Abstract
We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of \(\hbox {SLE}_\kappa \) loop measures for \(\kappa \in (0,8)\). First, we construct rooted \(\hbox {SLE}_\kappa \) loop measures in the Riemann sphere \(\widehat{\mathbb {C}}\), which satisfy Möbius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its \((1+\frac{\kappa }{8})\)-dimensional Minkowski content. Second, by integrating rooted \(\hbox {SLE}_\kappa \) loop measures, we construct the unrooted \(\hbox {SLE}_\kappa \) loop measure in \(\widehat{\mathbb {C}}\), which satisfies Möbius invariance and reversibility. Third, we use Brownian loop measures to extend the rooted and unrooted \(\hbox {SLE}_\kappa \) loop measures from \(\widehat{\mathbb {C}}\) to subdomains of \(\widehat{\mathbb {C}}\), which respectively satisfy conformal covariance and conformal invariance, and then further use the conformal invariance to extend unrooted \(\hbox {SLE}_\kappa \) loop measures to some Riemann surfaces. Finally, using a similar approach, we construct \(\hbox {SLE}_\kappa \) bubble measures in simply/multiply connected domains rooted at a boundary point. The space-time homogeneity of rooted \(\hbox {SLE}_\kappa \) loop measures in \(\widehat{\mathbb {C}}\) confirms a conjecture by Greg Lawler on the existence of such measures.
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Notes
Werner told the author privately that they were able to prove that the rooted loop measure is well defined and satisfies the conformal Markov property (CMP) as described in the current paper [Theorem 4.1 (ii)]. Given this fact, using the uniqueness statement [Theorem 4.1 (vii)], we see that the loop measures constructed in the current paper for \(\kappa \in (8/3,4]\) agree with Kemppainen–Werner’s measures.
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Acknowledgements
The author would like to thank Greg Lawler and Wendelin Werner for inspiring discussions, thank Laurie Field and an anonymous referee for pointing out mistakes in earlier drafts and providing important suggestions, thank Stéphane Benoist for explaining MKS loop measures, and thank Wei Wu and Yiling Wang for some useful comments. The author acknowledges the support from the National Science Foundation (DMS-1056840) and from the Simons Foundation (#396973). The author also thanks the Institut Mittag-Leffler and Columbia University, where part of this work was carried out during workshops held there.
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Research partially supported by NSF Grant DMS-1056840 and Simons Foundation Grant #396973.
Appendices
Chordal SLE in multiply connected domains
In the appendix, we review the definition of chordal SLE in multiply connected domains for \(\kappa \in (0,8)\). First, we review hulls, Loewner chains and chordal Loewner equations, which define chordal SLE in simply connected domains. The reader is referred to [19] for details.
A subset K of \({\mathbb {H}}\) is called an \({\mathbb {H}}\)-hull if it is bounded, relatively closed in \({\mathbb {H}}\), and \({\mathbb {H}}{\setminus }K\) is simply connected. For every \({\mathbb {H}}\)-hull K, there is are a unique \(c\ge 0\) and a unique \(g_K:{\mathbb {H}}{\setminus }K{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {H}}\) such that \(g_K(z)=z+\frac{c}{z}+O(\frac{1}{z^2})\) as \(z\rightarrow \infty \). The number c is called the \({\mathbb {H}}\)-capacity of K, and is denoted by \({{\,\mathrm{hcap}\,}}(K)\).
If \(K_1\subset K_2\) are two \({\mathbb {H}}\)-hulls, we define \(K_2/K_1=g_{K_1}(K_2{\setminus }K_1)\). Then \(K_2/ K_1\) is also an \({\mathbb {H}}\)-hull, and we have \({{\,\mathrm{hcap}\,}}(K_2/K_1)={{\,\mathrm{hcap}\,}}(K_2)-{{\,\mathrm{hcap}\,}}(K_1)\).
The following proposition is essentially Lemma 2.8 in [21].
Proposition A.1
Let W be a conformal map defined on a neighborhood of \(x_0\in {\mathbb {R}}\) such that an open real interval containing \(x_0\) is mapped into \({\mathbb {R}}\). Then
where \(H\rightarrow z_0\) means that \({{\,\mathrm{diam}\,}}(H\cup \{z_0\})\rightarrow 0\) with H being a nonempty \({\mathbb {H}}\)-hull.
Let \(T\in (0,\infty ]\) and \(\lambda \in C([0,T),{\mathbb {R}})\). The chordal Loewner equation driven by \(\lambda \) is
For each \(z\in \mathbb {C}\), let \(\tau _z\) be such that the maximal interval for \(t\mapsto g_t(z)\) is \([0,\tau _z)\). Let \(K_t=\{z\in {\mathbb {H}}:\tau _z\le t\}\), i.e., the set of \(z\in {\mathbb {H}}\) such that \(g_t(z)\) is not defined. Then \(g_t\) and \(K_t\), \(0\le t<T\), are called the chordal Loewner maps and hulls driven by \(\lambda \). It is known that \((K_t)\) is an increasing family of \({\mathbb {H}}\)-hulls with \({{\,\mathrm{hcap}\,}}(K_t)=2t\) and \(g_t=g_{K_t}\) for \(0\le t<T\). At \(t=0\), \(K_0=\emptyset \) and \(g_0={{\,\mathrm{id}\,}}\).
We say that \(\lambda \) generates a chordal Loewner curve \(\gamma \) if
exists for \(0\le t<T\), and \(\gamma \) is a continuous curve. We call such \(\gamma \) the chordal Loewner curve driven by \(\lambda \). If the such \(\gamma \) exists, then for each t, \({\mathbb {H}}{\setminus }K_t\) is the unbounded component of \({\mathbb {H}}{\setminus }\gamma ([0,t])\), and \(g_t^{-1}\) extends continuously from \({\mathbb {H}}\) to \({\mathbb {H}}\cup {\mathbb {R}}\). Since \({{\,\mathrm{hcap}\,}}(K_t)=2t\) for all t, we say that \(\gamma \) is parametrized by half-plane capacity.
Another way to characterize the chordal Loewner hulls \((K_t)\) is using the notation of \({\mathbb {H}}\)-Loewner chain. A family of \({\mathbb {H}}\)-hulls: \(K_t\), \(0\le t<T\), is called an \({\mathbb {H}}\)-Loewner chain if
-
1.
\(K_0=\emptyset \) and \(K_{t_1}\subsetneqq K_{t_2}\) whenever \(0\le t_1<t_2<T\);
-
2.
for any fixed \(a\in [0,T)\), the diameter of \(K_{t+\varepsilon }/ K_t\) tends to 0 as \(\varepsilon \rightarrow 0^+\), uniformly in \(t\in [0,a]\).
An \({\mathbb {H}}\)-Loewner chain \((K_t)\) is said to be normalized if \({{\,\mathrm{hcap}\,}}(K_t)=2t\) for each t. The following proposition is a result in [21].
Proposition A.2
Let \(T\in (0,\infty ]\). The following are equivalent.
-
(i)
\(K_t\), \(0\le t<T\), are chordal Loewner hulls driven by some \(\lambda \in C([0,T))\).
-
(ii)
\(K_t\), \(0\le t<T\), is a normalized \({\mathbb {H}}\)-Loewner chain.
If either of the above holds, we have
If \(K_t\), \(0\le t<T\), is any \({\mathbb {H}}\)-Loewner chain, then the function \(u(t){:}{=}{{\,\mathrm{hcap}\,}}(K_t)/2\), \(0\le t<T\), is continuous and strictly increasing with \(u(0)=0\), which implies that \(K_{u^{-1}(t)}\), \(0\le t<u(T)\), is a normalized \({\mathbb {H}}\)-Loewner chain.
For \(\kappa >0\), chordal \(\hbox {SLE}_\kappa \) is defined by solving the chordal Loewner equation with \(\lambda (t)=\sqrt{\kappa }B(t)\), where B(t) is a Brownian motion. The chordal Loewner curve \(\gamma \) driven by this driving function a.s. exists, and satisfies \(\lim _{t\rightarrow \infty }\gamma (t)=\infty \). So it is called a chordal \(\hbox {SLE}_\kappa \) curve in \({\mathbb {H}}\) from 0 to \(\infty \). It satisfies that, if \(\kappa \in (0,4]\), \(\gamma \) is simple, and \(K_t=\gamma ((0,t])\); if \(\kappa \ge 8\), \(\gamma \) is space-filling, i.e., visits every point in \(\overline{{\mathbb {H}}}\); if \(\kappa \in (4,8)\), \(\gamma \) is neither simple nor space-filling, and every bounded subset of \(\overline{{\mathbb {H}}}\) is contained in \(K_t\) for some finite \(t> 0\).
Via conformal maps, we may define an \(\hbox {SLE}_\kappa \) curve in any simply connected domain D from one prime end a to another prime end b. Recall that we use \(\mu ^\#_{D;a\rightarrow b}\) to denote the law of such a curve (modulo a time change).
Now we review the definition of chordal SLE in multiply connected domains in [18]. The laws of such SLE are no longer probability measures, but finite or \(\sigma \)-finite measures. We will use the following notation. Suppose D is a simply connected domain with two distinct prime ends a and b. Let \(U\subset D\) be an open neighborhood of both a and b in D. We define
Proposition A.3
Let U and V be open neighborhoods of \({\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\). Suppose \( W: U{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}V\) extends conformally across \({\mathbb {R}}\cup \{\infty \}\) such that \(W({\mathbb {R}})={\mathbb {R}}\) and \(W(\infty )=\infty \). Then for any \(x\in {\mathbb {R}}\),
where \( W'(\infty ){:}{=}(J\circ W\circ J)'(0)\) with \(J(z){:}{=}-1/z\).
Proof
This proposition was proved in [18, Section 4.1] for \(\kappa \in (0,4]\) by considering simply connected subdomains of U. In this proof, we assume that \(\kappa \in (4,8)\). The proof is similar to those of Theorem 5.1 and Lemma 3.4, and uses a standard argument that originated in [22]. WLOG, we may assume that \(x=0\) and \(W(0)=0\). Let \(P_a\) denote the multiplication map \(z\mapsto az\). By conformal invariance of chordal SLE and Brownian loop measure, we know that \(\mu ^{{\mathbb {H}}}_{ P_a(V); 0\rightarrow \infty }={P}_a(\mu ^{{\mathbb {H}}}_{ V; 0\rightarrow \infty })\) for any \(a>0\). Since \((aW)'(0)\cdot (aW)'(\infty )=W'(0)\cdot W'(\infty )\), we may assume that \(W'(\infty )=1\) by replacing W with aW for some \(a>0\).
Let \(\gamma \) be a chordal \(\hbox {SLE}_\kappa \) curve in \({\mathbb {H}}\) from 0 to \(\infty \) with driving function \(\lambda _t=\sqrt{\kappa }B_t\). Let \(g_t\) and \(K_t\), \(0\le t<\infty \), be the chordal Loewner maps and hulls, respectively, driven by \(\lambda \).
Let \(\tau _U\) be the first time that \(\gamma \) exits U. Then \(\beta (t){:}{=}W(\gamma (t))\) is well defined for \(0\le t<\tau _U\). For each \(0\le t<\tau _U\), let \(L_t\) be the \({\mathbb {H}}\)-hull such that \({\mathbb {H}}{\setminus }L_t\) is the unbounded connected component of \({\mathbb {H}}{\setminus }\beta ([0,t])\). If \(K_t\subset U\), then \(L_t=W(K_t)\). Since \(\kappa \in (4,8)\), \(K_t\) may swallow some relatively clopen subset of \({\mathbb {H}}{\setminus }U\) before the time \(\tau _U\), and \(W(K_t)\) is not defined at that time. Using the conformal invariance of extremal length, we can see that \((L_t)\) is an \({\mathbb {H}}\)-Loewner chain (even after \(K_t\) intersects \({\mathbb {H}}{\setminus }U\)). From Proposition A.2, we may reparametrize the family \((L_t)\) using the function \(u(t)={{\,\mathrm{hcap}\,}}(L_t)/2\) to get a family of chordal Loewner hulls. Let \(\sigma _s\), \(0\le s<S{:}{=}u(\tau _U)\), be the driving function for the normalized \((L_s)\). Let \(h_s\), \(0\le s<S\), be the corresponding chordal Loewner maps. We also reparametrize \(\beta \) using u. Then \(\beta \) is the chordal Loewner curve driven by \(\sigma \), and \(\beta _{u(t)}=W(\gamma (t))\), \(0\le t<\tau _U\).
For \(0\le t<\tau _U\), define \( U_t=g_t( U{\setminus }K_t)\), \( V_t=h_{u(t)}( V{\setminus }L_{u(t)})\), and \(W_t=h_{u(t)}\circ W\circ g_t^{-1}\). Then \( U_t\) and \( V_t\) are open neighborhoods of \({\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\), \( W_t: U_t{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}V_t\), and satisfies that, if \(z\in U_t\) tends to \({\mathbb {R}}\) or \(\infty \), then \( W_t\) tends to \({\mathbb {R}}\) or \(\infty \), respectively. By Schwarz reflection principle, \( W_t\) extends conformally across \({\mathbb {R}}\), and maps \({\mathbb {R}}\) onto \({\mathbb {R}}\). Since \(W,g_t,h_{u(t)}\) all fix \(\infty \), and have derivative 1 at \(\infty \), \(W_t\) also satisfies this property.
By the continuity of \( g_t\) and \( h_{u(t)}\) in t and the maximal principle, we know that the extended \( W_t\) is continuous in t (and z). Fix \(0\le t<\tau _U\). Let \(\varepsilon \in (0,\tau _U-t)\). Now \(K_{t+\varepsilon }/K_t\) is an \({\mathbb {H}}\)-hull with \({\mathbb {H}}\)-capacity being \(2\varepsilon \); and \(L_{u(t+\varepsilon )}/ L_{u(t)}\) is an \({\mathbb {H}}\)-hull with \({\mathbb {H}}\)-capacity being \(2u(t+\varepsilon )-2u(t)\). Since \( W_t(K_{t+\varepsilon }/K_t)=L_{u(t+\varepsilon )}/ L_{u(t)}\), using Propositions A.1 and A.2, we get
and \(u_+'(t)= W_t'({\lambda _t})^2\). Using the continuity of \(W_t\) in t, we get
Thus, \( h_{u(t)}\) satisfies the equation
From the definition of \( W_t\), we get the equality
Differentiating this equality w.r.t. t and using (A.1, A.6), we get
Combining this formula with (A.4, A.7) and replacing \(g_t(z)\) with w, we get
Letting \( U_t\ni w\rightarrow \lambda _t\) in (A.8), we get
Differentiating (A.8) w.r.t. w and letting \( U_t\ni w\rightarrow \lambda _t\), we get
Combining (A.4, A.9), and using Itô’s formula and that \(\lambda _t=\sqrt{\kappa }B_t\), we see that \(\sigma _{u(t)}\) satisfies the SDE
Combining (A.10) with \(\lambda _t=\sqrt{\kappa }B_t\) and using Itô’s formula, we get
Let \((Sf)(z)=\frac{f'''(z)}{f'(z)}-\frac{3}{2}(\frac{f''(z)}{f'(z)})^2\) be the Schwarzian derivative of f. Using (A.12) and Itô’s formula, we see that
So we get the following positive continuous local martingale
which satisfies the SDE
We claim that the following equality holds: for any \(0\le T<\tau _U\),
Note that this is similar to Lemma 3.4. To prove (A.16), we use the Brownian bubble analysis of Brownian loop measure. Let \(\mu ^{{{\,\mathrm{bb}\,}}}_{ {x_0}}\) denote the Brownian bubble measure in \({\mathbb {H}}\) rooted at \(x_0\in {\mathbb {R}}\) as defined in [25], from which we know, for any \(0\le T<\tau _U\),
If \(U^*\) is a subdomain of \({\mathbb {H}}\) that contains a neighborhood of \({\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\), we let \(P^{U^*}_{x_0}\) denote the Poisson kernel in \(U^*\) with the pole at \(x_0\in {\mathbb {R}}\). Especially, \(P^{{\mathbb {H}}}_{x_0}(z)={{\,\mathrm{Im }\,}}\frac{-1/\pi }{z-x_0}\). From [25] we know
Similarly, using (A.4) and that \(W_t:U_t{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}V_{u(t)}\), we get
Combining the above two formulas and using some tedious but straightforward computation involving power series expansions, we get
This together with (A.17, A.18) completes the proof of (A.16).
Since \(\gamma \) is continuous and tends to \(\infty \), from (2.1, A.16), we see that, on the event that \(\gamma \cap ({\mathbb {H}}{\setminus }U)=\emptyset \), the improper integral \(\int _0^\infty \frac{1}{6} S(W_s)(\lambda _s)ds\) converges to \(\frac{1}{2}\mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\beta ,{\mathbb {H}}{\setminus }V))-\frac{1}{2}\mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\gamma , {\mathbb {H}}{\setminus }U))\).
We claim that \(\lim _{t\rightarrow \infty } W_t'(\lambda _t)=1\) on the event that \(\gamma \cap {\mathbb {H}}{\setminus }U=\emptyset \). Since \(\kappa \in (4,8)\), there is \(t_0\in (0,\infty )\) such that \({\mathbb {H}}{\setminus }U\subset K_{t_0}\). Then for \(t\ge t_0\), \(U{\setminus }K_t={\mathbb {H}}{\setminus }K_t\), and so \(U_t={\mathbb {H}}\). Similarly, \(V_t={\mathbb {H}}\) for \(t\ge t_0\). Thus, for \(t\ge t_0\), \(W_t:({\mathbb {H}};\infty ){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}};\infty )\) and \(W_t'(\infty )=1\), which implies that \(W_t'(\lambda _t)=1\). So the claim is proved.
From the above we see that \(M_\infty {:}{=}\lim _{t\rightarrow \infty } M_t=e^{\frac{{{\,\mathrm{c}\,}}}{2} \mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\gamma , {\mathbb {H}}{\setminus }U))}/e^{\frac{{{\,\mathrm{c}\,}}}{2} \mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(W(\gamma ) ,{\mathbb {H}}{\setminus }V))}\) on the event that \(\gamma \cap ({\mathbb {H}}{\setminus }U)=\emptyset \). Thus, \(M_t\), \(0\le t<\infty \), is bounded on this event.
For \(n\in {\mathbb {N}}\), let \(T_n\) be the first time that \(\gamma \) hits \({\mathbb {H}}{\setminus }U\) or \(M_t\ge n\), whichever happens first. Then \(T_n\) is a stopping time, and \(M_t\) up to \(T_n\) is bounded by n. Thus, \({\mathbb {E}}[M_{T_n}]=M_0=W'(0)^{\frac{6-\kappa }{2\kappa }}\). Weighting the underlying probability measure by \(M_{T_n}/M_0\), we get a new probability measure. By Girsanov Theorem and (A.15), we find that
is a Brownian motion under the new probability measure. From (A.11), we get
From (A.5) we see that, under the new probability measure, \(\sigma _s/\sqrt{\kappa }\), \(0\le s< u(T_n)\), is a Brownian motion, and so \(\beta _s\), \(0\le s\le u(T_n)\), is a chordal \(\hbox {SLE}_\kappa \) curve in \({\mathbb {H}}\) from 0 to \(\infty \), stopped at \(u(T_n)\). let \(E_n\) denote the event that \(\gamma \cap ({\mathbb {H}}{\setminus }U)=\emptyset \) and \(M_t\le n\) for \(0\le t<\infty \); and let \(F_n\) denote the event that \(W^{-1}(\beta )\in E_n\). Then on the event \(E_n\), \(T_n=u(T_n)=\infty \), and \(M_{T_n}/M_0=M_\infty /W'(0)^{\frac{6-\kappa }{2\kappa }}\). From the above argument, we get
Since \(\mu ^\#_{{\mathbb {H}};0\rightarrow \infty }\)-a.s. \(\bigcup E_n=\{\cdot \cap {\mathbb {H}}{\setminus }U=\emptyset \}\) and \(\bigcup F_n=\{\cdot \cap {\mathbb {H}}{\setminus }V=\emptyset \}\), the above formula holds with \(E_n\) and \(F_n\) replaced by \(\{\cdot \cap {\mathbb {H}}{\setminus }U=\emptyset \}\) and \(\{\cdot \cap {\mathbb {H}}{\setminus }V=\emptyset \}\), respectively. The proposition now follows from this formula since we assumed that \(W'(\infty )=1\). \(\square \)
Remark A.4
The above proof also works for \(\kappa \in (0,4]\) except that the proof of the limit \(\lim _{t\rightarrow \infty } W_t'(\lambda _t)=1\) on the event \(\gamma \cap ({\mathbb {H}}{\setminus }U)=\emptyset \) requires a little bit more work.
Lemma A.5
Let K and L be two non-degenerate interior hulls. Let \(U,V\subset \widehat{\mathbb {C}}\) be open neighborhoods of K and L, respectively. Suppose \(W:(U;K){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}(V;L)\). Let a and b be distinct prime ends of \(\widehat{\mathbb {C}}{\setminus }K\). Then W(a) and W(b) are distinct prime ends of \(\widehat{\mathbb {C}}{\setminus }L\). Let \(g_K:\widehat{\mathbb {C}}{\setminus }K{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {D}}^*\) and \(g_L:\widehat{\mathbb {C}}{\setminus }L{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {D}}^*\). Suppose \(g_K(a)=e^{i\lambda }\), \(g_K(b)=e^{iq}\), \(g_L(W(a))=e^{i\sigma }\), and \(g_L(W(b))=e^{ip}\) for some \(\lambda ,q,\sigma ,p\in {\mathbb {R}}\). Let \(W_K=g_L\circ W\circ g_K^{-1}\). Extend \(W_K\) conformally across \({\mathbb {T}}\). Then we have
Proof
Let \(\phi (z)=i\frac{z+e^{iq}}{z-e^{iq}}\) and \(\psi (z)=i\frac{z+e^{ip}}{z-e^{ip}}\). Then \(\phi :({\mathbb {D}}^*;e^{i\lambda },e^{iq}){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}};\cot _2(\lambda -q),\infty )\) and \(\psi :({\mathbb {D}}^*;e^{i\sigma },e^{ip}){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}};\cot _2(\sigma -p),\infty )\). Let \(U_K=g_K(U{\setminus }K)\) and \(V_L=g_L(V{\setminus }L)\). Then \(U_K\) and \(V_L\) are open neighborhoods of \({\mathbb {T}}\) in \({\mathbb {D}}^*\), \(W_K:U_K{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}V_L\), and can be extended conformally across \({\mathbb {T}}\). The extended \(W_K\) maps \({\mathbb {T}}\) onto \({\mathbb {T}}\), and maps \(e^{i\lambda }\) and \(e^{iq}\) to \(e^{i\sigma }\) and \(e^{ip}\), respectively. Let \(\widehat{U}_K= \phi (U_K)\), \(\widehat{V}_L=\psi (V_L)\), and \(\widehat{W}_K=\psi \circ W_K\circ \phi ^{-1}\). Then \(\widehat{U}_K\) and \(\widehat{V}_L\) are open neighborhoods of \({\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\), and \(\widehat{W}_K:(\widehat{U}_K;{\mathbb {R}},\cot _2(\lambda -q),\infty ){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}(\widehat{V}_K;{\mathbb {R}},\cot _2(\sigma -p),\infty )\). From Proposition A.3, we have
We have \(\phi \circ g_K:(\widehat{\mathbb {C}}{\setminus }K,U{\setminus }K;a,b){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}},\widehat{U}_K;\cot _2(\lambda -q),\infty )\) and \(\psi \circ g_L:(\widehat{\mathbb {C}}{\setminus }L,V{\setminus }L;W(a),W(b)){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}},\widehat{V}_L;\cot _2(\sigma -p),\infty )\). From the conformal invariance of chordal SLE and Brownian loop measure, we have
Combining the above displayed formulas and the fact that \(\widehat{W}_K=\psi \circ g_L\circ W\circ g_K^{-1}\circ \phi ^{-1}\), we see that it suffices to prove that
To see this, one may check that \(|\phi '(e^{i\lambda })|=|\sin _2(\lambda -q)|^{-2}/2\), \(|\psi '(e^{i\sigma })|=|\sin _2(\sigma -p)|^{-2}/2\); and with \(J(z){:}{=}-1/z\), \(|(J\circ \phi )'(e^{iq})|=|(J\circ \psi )'(e^{ip})|=1/2\). \(\square \)
Lemma A.6
Let K and L be two \({\mathbb {H}}\)-hulls. Let U and V be open neighborhoods of \( {\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\) such that \(K\subset U\) and \(L\subset V\). Suppose \(W:(U;{\mathbb {R}},\infty ,K){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}(V;{\mathbb {R}},\infty ,L)\). Let a and b be distinct prime ends of \({\mathbb {H}}{\setminus }K\) that lie on \(\partial K\). Then W(a) and W(b) are distinct prime ends of \({\mathbb {H}}{\setminus }L\) that lie on \(\partial L\). Let \(g_K:{\mathbb {H}}{\setminus }K{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {H}}\) and \(g_L:{\mathbb {H}}{\setminus }L{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {H}}\). Suppose \(g_K(a)={\lambda }\), \(g_K(b)={q}\), \(g_L(W(a))={\sigma }\), and \(g_L(W(b))={p}\) for some \(\lambda ,q,\sigma ,p\in {\mathbb {R}}\). Let \(W_K=g_L\circ W\circ g_K^{-1}\). Extend \(W_K\) conformally across \({\mathbb {R}}\). Then we have
Proof
The proof is similar to that of Lemma A.5 except that here we use the functions \(\phi (z)=-\frac{z+q}{z-q}\) and \(\psi (z)=-\frac{z+p}{z-p}\), which map \({\mathbb {H}}\) conformally onto \({\mathbb {H}}\). \(\square \)
Image of an interior hull under a conformal map
Proposition B.1
Let \({{\mathcal {H}}}_0\) denote the set of all interior hulls that contain 0. For \(K\in {{\mathcal {H}}}_0\), let \(r_K\) denote \(e^{{{\,\mathrm{cap}\,}}(K)}\). Suppose W is a conformal map defined on a neighborhood U of 0, and \(W(0)=0\). If \(K\in {{\mathcal {H}}}_0\) is such that \(r_K\) is small enough, then \(W(K)\in {{\mathcal {H}}}_0\). Moreover, we have
where the implicit constants depend only on W and U.
Proof
Since \(r_{aK}=|a| r_K\), we may assume that \(W'(0)=1\). Then there exist \(r,C\in (0,\infty )\) such that \(\{|z|\le r\}\subset U\), and for any \(|z|\le r\), \(|W(z)-z|\le C|z|^2\). We may assume that \(r\le 1\) and \(rC\le \frac{1}{2}\). Then \(|z|/2\le |W(z)|\le 2|z|\) when \(|z|\le r\). Recall that by Koebe 1/4 theorem, for \(K\in {{\mathcal {H}}}_0\), \(r_K\le \max \{|z|:z\in K\}\le 4 r_K\). Thus, if \(K\in {{\mathcal {H}}}_0\) and \(r_K\le r/4\), then \(K\subset \{|z|\le r\}\), and so \(W(K)\in {{\mathcal {H}}}_0\), and \(r_K/8\le r_{W(K)}\le 8r_K\).
Let \(K\in {{\mathcal {H}}}_0\). Applying Koebe distortion theorem to \(r_K*J\circ g_K^{-1}\circ J\), where \(J(z){:}{=}1/z\), we find that for any \(z\in \mathbb {C}\) with \(|z|>1\),
Suppose that \(r_K\le r^2/16\) and \(1<|z|\le r_K^{-1/2}\). We get by (B.2)
So \(g_K^{-1}(\{1<|z|\le r_K^{-1/2}\})\subset \{|z|\le r\}\subset U\). Let \(W_K= g_{W(K)}\circ W\circ g_K^{-1}\). Then \(W_K\) is well defined on \(\{1<|z|\le r_K^{-1/2}\}\), and maps the annulus \(A{:}{=}\{1<|z|< r_K^{-1/2}\}\) conformally onto the doubly connected domain D bounded by \(\{|z|=1\}\) and \(J{:}{=}W_K(\{|z|=r_K^{-1/2}\})\), where J is a Jordan curve surrounding \(\{|z|=1\}\). Since D has the same modulus as A, J must intersect the circle \(\{|z|=r_K^{-1/2}\})\). So there is \(z_0\) with \(|z_0|=r_K^{-1/2}\) such that \(|W_K(z_0)|=r_K^{-1/2}\). Let \(z_0'=W_K(z_0)\), \(z_K=g_K^{-1}(z_0)\), and \(z_W'=W(z_K)=g_{W(K)}^{-1}(z_0')\). Applying (B.2) to \((K,z_0)\) and \((W(K),z_0')\), respectively, we get
The latter inequality implies that
Since \(z_W'=W(z_K)\) and by (B.3), \(|z_K|\le 4 r_K^{1/2}\le r\), we get \(|z_W'-z_K|\le C|z_K|^2\), which implies that \(||z_W'|^{1/2}-|z_K|^{1/2}|\le C|z_K|^{3/2}\le 8C r_K^{3/4}\). Thus, \(|r_K^{-1/2} |z_K|^{1/2}-r_{K}^{-1/2} |z_W'|^{1/2}|\le 8C r_K^{1/4}\). Combining this with (B.4), we get \(|r_{W(K)}^{-1/2} |z_W'|^{1/2}-r_K^{-1/2} |z_W'|^{1/2}|\le (2+8C) r_K^{1/4}\), which combined with (B.5) implies that
when \(r_K\le r^2/16\). So we get the estimate (B.1). \(\square \)
Remark B.2
This proposition resembles [21, Lemma 2.8], which concerns the growth rates of expanding \({\mathbb {H}}\)-hulls. The estimate (B.1) is not sharp, but is sufficient for the application in (3.7). We at least have the following estimate: \(r_{W(K)}=|W'(0)|r_K(1+O(r_K |\log (r_K)|))\). Here is a sketchy proof. By approximation we may assume that there is a family of whole-plane Loewner hulls \(K_t\), \(-\infty <t\le a=\log (r_K)\), driven by \(\lambda \), such that \(K=K_a\). Recall that \({{\,\mathrm{cap}\,}}(K_t)=t\). Then \(W(K_t)\), \(-\infty <t\le a\), is a time-change of a family of whole-plane Loewner hulls. Using a radial counterpart of [21, Lemma 2.8], we know that \(\frac{d}{dt} {{\,\mathrm{cap}\,}}(W(K_t))=|W_t'(e^{i\lambda _t})|^2\), where \(W_t{:}{=}g_{W(K_t)}\circ W\circ g_{K_t}^{-1}\). By [41, Lemma 4.4], \(W_t'(e^{i\lambda _t})=1+O(|t|e^t)\). Thus,
By Proposition B.1, \({{\,\mathrm{cap}\,}}(W(K_s))=\log |W'(0)|+s+O(e^{s/2})\). Sending \(s\rightarrow -\infty \) in the displayed formula, we get \({{\,\mathrm{cap}\,}}(W(K))=a+O(|a|e^a)\), which implies the improved estimate.
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Zhan, D. SLE loop measures. Probab. Theory Relat. Fields 179, 345–406 (2021). https://doi.org/10.1007/s00440-020-01011-7
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DOI: https://doi.org/10.1007/s00440-020-01011-7
Mathematics Subject Classification
- 60D
- 30C