$. We will first consider $\gamma _\mathbb {H}(t)$, $t \in [0,n]$, and its driving term W(t), $t \in [0,n]$. By Proposition 6.12, for any $\alpha >0$, there exists a random variable $C_{n,\alpha }$ such that $|W(t) - W(s)| \le C_{n,\alpha } \, |t-s|^\alpha$ for all $s,t \in [0,n]$. Notice also that $W(0)=0$. Choose $m_{n,1}>0$ such that $\mu _{(U,a,b,\delta )} [ C_{n,\alpha } \le m_{n,1} ] \ge 1 - \varepsilon 2^{-n-1}$ for all $(U,a,b,\delta ) \in \mathscr {M}$. Then $\mu _{(U,a,b,\delta )} [ C_{n,\alpha } \le m_{n,1} \text { for all } n ] \ge 1 - \varepsilon \sum _{n=1}^\infty 2^{-n-1} \ge 1 - \varepsilon 2^{-1}$.

Let $f:(t,z) \rightarrow \mathbb {H}$ be the inverse (for fixed t) of the Loewner map $g:(t,z) \rightarrow \mathbb {H}$ as usual and $F(t,y) = f(t,W(t) + {\mathrm {i}}y)$. By Proposition 6.11 and Theorem 6.7, there exists a function $\mu :\mathbb {R}_{>0}\rightarrow \mathbb {R}_{>0}$ and a random variable $\tilde{C}_n$ such that $\lim _{y \rightarrow 0} \mu (y) = 0$ and $|F(t,y) - \gamma _\mathbb {H}(t)| \le \tilde{C}_n \mu (y)$ for all $y \in \mathbb {R}_{>0}$ and $t \in [0,n]$. Choose $m_{n,2}$ such that $\mu _{(U,a,b,\delta )} [ \tilde{C}_{n} \le m_{n,2} ] \ge 1 - \varepsilon 2^{-n-1}$ for all $(U,a,b,\delta ) \in \mathscr {M}$. Then it holds that ${\mu _{(U,a,b,\delta )} [ \tilde{C}_{n} \le m_{n,2} \text { for all } n ]} \ge {1 - \varepsilon \sum _{n=1}^\infty 2^{-n-1} \ge 1 - \varepsilon 2^{-1}}$.

Let E be the event that $\gamma _\mathbb {H}$ satisfies $C_{n,\alpha } \le m_{n,1}$ and $\tilde{C}_{n} \le m_{n,2}$ for all n. By above, $\mu _{(U,a,b,\delta )} [E] \ge 1-\varepsilon$. The claim of the theorem follows if we manage to show that E is relatively compact.

Let $\gamma _k$ be a sequence in E. Since $W_k$ are all $\alpha$-Hölder continuous with the Hölder norm bounded by $m_{n,1}$, we can extract a converging subsequence $W_{k_j}$ by the Arzelà–Ascoli theorem. Furthermore, by a standard diagonal argument we can suppose that $W_{k_j}$ converges uniformly on each [0, n]. In particular, $W_{k_j}$ is a Cauchy sequence on each [0, n].

Let $\tilde{\varepsilon }>0$. Let $\delta >0$ and $n_0 \in \mathbb {Z}_{>0}$ be such that $\lambda (\delta ) <\tilde{\varepsilon }$ and $2^{-n_0} <\tilde{\varepsilon }$, where $\lambda (\delta )$ is as in Lemma 6.4. By passing to a subsequence, we can suppose that for any $i,j \ge n_0$ and $n \le n_0$ it holds that $\Vert W_{k_i} - W_{k_j} \Vert _{[0,n],\infty } \le (C(n,\delta ))^{-1} \tilde{\varepsilon }$ where $C(n,\delta )$ is as in Lemma 6.4. By Lemma 6.4,

$$\begin{aligned} \Vert \gamma _{k_i} - \gamma _{k_j} \Vert _{\infty , [0,n]} \le C(n,\delta ) \Vert W_{k_i} - W_{k_j} \Vert _{[0,n],\infty } + 2 \lambda (\delta ) . \end{aligned}$$

Therefore

$$\begin{aligned} {\mathrm {d}}(\gamma _{k_i},\gamma _{k_j} )&\le 2^{-n_0} + \sum _{n=1}^{n_0} 2^{-n}( C(n,\delta ) \Vert W_{k_i} - W_{k_j} \Vert _{[0,n],\infty } + 2 \lambda (\delta ) ) \le 4 \tilde{\varepsilon } . \end{aligned}$$

Hence $\gamma _{k_j}$ is a Cauchy sequence and converges in $C(\mathbb {R}_{\ge 0},\mathbb {C})$ which is a complete metric space. Thus E is a relatively compact set. The claim follows. $\square$

6.3.6 Cardy–Smirnov Formula of a Crossing Probability

In this section, we present the full argument showing the convergence of the percolation interface $\gamma _{\delta _n}$ to a SLE$(6)$ random curve $\gamma$. We use Smirnov’s very readable original papers [10, 11] and Beffara’s equally excellent note [2] on the convergence of so called discrete martingale observables of the site percolation model. We take the liberty to omit some details, but we will state explicitly when we do so. We are careful in particular on the mode of convergence of the observable which is needed in passing to the limit with a martingale property.

6.3.6.1 Introduction: Boundary Value Problems and Martingales

We have already developed the theory of regularity of random curves, which ensures the convergence of the random curves along subsequences by a compactness argument. The convergence of the entire sequence is thus equivalent to the uniqueness of the limit. Hence to complete the approach we need for each ${t \in \mathbb {R}_{\ge 0}}$, a random variable

$$\begin{aligned} \gamma \mapsto X(t,z; \, \gamma ) \end{aligned}$$

where $z \in \mathbb {C}$ is a free variable which we can vary, such that $X(t,z; \, \gamma )$ depends non-trivially on z and $\gamma [0,t]$ and its law can be efficiently written and analyzed. Not surprisingly, if you compare this approach to that of the characteristic function of a real-valued random variable, this is sufficient for characterizing the law of $\gamma$.

In many cases including the case of the percolation interface, the observable $X(t,z; \, \gamma )$ can be written as a solution of a boundary value problem as $X(t,z; \, \gamma ) = h(z)$ where h is defined in the following way. First of all, h is the scaling limit $h = \lim _{\delta _n \rightarrow 0} H_{\delta _n}$ of a discrete observable $H_{\delta _n}$ , which solves a corresponding discrete boundary value problem and is a natural percolation quantity, namely, a crossing probability. Secondly, the continuum boundary value problem solved by h can be formulated in the following way. Denote by $U_t$ the connected component containing a neighborhood of the arc bc in $U \setminus \gamma (0,t]$ and by $a_t$ the point $\gamma (t)$. The continuum boundary value problem for percolation is the following

$$\begin{aligned} {\left\{ \begin{array}{ll} \varDelta h = 0 &{}\text {, in} U_t \\ h(a_t) = 1 &{} \\ h = 0 &{}\text {, in }bc \\ \partial _{u} h = 0 &{}\text {, in} (a_t b) \cup (c a_t) \\ \end{array}\right. } \end{aligned}$$

(6.38)

where $\partial _u$ is the directional derivative in direction $u = (\tau /\nu )^{\frac{1}{3}} \, \nu$ where $\nu$ is the outward normal and $\tau$ is the tangent of $\partial U_t$ at the boundary point; the boundary is oriented from $a_t$ to bc on both arcs $a_t b$ and $c a_t$ and the third root is defined such that $(e^{{\mathrm {i}}\theta })^{\frac{1}{3}} = e^{\frac{{\mathrm {i}}\theta }{3}}$ when $\theta \in (-\pi ,\pi )$.

An essential property of $X(t,z;\,\gamma )$ that we need below, is that it satisfies a martingale property. For the discrete observable $X_n(\tau _n(t),z)=H_{\delta _n}(z)$, as we shall see, it holds that

$$\begin{aligned} \mathsf {E}_n[ X_n(\tau _n(t),z) \,|\, \mathscr {F}_s] = X_n(\tau _n(s),z) \end{aligned}$$

(6.39)

where $\tau _n(t)$ is the least discrete time such that the path $\gamma _\mathbb {H}$ has capacity greater or equal to t.

Let’s extend the martingale property to the scaling limit of the observable. For that we need some assumptions on the mode of its convergence. Let E be an event such that

$$\begin{aligned} \mathsf {P}_n[E] > 1 -\varepsilon \end{aligned}$$

(6.40)

for all $n \ge n_0$ and suppose that on the event E it holds that

$$\begin{aligned} \sup _{\gamma \in supp (\mathsf {P}_n) \cap E}| X_n(\tau _n(t),z_n;\,\gamma ) - X(t,z;\,\gamma ) | < \varepsilon \end{aligned}$$

(6.41)

for all $n \ge n_0$. We can furthermore set $E_n = \text {supp}(\mathsf {P}_n) \cap E$ and then (6.40) holds when E is replaced by $E_n$.

Let $u \in [s,\infty )$ and f be a continuous, bounded $\mathscr {F}_s$-measurable random variable. By scaling we may assume that $|f| \le 1$, $|X_n| \le 1$ and $|X| \le 1$. Then

$$\begin{aligned}&\int X_n(\tau _n(u),z) \, f \, {\mathrm {d}}P_n = \int \mathbbm {1}_{E_n} \, X_n(\tau _n(u),z) \, f \, {\mathrm {d}}P_n + \text {error} \nonumber \\ =&\int \mathbbm {1}_{E_n} \, X(u,z) \, f \, {\mathrm {d}}P_n + \text {error} = \int X(u,z) \, f \, {\mathrm {d}}P_n + \text {error} \nonumber \\ =&\int X(u,z) \, f \, {\mathrm {d}}P + \text {error} . \end{aligned}$$

(6.42)

On each line, the error is at most of the order $\varepsilon$ (bounded from above in absolute value by a universal constant times $\varepsilon$). The first and third equality uses (6.40). The second one uses (6.41) and the fourth one follows from the convergence of $\mathsf {P}_n$ to $\mathsf {P}$ in the sense of weak convergence of probability measures. The martingale property (6.39) is by definition equivalent to the fact that

$$\begin{aligned} \int X_n(\tau _n(t),z) \, f \, {\mathrm {d}}P_n = \int X_n(\tau _n(s),z) \, f \, {\mathrm {d}}P_n \end{aligned}$$

(6.43)

holds for all continuous, bounded $\mathscr {F}_s$-measurable random variables h. Using the estimate (6.42) for both $u=t$ and $u=s$, we find that

$$\begin{aligned} \left| \int X(t,z) \, f \, {\mathrm {d}}P - \int X(s,z) \, f \, {\mathrm {d}}P\right| \le C \Vert f \Vert _\infty \varepsilon \end{aligned}$$

(6.44)

holds for all continuous, bounded $\mathscr {F}_s$-measurable random variables f. Here C is a absolute constant. Since $\varepsilon$ is arbitrary it holds that,

$$\begin{aligned} \mathsf {E}[ X(t,z) \,|\, \mathscr {F}_s] = X(s,z) . \end{aligned}$$

(6.45)

That is, X(t, z) as a stochastic process is a martingale.

In next subsections, we define the observable more carefully and establish most important properties, namely, the martingale property (6.39) and the uniform convergence (6.41).

6.3.6.2 The Discretization of the Domain

We aim to define a curve $\gamma$ in a domain U from a boundary point a to another boundary point b. We need further third boundary point c and a point z in $\overline{U}$ to characterize the law of $\gamma$. We proceed in the following steps:

• Let $\mathbb {L}_ tri$ be the triangular lattice and $\mathbb {L}_ hex$ be its dual, the hexagonal lattice. Let

  $$\tau = e^{{\mathrm {i}}2\pi /3}.$$

• Let U be any bounded simply connected domain and a, b its distinct boundary points.

• Choose a sequence $(U_{\delta _n}, a_{\delta _n}, b_{\delta _n})$ where $\delta _n$ is the lattice mesh tending to 0 as n tends to $\infty$ and $U_{\delta _n}$ a bounded, simply connected domain such that $\partial U_{\delta _n}$ is a lattice path on $\delta _n \mathbb {L}_ hex$ and $a_{\delta _n}, b_{\delta _n})$ its boundary points which are assumed to be sites on $\delta _n \mathbb {L}_ hex$.

• Map next (U, a, b) conformally onto $(\mathbb {H},0,\infty )$.

• Pick $-l \in \mathbb {R}_{<0}$ and $m \in \mathbb {H}\cup \mathbb {R}_{>0}$. Then map back to U using the inverse of the conformal map. Denote the image of $-l$ by c and m by z.

• Choose $c_{\delta _n}$ in the arc $b_{\delta _n}a_{\delta _n}$ and $z_{\delta _n}$ in the union of the arc $a_{\delta _n}b_{\delta _n}$ and $U_n$ such that they are sites on $\delta _n \mathbb {L}_ hex$.

• Let $\phi _{\delta _n}$ and $\phi$ be the conformal and onto maps from $(\mathbb {D},1,\tau ,\tau ^2)$ onto the domains $(U_{\delta _n}, a_{\delta _n}, b_{\delta _n}, c_{\delta _n})$ and (U, a, b, c), respectively. Those maps are unique. Next we make a critical assumption that $(U_{\delta _n}, a_{\delta _n}, b_{\delta _n}, c_{\delta _n})$ converges to (U, a, b, c) in the Carathéodory sense, that is, $\phi =\lim _{n \rightarrow \infty } \phi _{\delta _n}$. Assume also that $z = \lim _{n \rightarrow \infty } z_{\delta _n}$.

• Let $V_{\delta _n}$ and $V_{1,\delta _n}$ be as in Sect. 6.3.1.1. Then $V_{\delta _n}$ is the set of sites where we consider the percolation configuration and $V_{1,\delta _n}$ is the set of boundary sites where we apply the chosen boundary conditions.

For illustration see Fig. 6.5. Let us shorten the notation by dropping n from $\delta _n$.

Fig. 6.5

The microscopic details of discrete domain. Notice that we arrange so that a and b are boundary points—this is needed, if we want the interface to be well-described by a chordal Loewner chain. The points c, z or x are interior points, but c and z, when $z=x$, are points “next to the boundary”

6.3.6.3 The Crossing Event of Cardy–Smirnov Formula

Let $U,a,b,c,z,U_{\delta }$ and $a_{\delta }, b_{\delta }, c_{\delta },z_{\delta }$ be as above. Assume first that $z_\delta$ is a vertex of a hexagon in $V_{\delta }$.

Definition 6.8

Define an event $E_{a,\delta }(z_{\delta })$ that there exists a simple open path on $V_\delta$ that separates $a_\delta$ and $z_{\delta }$ from $b_\delta$ and $c_\delta$. More specifically, this means that there exists a simple path $\pi = (x_0,x_1,\ldots ,x_n,x_{n+1})$ such that (i) $x_1,\ldots ,x_n$ are in $V_\delta$ and their state is open, (ii) $x_0$ and $x_{n+1}$ are in $V_{1,\delta }$, the edge $\{x_0,x_1\}$ crosses the arc $a_\delta b_\delta$ and the edge $\{x_n,x_{n+1}\}$ crosses the arc $c_\delta a_\delta$, and (iii) the union of the hexagons with centers $x_0,x_1,\ldots ,x_n,x_{n+1}$ disconnects $z_\delta$ from $b_\delta c_\delta$ in $U_\delta$. Define similarly $E_{b,\delta }(z)$ and $E_{c,\delta }(z)$ by cyclically permuting the points $a_\delta ,b_\delta ,c_\delta$.

Define $H_{a,\delta }(z_\delta )$ as the probability of the event $E_{a,\delta }(z_\delta )$ and similarly $H_{b,\delta }(z_\delta )$ and $H_{c,\delta }(z_\delta )$. Then $H_{a,\delta }=0$ on the boundary arc $b_\delta c_\delta$ and as we shall see $H_{a,\delta }(a_\delta ) \approx 1$.

Suppose that the set of $z_\delta$’s, on which we have defined $H_{a,\delta }$, is denoted by W. When $z_\delta$ is a vertex of some hexagon whose center is in $V_{1,\delta }$ and no hexagon whose center is in $V_\delta$, define $H_{a,\delta }(z_\delta )$ to be equal to the value of $H_{a,\delta }$ at the neighboring site in W, if the site exists (there is at most one). If it doesn’t exists, fix an arbitrary rule (which can depend for instance on the local shape of the boundary) for calculating the value as a convex combination of the values of $H_{a,\delta }$ on the sites of W in the hexagon of $z_\delta$. Threat $z_\delta$ in this case as a generalized boundary point (as a limit of a sequence of interior points in a topology that separates the different sides of the boundary). As a consequence $H_{a,\delta }$ is now defined on all vertices of the hexagons whose centers are in $V_{\delta } \cup V_{1,\delta }$ and we are in good shape to define it in all points of $\overline{U_\delta }$ as we will do in the next subsection.

Define functions

$$\begin{aligned} H_\delta = H_{a,\delta } + \tau H_{b,\delta } + \tau ^2 H_{c,\delta } , \qquad S_\delta = H_{a,\delta } + H_{b,\delta } + H_{c,\delta } . \end{aligned}$$

(6.46)

Suppose now that z is on the boundary arc $a_\delta b_\delta$ (similar statement holds for $c_\delta a_\delta$). Then $H_{a,\delta }(z)$ is equal to the probability that in $V_\delta$ there exists an open path from the arc $c_\delta a_\delta$ to the arc $z b_\delta$. The next lemma follows since for instance, on $a_\delta b_\delta$, $E_{a,\delta }(z)$ and $E_{b,\delta }(z)$ are almost complementary events.

6.3.6.4 A Continuous Extension of a Discrete Function

Use the following (one of many) construction that extends a function f defined on a set of sites of a simply connected subgraph of a planar lattice to the closed set of points consisting of the union of the closed faces of the graph. For a neighboring pair of sites x, y, extend f on the edge (line segment) between them linearly, that is, $f( \, t x + (1-t) \, y) = t \, f(x) + (1-t) \, f(y)$ for any $t \in [0,1]$. After this step, f is defined on all edges of the graph and thus on the boundary of any face of the graph.

Extend f inside each of the faces using only the values on the boundary of that face. Explicitly, we can use the harmonic extension of f inside each face. A property of the chosen extension is that $|f(x)-f(y)|$ is maximized over a face when x and y are sites of the lattice. Since the diameter of a hexagon in $\delta \mathbb {L}_ hex$ is equal to $2\delta /\sqrt{3}$, we get the following lemma.

Lemma 6.16

For any $r >0$ and $z_0 \in \mathbb {C}$,

$$\sup _{x,y \in B(z_0,r)} |f(y)-f(x)| \le \max _{v,w \in V( \delta \mathbb {L}_ hex ) \cap B(z_0,r+2\delta /\sqrt{3})} |f(v)-f(w)|. $$

6.3.6.5 Equicontinuity of Discrete Observables and Uniform Convergence

Next we will establish the convergence of $H_{a,\delta },H_\delta ,S_\delta , \ldots$ as $\delta$ tends to 0. First we will establish equicontinuity. For a domain (U, a, b, c) and $z \in U$, define

$$\begin{aligned} d_{(U,a,b,c)} (z) \mathrel {\mathop :}= \max \{ {{\mathrm{dist}}}(z,AB), {{\mathrm{dist}}}(z,BC), {{\mathrm{dist}}}(z,CA) \} . \end{aligned}$$

(6.47)

Lemma 6.17

Let $0<r<R$ and $0<m<\frac{1}{100}$. For any $\varepsilon >0$, there exists $\eta >0$ such that the following holds. If $(U,a,b,c; \delta )$ is a discrete domain such that $\delta < m r$, $\mathrm{diam}(U)<R$ and $\inf \{ d_{(U,a,b,c)} (z) \,:\, z \in \overline{U} \} > r$, and $\phi : \mathbb {D}\rightarrow U$ a conformal and onto map such that $\phi (1)=a$, $\phi (\tau )=b$ and $\phi (\tau ^2)=c$, then

$$\begin{aligned} |H_{A,\delta }\circ \phi (z) - H_{A,\delta }\circ \phi (w)| < \varepsilon \end{aligned}$$

(6.48)

for all $z,w \in \overline{\mathbb {D}}$ such that $|z-w| < \eta$.

Proof

Let $\tilde{\varepsilon }>0$ be much smaller than r and let $0< \eta <\frac{1}{2}$ be such that

$$2\pi R /\sqrt{\log (1/\eta )} < \tilde{\varepsilon }. $$

Then by Lemma 4.6, it holds that ${\text {Length}}[\phi (\mathbb {D}\cap \partial B(z,r) )] < \tilde{\varepsilon }$ for some $\rho \in [\eta ,\frac{1}{2}]$. Denote $\phi (\mathbb {D}\cap \partial B(z,\rho ))$ by C. Let $z_0 \in C$. Then $C \subset B(z_0,\tilde{\varepsilon })$.

It is sufficient to establish that when $\tilde{\varepsilon }> \delta$, there are universal constants K and $\varDelta$ such that

$$\begin{aligned} |H_{A,\delta }\circ \phi (z) - H_{A,\delta }\circ \phi (w)| \le K \left( \frac{\tilde{\varepsilon }}{r} \right) ^\varDelta . \end{aligned}$$

(6.49)

Namely, if $\tilde{\varepsilon }$ is chosen to be less than $r \, (\varepsilon /K)^{1/\varDelta }$ and $\delta < \tilde{\varepsilon }$, the claim follows directly. On the other hand, if $\delta \ge \tilde{\varepsilon }$, then C is fully contained in the union of at most three neighboring hexagons (meeting at a common vertex). In that case, there are two options: either C is a closed loop and $\phi (z)$ and $\phi (w)$ are in its interior or C is an open path with endpoints on the boundary of the domain (and on boundaries of hexagons) and $\phi (z)$ and $\phi (w)$ are contained in the interior of the concatenation of C and the shortest path along the hexagonal boundaries connecting the endpoints. In both cases, it follows that $\phi (z)$ and $\phi (w)$ are in the union of those three hexagons mentioned above. Consequently, C can be replaced by a lattice path with identical properties, except that its length is of the order $\delta$. If (6.49) holds for z and w replaced by (points mapped to) vertices of the hexagonal lattice (of mesh size $\delta$) and $\tilde{\varepsilon }$ by $\delta$ on its right, then using harmonic extension property, the claim follows for the case $\delta \ge \tilde{\varepsilon }$. The bound (6.49) follows in this case in the same way as we will show it below for the other case. We leave filling the full details for the case $\delta \ge \tilde{\varepsilon }$ to the reader.

Since $d_{(U,a,b,c)} (z_0) \ge r$, we can assume that either ${{\mathrm{dist}}}(z_0,AB) \ge r$ or ${{\mathrm{dist}}}(z_0,BC) \ge r$. The case ${{\mathrm{dist}}}(z_0,CA) \ge r$ is symmetric to the former one.

Suppose that ${{\mathrm{dist}}}(z_0,AB) \ge r$. Then in the annulus $A(z_0,\tilde{\varepsilon }, r)$ there is a closed non-trivial loop in the full-plane percolation configuration with high probability. This closed path disconnects $\phi (z)$ and $\phi (w)$ from AB. Consequently on that event, either $E_{A,\delta }(\phi (z))$ and $E_{A,\delta }(\phi (w))$ both occur or they both fail to occur. Thus the bound (6.49) follows from the RSW estimate for open crossings in annuli.

Suppose next that ${{\mathrm{dist}}}(z_0,BC) \ge r$. There is an open non-trivial loop in $A(z_0,\tilde{\varepsilon }, r)$ with high probability and thus similarly as above, (6.49) follows. $\square$

The very same proof gives the following lemma which is a key lemma for verifying the boundary conditions of $H_{a,\delta }$, $H_\delta$ and $S_\delta$.

Lemma 6.18

For each r, R, m as in the previous lemma, there exists a constant C such that the following holds. Let $(U,a,b,c; \delta )$ be as in the previous lemma. Then $0 \le H_{a,\delta } \circ \phi (z) \le C (\log \frac{1}{\varepsilon })^{-1/2}$ for all $z \in \overline{\mathbb {D}}$ such that $|z|>1-\varepsilon$ and $\arg z \in (2\pi /3 - \varepsilon , 4\pi /3 + \varepsilon )$ and $1- C (\log \frac{1}{\varepsilon })^{-1/2} \le H_{a,\delta } \circ \phi (z) \le 1$ for all $z \in \overline{\mathbb {D}}$ such that $|z|>1-\varepsilon$ and $\arg z \in ( - \varepsilon , + \varepsilon )$.

This implies the next result.

Lemma 6.19

On the boundary, $S_\delta =1 + o(1)$ uniformly as $\delta \rightarrow 0$. Values of $H_\delta$ at boundary points are at most at the distance $o(1)$ from the boundary of equilateral triangle $(1,\tau ,\tau ^2)$ and the values are in the same order as the boundary points and include the three vertices of the triangle.

Remark 6.13

Define the set of points $W_\delta$ such that $\phi _\delta (z)$ is a boundary point of the union of the hexagons with centers in $V_{\delta }$ for any $z \in W_\delta$. Then for $z \in W_\delta$, $|z|=1 + o(1)$ uniformly as $\delta \rightarrow 0$, for consecutive points, $|z_{k+1} - z_k| = o(1)$ and $H_{\delta }$ is monotonic on the boundary in the following sense: if $z_k$, $k = 1,2, \ldots , n$ corresponds to $a_\delta b_\delta$, then $H_{a,\delta } \circ \phi _\delta (z_k)$ is monotonic decreasing and $H_{b,\delta } \circ \phi _\delta (z_k)$ is monotonic increasing for $k = 1,2, \ldots , n$. Similarly for the other boundary arcs.

Finally, the next theorem gives the convergence of the observables. We refer to [2, 10, 11] for the combinatorial results, on which this result is based.

Theorem 6.9

Let $R>0$. Let (U, a, b, c) is a bounded domain and $(U_\delta ,a_\delta ,b_\delta ,c_\delta ; \delta )$ is a discrete domain such that $U, U_\delta \subset B(0,R)$ If $(U_{\delta _n},a_{\delta _n},b_{\delta _n},c_{\delta _n})$ converges to (U, a, b, c) and $\phi _n: \mathbb {D}\rightarrow U$ are the corresponding conformal and onto maps (such that $\phi _n(1)=a_{\delta _n}$, $\phi _n(\tau )=b_{\delta _n}$ and $\phi _n(\tau ^2)=c_{\delta _n}$), then $H_{a,\delta _n} \circ \phi _n$ converges uniformly on $\overline{\mathbb {D}}$ to $h_a$ which is equal to $(2\mathrm{{Re}}\varPhi (z)+1)/3$.

Proof

Let (U, a, b, c) is a bounded domain $U \subset B(0,R)$. Then there is $r \in (0,R)$ such that $\mathrm{diam}(U)<R$ and $\inf \{ d_{(U,a,b,c)} (z) \,:\, z \in \overline{U} \} > r$. Therefore for a sequence $(U_{\delta _n},a_{\delta _n},b_{\delta _n},c_{\delta _n})$ of discrete domains such that $U_{\delta _n} \subset B(0,R)$ it holds for n large enough that $\inf \{ d_{(U_{\delta _n},a_{\delta _n},b_{\delta _n},c_{\delta _n})} (z) \,:\, z \in \overline{U_{\delta _n}} \} > r$.

By Lemma 6.17, the sequence $H_{a,\delta _n} \circ \phi _n$ is an equicontinuous family of functions on the compact set $\overline{\mathbb {D}}$ and it is also uniformly bounded as $0 \le H_{a,\delta _n} \le 1$. Thus by Arzelà–Ascoli theorem, there exists a subsequence $H_{a,\delta _{n_k}} \circ \phi _{n_k}$ such that it converges to a continuous function $h_a$ uniformly on $\overline{\mathbb {D}}$. By the same argument, we can suppose that $H_{b,\delta _{n_k}} \circ \phi _{n_k}$ and $H_{c,\delta _{n_k}} \circ \phi _{n_k}$ converge uniformly to continuous functions $h_b$ and $h_c$, respectively.

By Lemma 6.18, $h_a(1)=1$ and $h_n(e^{{\mathrm {i}}\theta })=0$, $\theta \in [2\pi /3,4\pi /3]$. Moreover $s= \lim _{k \rightarrow \infty } S_{\delta _{n_k}}$ and $h= \lim _{k \rightarrow \infty } H_{\delta _{n_k}}$ satisfy $s=1$ on $\partial \mathbb {D}$ and $\theta \mapsto h(e^{{\mathrm {i}}\theta })$, $\theta \in [0,2\pi ]$, is closed loop whose trace is the equilateral triangle $(1,\tau ,\tau ^2)$ and which winds around the origin once in the counterclockwise direction.

By the argument presented in Beffara’s note [2], the functions s and h are holomorphic in $\mathbb {D}$. Consequently, $s=1$ everywhere and h is the conformal map from $\mathbb {D}$ onto the equilateral triangle $(1,\tau ,\tau ^2)$ such that $h(\tau ^k) = \tau ^k$ for $k=0,1,2$. We skip presenting that argument by Smirnov, however we emphasize the importance of that combinatorial result which implies that $S_\delta$ and $H_\delta$ are approximately holomorphic.

We deduce that $h=\varPhi$. Therefore from $h_a + h_b + h_c = 1$ and $h_a + \tau h_b + \tau ^2 h_c = \varPhi$. Thus $\mathrm{{Re}}\varPhi = h_a - \frac{1}{2} ( h_b + h_c)=\frac{1}{2}(3h_a-1)$ or equivalently $h_a = \frac{1}{3}(2 \mathrm{{Re}}\varPhi +1)$. Thus the limit is independent of the choice of subsequence. Consequently, $\lim _{n \rightarrow \infty } H_{a,\delta _n} = \frac{1}{3}(2 \mathrm{{Re}}\varPhi +1)$. $\square$

Corollary 6.7

The convergence is uniform over domains in the following sense. Let C be a crosscut in $\mathbb {D}$ connecting the arc $1\, \tau$ to the arc $\tau ^2 \,1$, $V_C$ is the connected component of $\overline{\mathbb {D}} \setminus \overline{C}$ that contains the arc $\tau \, \tau ^2$. Then for any converging sequence of discrete domains $(U_n,a_n,b_n,c_n ; \delta _n)$ for any sequence of curves $\gamma _n$ such that $\gamma _n \subset V_C$, $\gamma _n$ converges in the Carathéodory sense and $\phi _n \circ \gamma _n$ is a lattice path, the sequence $H_{a,\delta _n} \circ \phi _n$ converges uniformly on $\overline{\mathbb {D}}$ to $h_a$ which is equal to $(2\mathrm{{Re}}\varPhi (z)+1)/3$ and the rate of convergence (as a function of $\delta _n$) is uniform over all sequences $\gamma _n$.

6.3.7 The Characterization of the Percolation Interface Scaling Limit

Theorem 6.10

Let $\mu _{n}$ be the law of the percolation interface in $(U_{\delta _n},a_{\delta _n},b_{\delta _n})$ and $\phi _n$ be a conformal map from $(U_{\delta _n},a_{\delta _n},b_{\delta _n})$ onto $(\mathbb {H},0,\infty )$ and assume that $\phi _n^* \mu _n$ converges in the weak sense to $\mu$ in the topology described in Theorem 6.8. Then $\mu$ is the law of SLE$(6)$ in $\mathbb {H}$.

Remark 6.14

Together with Theorem 6.8 this shows that any sequence of laws $\mu _{n}$ of the percolation interface in $(U_{\delta _n},a_{\delta _n},b_{\delta _n})$ converges to the law of SLE$(6)$.

Proof

Take any subsequence of laws of percolation interfaces that converges in the sense of Theorem 6.8. The convergent subsequence exists by tightness. It is sufficient to show that the limit of the subsequence is SLE(6).

Let $\hat{h}_A$ be the scaling limit of the discrete observable transformed to the upper half-plane with $a=1$, $b=\infty$ and $c=0$. Then $\hat{h}_A=0$ on $\mathbb {R}_{<0}$ and $\hat{h}_A(1)=1$ and on (0, 1) and $(1,\infty )$ it satisfies the correct Neumann-type boundary condition (derivative to the direction of the tangent rotated by $\pm \pi /6$ vanishes). Then we can write the martingale observable X(t, z) as

$$\begin{aligned} X (t,z) = \hat{h}_A \left( \frac{g_t(z)-g_t(c)}{W_t - g_t(c)} \right) . \end{aligned}$$

(6.50)

By the discussion in Sect. 6.3.6.1, $(X(t,x))_{t \in \mathbb {R}_{\ge 0}}$ is a martingale for any subsequent scaling limit of percolation laws.

It is easy to verify based on Example 3.2 that the observable can be written in the form

$$\begin{aligned} \hat{h}_A (z) = L \;\mathrm{{Re}}\left[ -{\mathrm {i}}\tau ^2 \int _0^z (\zeta -1)^{-\frac{2}{3}} \, \zeta ^{-\frac{2}{3}} \, {\mathrm {d}}\zeta \right] . \end{aligned}$$

(6.51)

Here the branches of the integrand are chosen so that for $\zeta >1$ it is real and positive and then extended continuously to $\overline{\mathbb {H}} \setminus \{0,1\}$. The constant L is positive and we omit here it’s value, which can be written explicitly.

Let $\lambda \in \mathbb {R}_{>0}$. Expand for parameters $z=l$, $c=-\lambda l$, where $l>0$, the expression

$$\begin{aligned} \frac{g_t(z)-g_t(c)}{W_t - g_t(c)}&= \frac{l(1+\lambda ) + \frac{2t}{l} \left( 1+\frac{1}{\lambda }\right) + \cdots }{ l \lambda + W_t + \frac{2t}{l} \frac{1}{\lambda } + \cdots } \nonumber \\&= \frac{1+\lambda }{\lambda } \left( 1 - \frac{W_t}{l \lambda } + \frac{1}{l^2 \lambda ^2}(W_t^2 + (\lambda -1)\,2t) +\cdots \right) \end{aligned}$$

(6.52)

in powers of l, as $l \rightarrow \infty$. Write $\hat{\varPhi }(z)= \int _0^z (\zeta -1)^{-\frac{2}{3}} \, \zeta ^{-\frac{2}{3}} \, {\mathrm {d}}\zeta$. Notice then that for $x>1$, it holds that $\mathrm{{Re}}[-{\mathrm {i}}\tau ^2\hat{\varPhi }'(x)] \ne 0$ and $\hat{\varPhi }'' (x) / \hat{\varPhi }' (x) = - \frac{2}{3} \frac{2 x -1}{x(x-1)}$. If we combine these with (6.50), (6.51), and(6.52) we get after an easy calculation that

$$\begin{aligned} X (t,z) = C_1 + C_2 \left( \left( - \frac{1+\lambda }{\lambda ^2} \right) \frac{W_t}{l} + \left( \frac{1-\lambda ^2}{3\lambda ^3}\right) \frac{W_t^2-6 t}{l^2} \right) + \mathscr {O}(l^{-3}) \end{aligned}$$

(6.53)

where $C_1=\mathrm{{Re}}[-{\mathrm {i}}\tau ^2\hat{\varPhi } (1+\lambda ^{-1})]$ and $C_2 = \mathrm{{Re}}[-{\mathrm {i}}\tau ^2\hat{\varPhi }'(1+\lambda ^{-1})]$.

Since $H_t(z)$ as a process in the time variable t is a martingale, we deduce from (6.53) that the processes $W_t$ and $W_t^2 - 6t$ are martingales. From Lévy’s martingale characterization theorem (Theorem 2.6) we deduce that $W_t = \sqrt{6} B_t$ for some standard Brownian motion $(B_t)_{t \in \mathbb {R}_{\ge 0}}$. Thus the limit of the sequence of the percolation interfaces is SLE(6). $\square$