# Existence and Uniqueness of Nonsimple Multiple SLE

Dapeng Zhan  Michigan State University

###### Abstract

We prove the existence and uniqueness of multiple SLEκ associated with any
given link pattern for $\kappa\in(4,6]$. We also have the uniqueness for
$\kappa\in(6,8)$. The multiple SLEκ law is constructed by first inductively
constructing a $\sigma$-finite multiple SLEκ measure and then normalizing the
measure whenever it is finite. The total mass of the measure satisfies the
conformal covariance, asymptotics and PDE for multiple SLEκ partition
functions in the literature subject to the assumption that it is smooth.

##  1 Introduction

The Schramm-Loewner evolution (SLE) is a one-parameter ($\kappa\in(0,\infty)$)
family of random curves, which are known to be the scaling limits of various
two-dimensional discrete models. A chordal SLE curve grows in a simply
connected domain connecting two distinct boundary points (more precisely,
prime ends). It arises naturally as the candidate of the scaling limit of some
lattice model with Dobrushin boundary condition.

When the Dobrushin boundary condition is replaced by alternative boundary
condition, the scaling limit is expected to be multiple SLE. In a multiple
$N$-SLEκ configuration, there are $N$ random curves connecting $2N$ distinct
marked boundary points of a simply connected domain with the property that the
conditional law of any curve given all other curves is a chordal SLEκ curve in
one connected component of the complement of the other curves.

When $\kappa\in(0,4]$, multiple SLEκ associated with any given link pattern
exists (cf. [4],[6],[12]) and is unique in law (cf. [1]). This is also true
for $\kappa\in\\{16/3,6\\}$ ([1]). When $N=2$, the existence and uniqueness of
$2$-SLEκ are known for all $\kappa\in(0,8)$ (cf. [15, 11]).

In this paper we study multiple SLEκ for $\kappa\in(4,8)$. The main result is
the existence and uniqueness for any $\kappa\in(4,6]$. We also prove the
uniqueness in the case $\kappa\in(6,8)$. In fact, for all $\kappa\in(0,8)$,
given any link pattern $\alpha$ in a domain $D$, we construct a
$\sigma$-finite measure $Q^{D}_{\alpha}$ on the space of $N$-tuples of curves
associated with $\alpha$ using a cascade relation. When $\kappa\leq 6$,
$Q^{D}_{\alpha}$ is finite, and its normalization is the law of multiple SLE
associated with $\alpha$. For $\kappa\in(6,8)$, the same statement holds as
long as $Q^{D}_{\alpha}$ is known to be finite.

We call the total mass $H^{D}_{\alpha}=|Q^{D}_{\alpha}|$ a multiple SLEκ
partition function. We focus on the case $D=\mathbb{H}$, in which case
$H(\alpha):=H^{\mathbb{H}}_{\alpha}$ is a function on an open subset of
$\mathbb{R}^{2N}$. Using the construction of $Q^{D}_{\alpha}$, we prove that
$H$ satisfies conformal covariance and asymptotics for multiple SLE pure
partition functions (cf. [7]). We construct a continuous local martingale
using $H$. If $H$ is known to be smooth, then Itô’s calculus shows that $H$
satisfies a system of PDEs for multiple SLE partition function introduced by
Dubédat in [4].

## Acknowledgement

The author thanks Xin Sun for rasing the question about nonsimple multiple
SLE, and Pu Yu for introducing the irreducible Markov chain theory to the
author.

##  2 Preliminary

###  2.1 Measures and kernels

We use $f(\mu)$ (instead of $f_{*}(\mu)$) to denote the pushforward of a
measure $\mu$ under a measurable map $f$. We use $g\cdot\mu$ or
$g(\omega)\cdot\mu(d\omega)$ to denote a measure obtained by weighting a
measure $\mu$ by a nonnegative function $g$.

Let $(\Omega_{j},{\cal F}_{j})$, $j=1,2$, be measurable spaces. A function
$\nu:\Omega_{1}\times{\cal F}_{2}\to[0,\infty)$ is called a finite kernel from
$\Omega_{1}$ to $\Omega_{2}$ if (i) for any $\omega_{1}\in\Omega_{1}$,
$\nu(\omega_{1},\cdot)$ is a finite measure on $(\Omega_{2},{\cal F}_{2})$;
and (ii) for any $B\in{\cal F}_{2}$, $\nu(\cdot,B)$ is a measurable function
on $(\Omega_{1},{\cal F}_{1})$. Suppose $\mu$ is a $\sigma$-finite measure on
$\Omega_{1}$, and $\nu$ is a finite kernel from $\Omega_{1}$ to $\Omega_{2}$.
Using Carathéodory’s extension theorem, we define $\mu{\otimes}\nu$ to be the
unique $\sigma$-finite measure on $\Omega_{1}\times\Omega_{2}$, which
satisfies that,

$\mu{\otimes}\nu(B_{1}\times
B_{2})=\int_{B_{1}}\mu(d\omega_{1})\nu(\omega_{1},B_{2}),\quad B_{1}\in{\cal
F}_{1},\quad B_{2}\in{\cal F}_{2}.$ (2.1)

Note that $\mu\otimes\nu$ is well defined even if $\nu(\omega_{1},\cdot)$ is
not defined for $\omega_{1}$ in some $\mu$-null set. Sometimes we write
$\mu{\otimes}\nu$ as $\mu(d\omega_{1}){\otimes}\nu(\omega_{1},d\omega_{2})$ to
emphasize the dependence of $\nu$ on $\omega_{1}$.

We list a few simple facts.

  * •

Let $P_{1}$ be the projection $(\omega_{1},\omega_{2})\mapsto\omega_{1}$. Then

$P_{1}(\mu{\otimes}\nu)=\nu(\cdot,\Omega_{2})\cdot\mu\ll\mu.$ (2.2)

  * •

If $f$ is a nonnegative measurable function on $\Omega_{1}$, then

$f(\omega_{1})\cdot\mu(d\omega_{1}){\otimes}\nu(\omega_{1},d\omega_{2})=\mu(d\omega_{1}){\otimes}f(\omega_{1})\nu(\omega_{1},d\omega_{2}).$
(2.3)

  * •

Let $\mu$ be a $\sigma$-finite measure on $(\Omega_{1},{\cal F}_{1})$, and
$\nu,\nu^{\\#}$ be finite kernels from $(\Omega_{1},{\cal F}_{1})$ to
$(\Omega_{2},{\cal F}_{2})$, which satisfy
$\nu(\omega_{1},\cdot)=\nu(\omega_{1},\Omega_{2})\nu^{\\#}(\omega_{1},\cdot)$.
Let $P_{1}$ be the projection $(\omega_{1},\omega_{2})\mapsto\omega_{1}$. Then

$\mu{\otimes}\nu=P_{1}(\mu{\otimes}\nu){\otimes}\nu^{\\#}.$ (2.4)

  * •

An $\Omega_{1}\times\Omega_{2}$-valued random element $(X,Y)$ satisfies that
the conditional law of $Y$ given $X$ is $\nu(X,\cdot)$ iff the law $\mu$ of
$(X,Y)$ satisfies

$\mu=P_{1}(\mu)(dx)\otimes\nu(x,dy).$ (2.5)

###  2.2 Schramm-Loewner evolution

We focus on chordal SLE, which is first defined using chordal Loewner
equation. Below we provide a brief review (cf. [8]). Let $W\in
C([0,T);\mathbb{R})$ for some $T\in(0,\infty]$. The chordal Loewner equation
driven by $W$ is

$\partial_{t}g_{t}(z)=\frac{2}{g_{t}(z)-W_{t}},\quad g_{0}(z)=z.$ (2.6)

For every $z\in\mathbb{C}$, there is $\tau(z)\in[0,T]$ such that $[0,\tau(z))$
is the biggest interval on which $t\mapsto g_{t}(z)$ is defined. Let
$K_{t}=\\{z\in\mathbb{H}:\tau(z)\leq t\\}$, $0\leq t<T$, where
$\mathbb{H}:=\\{z\in\mathbb{C}:\operatorname{Im}z>0\\}$. The $g_{t}$ and
$K_{t}$ are respectively called chordal Loewner maps and hulls driven by $W$.
It turns out that, for each $t$, $g_{t}$ maps $\mathbb{H}\setminus K_{t}$
conformally onto $\mathbb{H}$, and fixes $\infty$.

If we differentiate (2.6) w.r.t. $z$, then we get

$\partial_{t}g_{t}^{\prime}(z)=\frac{-2g_{t}^{\prime}(z)}{g_{t}(z)-W_{t}},\quad
g_{0}^{\prime}(z)=1.$ (2.7)

If the map $(t,z)\mapsto g_{t}^{-1}(z)$ extends continuously from
$[0,T)\times\mathbb{H}$ to $[0,T)\times\overline{\mathbb{H}}$, then the
continuous curve $\eta(t):=g_{t}^{-1}(W_{t})$, $0\leq t<T$, in
$\overline{\mathbb{H}}$, is called the chordal Loewner curve driven by $W$.
Such $\eta$ may not exist in general. When it exists, then for any
$t\in[0,T)$, $\mathbb{H}\setminus K_{t}$ is the unbounded component of
$\mathbb{H}\setminus\eta[0,t]$.

Let $W_{t}=\sqrt{\kappa}B_{t}$, where $\kappa>0$ and $B_{t}$ is a standard
Brownian motion. Then the chordal Loewner curve $\eta$ driven by $W$ is known
to exist ([13]), and is called a chordal SLEκ in $\mathbb{H}$ from $0$ to
$\infty$. If $D$ is a simply connected domain with two distinct prime ends
$a,b$, we may find a conformal map $f$ from $\mathbb{H}$ onto $D$, which sends
$0,\infty$ to $a,b$. Then $f\circ\eta$ modulo time parametrization (due to the
non-uniqueness of $f$) is called a chordal SLEκ in $D$ from $a$ to $b$. Such
$f\circ\eta$ lies in the prime end closure of $D$, and when $\partial D$ is
locally connected, lies in $\overline{D}$.

An SLEκ curve is simple if $\kappa\in(0,4]$; space-filling if $\kappa\geq 8$;
non-simple and non-space-filling if $\kappa\in(4,8)$. For the rest of the
paper we assume $\kappa\in(0,8)$. The following proposition follows
immediately from [1, Lemma 3.6].

######  Proposition 2.1.

Let $D^{\prime}\subset D$ be two simply connected domains with locally
connected boundary that share two distinct prime ends $a,b$. Let $\mathbb{P}$
be the law of chordal SLEκ in $D$ from $a$ to $b$. Then
$\mathbb{P}[\eta\subset\overline{D^{\prime}}]>0$.

###  2.3 Multiple SLE

Let $D$ be a simply connected domain with locally connected boundary and $2N$
distinct prime ends $a_{1},b_{1},\dots,a_{N},b_{N}$. An $N$-tuple
$(\eta_{1},\dots,\eta_{N})$ of random curves in $\overline{D}$ is called a
multiple $N$-SLEκ in $D$ associated with
$\alpha:=((a_{1},b_{1}),\dots,(a_{N},b_{N}))$ if, for any fixed
$r\in\\{1,\dots,N\\}$, the conditional law of $\eta_{r}$ given all $\eta_{j}$,
$j\neq r$, is an SLEκ curve from $a_{r}$ to $b_{r}$ in the connected component
of $D\setminus\bigcup_{j\neq r}\eta_{j}$ which contains neighborhoods of
$a_{r},b_{r}$ in $D$. A necessary condition for the existence of these curves
is that there exist $N$ mutually disjoint simple curves
$\gamma_{1},\dots,\gamma_{N}$ in $D$ such that $\gamma_{j}$ connects $a_{j}$
with $b_{j}$, in which case $\alpha$ is called a link pattern.

By conformal invariance of SLE, one may assume that $\partial D$ is analytic
at the points in $\alpha$. In the case $\kappa\leq 4$, the $N$-SLEκ associated
with any link pattern is known to exist (cf. [4],[6],[12]). The important idea
in [6] is that it is useful to view SLEκ as a non-probability measure, which
is the SLEκ law (probability measure) multiplied by its partition function.

Let $\mu^{D}_{(a,b)}$ denote the chordal SLEκ law in $D$ from $a$ to $b$.
Assume that $\partial D$ is analytic at $a,b$. The boundary Poisson kernel in
$D$ at $(a,b)$ is defined by

$P^{D}_{(a,b)}=\lim_{\varepsilon\to
0^{+}}\frac{\pi}{\varepsilon^{2}}G_{D}(a+\varepsilon{\bf
n}_{a},b+\varepsilon{\bf n}_{b}),$

where $G_{D}(\cdot,\cdot)$ is the Dirichlet Green’s function for $D$, and
${\bf n}_{a},{\bf n}_{b}$ are inward normals.

Define two constants depending on $\kappa$:

$\operatorname{b}=\frac{6-\kappa}{2\kappa},\qquad\operatorname{c}=\frac{(6-\kappa)(3\kappa-8)}{2\kappa},$
(2.8)

which are respectively called the boundary scaling exponent and central charge
of SLEκ. The $H^{D}_{(a,b)}:=(P^{D}_{(a,b)})^{\operatorname{b}}$ and
$Q^{D}_{(a,b)}:=H^{D}_{(a,b)}\mu^{D}_{(a,b)}$ are respectively called the SLEκ
partition function and SLEκ measure for $(D;a,b)$. The partition function
satisfies the conformal covariance: if $f$ is a conformal map on $D$ that is
extended in neighborhoods of $a,b$, then

$H^{D}_{(a,b)}=|f^{\prime}(a)|^{\operatorname{b}}|f^{\prime}(b)|^{\operatorname{b}}H^{f(D)}_{(f(a),f(b))}.$
(2.9)

When $D=\mathbb{H}$, we have the special value
$H^{\mathbb{H}}_{(a,b)}=|a-b|^{-2\operatorname{b}}$. If $\kappa\leq 6$, then
$\operatorname{b}\geq 0$, and $H^{D}_{(a,b)}$ satisfies the monotonicity: if
$D^{\prime}\subset D$ contains neighborhoods of $a,b$ in $D$, then
$H^{D^{\prime}}_{(a,b)}\leq H^{D}_{(a,b)}$. The direction of the inequality is
reversed if $\kappa>6$.

Let $\alpha=((a_{1},b_{1}),\dots,(a_{N},b_{N}))$ be a link pattern in $D$.
Assume that $\partial D$ is analytic at points in $\alpha$. Let
$\kappa\in(0,4]$. The $N$-SLEκ measure in $D$ with link pattern $\alpha$ is
defined as a measure $Q^{D}_{\alpha}$ defined by

$Q^{D}_{\alpha}(d\underline{\eta})=Q^{D}_{\alpha}(d(\eta_{1},\dots,\eta_{N}))=Y(\underline{\eta})\cdot\prod_{j=1}^{N}Q^{D}_{(a_{j},b_{j})}(d\eta_{j}),$
(2.10)

where

$Y(\underline{\eta})=I(\underline{\eta})\exp\Big{(}\frac{\operatorname{c}}{2}\sum_{j=2}^{N}m_{D}(K_{j}(\underline{\eta}))\Big{)},$

$I(\underline{\eta})$ is the indicator of the event that
$\eta_{1},\dots,\eta_{N}$ are mutually disjoint, $\operatorname{c}$ is given
by (2.8), $m_{D}$ is the Brownian loop measure in $D$ (cf. [9]), and
$K_{j}(\underline{\eta})$ is the collection of loops that intersect at least
$j$ of the curves $\eta_{1},\dots,\eta_{N}$. Such $Q^{D}_{\alpha}$ is a finite
measure, and its normalization is an $N$-SLEκ law in $D$ with link pattern
$\alpha$.

The uniqueness of multiple SLEκ for $\kappa\leq 4$ is established in [1],
which also proves the existence and uniqueness of multiple SLE16/3 using the
convergence of FK Ising model. The same argument works for multiple SLE6 as
scaling limit of critical percolation. The work also studies multiple SLEκ for
other $\kappa\in(4,6]$ and proves the existence and uniqueness based on a
conjecture about the convergence of some random cluster model. When there are
only $2$ curves, the existence and uniqueness of $2$-SLEκ hold for all
$\kappa\in(0,8)$ (cf. [15, 11]).

##  3 Existence

The main goal of the paper is to prove the existence and uniqueness of
multiple SLEκ associated with any given link pattern for $\kappa\in(4,8)$, in
which case the curves are nonsimple and have positive probability to
intersect. We do not consider the case $\kappa\geq 8$ because in that case the
existence is trivial and the uniqueness does not hold.

If one uses (2.10) to construct $Q^{D}_{\alpha}$ for $\kappa\in(4,8)$, then by
[1, Lemma 2.3] its normalization is not the law of multiple SLEκ but the law
of $(\eta_{1},\dots,\eta_{N})$ with the properties that, every $\eta_{r}$
conditional on all other $\eta_{j}$’s is an SLEκ curve from $a_{r}$ to $b_{r}$
in some complement domain of the given curves conditioned to avoid these
curves.

Fortunately, [6, Proposition 3.2] provides an cascade relation of
$Q^{D}_{\alpha}$, which could be used for the construction. It says that the
marginal measure on $(\eta_{1},\dots,\eta_{N-1})$ in $Q^{D}_{\alpha}$ is
absolutely continuous w.r.t. $Q^{D}_{\alpha^{\prime}}$, $\alpha^{\prime}$
being the $\alpha$ without the link $(a_{N},b_{N})$, with Radon-Nikodym
derivative $H^{D^{\prime}}_{(a_{N},b_{N})}$, where $D^{\prime}$ is the
connected component of $D\setminus\cup_{k=1}^{N-1}\eta_{k}$ whose boundary
contains $a_{N},b_{N}$. Since the $\eta_{N}$ in $Q^{U}_{\alpha}$ conditionally
on $(\eta_{1},\dots,\eta_{N-1})$ has the law of
$\mu^{D^{\prime}}_{(a_{N},b_{N})}$, we may obtain $Q^{D}_{\alpha}$ by first
weighting $Q^{D}_{\alpha^{\prime}}$ by $H^{D^{\prime}}_{(a_{N},b_{N})}$ and
then sampling $\eta_{N}$ according to $\mu^{D^{\prime}}_{(a_{N},b_{N})}$.

We will follow the above approach to construct $Q^{D}_{\alpha}$ for
$\kappa\in(4,8)$ inductively on $N$. It is useful to construct multiple SLE in
possibly disconnected open sets. Let $\cal U$ denote the collection of open
subsets of $\mathbb{C}$. For $U\in\cal U$ and $N\in\mathbb{N}$, let
$\widehat{\operatorname{LP}}_{N}^{U}$ denote the set of tuples
$\alpha=((a_{1},b_{1}),\dots,(a_{N},b_{N}))$ of $2N$ distinct points on
$\partial U$, such that each of the $2N$ points lies on the boundary of a
unique connected component of $U$, which is simply connected with locally
connected boundary and whose boundary is analytic at this point. We write
$\alpha_{j}$ for the couple $(a_{j},b_{j})$ or the set $\\{a_{j},b_{j}\\}$. If
there exist $N$ mutually disjoint simple curves $\gamma_{j}$, $1\leq j\leq N$,
in $U$, such that each $\gamma_{j}$ connects points of $\alpha_{j}$, then
$\alpha$ is called a link pattern of $U$ of size $N$. Let
$\operatorname{LP}^{U}_{N}$ denote the collection of all such $\alpha$. We are
going to construct a measure $Q^{U}_{\alpha}$ for any
$\alpha\in\widehat{\operatorname{LP}}^{U}_{N}$ on the space
$X^{U}_{\alpha}:=\prod_{j=1}^{N}X^{U}_{\alpha_{j}}$, where
$X^{U}_{\alpha_{j}}$ is the space of unparametrized curves from $a_{j}$ to
$b_{j}$.

If $\alpha=(a,b)\in\operatorname{LP}^{U}_{1}$, there exists a unique connected
simply connected component $D$ of $U$, whose boundary contains and is analytic
at $a,b$. Then we define $\mu^{U}_{\alpha}=\mu^{D}_{\alpha}$,
$H^{U}_{\alpha}=H^{D}_{\alpha}$, and $Q^{U}_{\alpha}=Q^{D}_{\alpha}$. Let
$\mu^{U}_{\alpha}=H^{U}_{\alpha}=Q^{D}_{\alpha}=0$ if
$\alpha\in\widehat{\operatorname{LP}}^{U}_{1}\setminus\operatorname{LP}^{U}_{1}$.

Using the above convention and (2.3), in the case that $U$ is a simply
connected domain $D$, we may express the cascade relation of $Q^{U}_{\alpha}$
for $\kappa\in(0,4]$ symbolically as

$Q^{U}_{\alpha}=Q^{U}_{P_{\widehat{N}}(\alpha)}(d(\eta_{1},\dots\eta_{N-1})){\otimes}Q^{U\setminus\bigcup_{j=1}^{N-1}\eta_{j}}_{\alpha_{N}}(d\eta_{N}),$
(3.1)

where $P_{\widehat{N}}(\alpha)$ is the $\alpha$ without the $N$-th component.

We introduce some symbols. Let $\mathbb{N}_{N}=\\{j\in\mathbb{N}:j\leq N\\}$
for $N\in\mathbb{N}$, and $S_{N}$ be the symmetric group of $\mathbb{N}_{N}$.
For a vector $\underline{x}=(x_{1},\dots,x_{N})$ and $j\in\mathbb{N}_{N}$, we
define

$P_{\widehat{j}}(\underline{x})=(x_{1},\dots,\widehat{x}_{j},\dots,x_{N}),\quad
P_{\leq j}(\underline{x})=(x_{1},\dots,x_{j}),$

where $\widehat{x}_{j}$ means that $x_{j}$ is missing in the list. For
$\sigma\in S_{N}$, let

$\sigma(\underline{x})=(x_{\sigma^{-1}(1)},\dots,x_{\sigma^{-1}(N)}).$

Note that
$\sigma_{1}(\sigma_{2}(\underline{x}))=(\sigma_{1}\sigma_{2})(\underline{x})$
for any $\sigma_{1},\sigma_{2}\in S_{N}$. We will view these
$P_{\widehat{j}}$, $P_{\leq j}$ and $\sigma$ as functions on
$\widehat{\operatorname{LP}}^{U}_{N}$ or on $X^{U}_{\alpha}$ for some
$\alpha\in\widehat{\operatorname{LP}}^{U}_{N}$.

######  Definition 3.1.

For any $\kappa\in(0,8)$, $U\in{\cal U}$ and
$\alpha\in\widehat{\operatorname{LP}}^{U}_{N}$, $N\in\mathbb{N}$, the
$Q^{U}_{\alpha}$ is a $\sigma$-finite measure on $X^{U}_{\alpha}$ defined
inductively on $N$ using the induction basis: the definition of
$Q^{U}_{(a,b)}$ when $(a,b)\in\widehat{\operatorname{LP}}^{U}_{1}$ and the
induction step: (3.1).

We remark that (3.1) makes sense because from $\kappa<8$ we know that
$\eta_{1},\dots,\eta_{N-1}$ following the law
$Q^{U}_{P_{\widehat{N}}(\alpha)}$ a.s. do not pass through points of
$\alpha_{N}$, and so
$\alpha_{N}\in\widehat{\operatorname{LP}}^{U\setminus\bigcup_{j=1}^{N-1}\eta_{j}}_{1}$.
Note that $U\setminus\bigcup_{j=1}^{N-1}\eta_{j}$ is not connected, and
$\alpha_{N}$ may not belong to
$\operatorname{LP}^{U\setminus\bigcup_{j=1}^{N-1}\eta_{j}}_{1}$ even if
$\alpha\in\operatorname{LP}^{U}_{N}$.

The $N$-SLEκ law could also be described using the language of kernels. For
$j\leq N$, we use $\sigma_{[j,N]}$ to denote the cyclic permutation
$(j,j+1,\dots,N)\in S_{N}$. Then

$P_{\widehat{N}}=P_{\widehat{j}}\circ\sigma_{[j,N]}.$ (3.2)

In view of (2.5), it is easy to see that $(\eta_{1},\dots,\eta_{N})$ is an
$N$-SLEκ in $U$ with link pattern $\alpha$ iff its law $\mu$, a probability
measure on $X^{U}_{\alpha}$, satisfies that

$\mu=\sigma_{[j,N]}\Big{(}P_{\widehat{j}}(\mu)(d(\eta_{1},\dots,\widehat{\eta}_{j},\dots,\eta_{N})){\otimes}\mu^{U\setminus\cup_{k\neq
j}\eta_{k}}_{(a_{j},b_{j})}(d\eta_{j})\Big{)},\quad j\in\mathbb{N}_{N}.$ (3.3)

######  Definition 3.2.

A $\sigma$-finite measure $\mu$ on $X^{U}_{\alpha}$ is called a multiple SLEκ
measure in $U$ with link pattern $\alpha$ if it satisfies (3.3). Clearly, if
such $\mu$ is finite and nontrivial, then its normalization $\mu/|\mu|$ is an
$N$-SLEκ law in $U$ with link pattern $\alpha$.

In the following we are going to show that $Q^{U}_{\alpha}$ satisfies (3.3)
and study its trivialness and finiteness. Let $\alpha^{k}=P_{\leq k}(\alpha)$,
$1\leq k\leq N$. Then $\alpha=\alpha^{N}$, and

$Q^{U}_{\alpha^{k}}=Q^{U}_{\alpha^{k-1}}(d(\eta_{1},\dots\eta_{k-1})){\otimes}Q^{U\setminus\bigcup_{j=1}^{k-1}\eta_{j}}_{\alpha_{k}}(d\eta_{k}),\quad
2\leq k\leq N.$ (3.4)

######  Lemma 3.3.

$Q^{U}_{\alpha}$ is not trivial iff $\alpha\in\operatorname{LP}^{U}_{N}$.

###### Proof.

If
$\alpha\in\widehat{\operatorname{LP}}^{U}_{N}\setminus\operatorname{LP}^{U}_{N}$,
then there are $j<k\in\mathbb{N}_{N}$ such that $\alpha_{j},\alpha_{k}$ lie on
the boundary of the same component $D$ of $U$, and if $\eta_{j}$ is an SLEκ in
$\overline{D}$ connecting $\alpha_{j}$, then $\eta_{j}$ disconnects points of
$\alpha_{k}$ in $D$. Thus,
$\alpha_{k}\not\in\operatorname{LP}_{1}^{U\setminus\bigcup_{s=1}^{k-1}\eta_{s}}$,
which implies that $Q^{U}_{\alpha^{k-1}}$-a.s.
$Q^{U\setminus\bigcup_{j=1}^{k-1}\eta_{j}}_{\alpha_{k}}=0$. By (3.4) we get
$0=Q^{U}_{\alpha^{k}}=Q^{U}_{\alpha^{k+1}}=\cdots=Q^{U}_{\alpha^{N}}=Q^{U}_{\alpha}$.

Suppose $\alpha\in\operatorname{LP}^{U}_{N}$. We call $T$ a tube in $U$
connecting $\alpha_{j}$ if it is the closure of a Jordan subdomain of one
component $D_{j}$ of $U$, whose boundary is a union of the closure of two
crosscuts of $D_{j}$ and two disjoint open boundary arcs, which respectively
contain points of $\alpha_{j}$. Since $\alpha\in\operatorname{LP}^{U}_{N}$,
there exist mutually disjoint tubes $T_{j}$ in $U$ connecting $\alpha_{j}$,
$1\leq j\leq N$. Let ${\cal T}_{j}=\\{\eta\in X^{U}_{j}:\eta\subset T_{j}\\}$.
By Proposition 2.1, $Q^{U}_{\alpha^{1}}({\cal T}_{1})>0$ and for any $2\leq
k\leq N$, when $(\eta_{1},\dots,\eta_{k-1})\in\prod_{j=1}^{k-1}{\cal T}_{j}$,
$Q^{U\setminus\bigcup_{j=1}^{k-1}\eta_{j}}_{\alpha_{k}}({\cal T}_{k})>0$.
Thus, by (3.1) and induction, $Q^{U}_{\alpha^{k}}(\prod_{j=1}^{k}{\cal
T}_{j})>0$ for $1\leq k\leq N$, which means that
$Q^{U}_{\alpha}=Q^{U}_{\alpha^{N}}$ is not trivial. ∎

######  Lemma 3.4.

If $\kappa\leq 6$, then $Q^{U}_{\alpha}$ is finite.

###### Proof.

By (3.1) and the monotonicity of $H^{D}_{(a,b)}$ when $\kappa\leq 6$,

$|Q^{U}_{\alpha^{k}}|\leq\int
Q^{U}_{\alpha^{k-1}}(d(\eta_{1},\dots,\eta_{k-1}))|Q^{U}_{(a_{k},b_{k})}|=|Q^{U}_{\alpha^{k-1}}|H^{U}_{(a_{k},b_{k})},\quad
2\leq k\leq N,$

which implies that
$|Q^{U}_{\alpha}|\leq\prod_{k=1}^{N}H^{U}_{(a_{k},b_{k})}<\infty$. ∎

The following lemma follows from Wu’s work [15].

######  Lemma 3.5.

For any $\alpha\in\widehat{\operatorname{LP}}^{U}_{2}$, $Q^{U}_{\alpha}$ is
finite, and $\sigma(Q^{U}_{\alpha})=Q^{U}_{\sigma(\alpha)}$ for
$\sigma=(1,2)\in S_{2}$.

###### Proof.

If $\alpha\not\in\operatorname{LP}^{U}_{2}$, then
$\sigma(\alpha)\not\in\operatorname{LP}^{U}_{2}$. By Lemma 3.3,
$Q^{U}_{\alpha}=Q^{U}_{\sigma(\alpha)}=0$. Suppose now
$\alpha\in\operatorname{LP}^{U}_{2}$. For $j=1,2$, there is a simply connected
component $D_{j}$ of $U$ whose boundary contains $\alpha_{j}$. Then
$Q^{U}_{\alpha_{1}}=Q^{D_{1}}_{\alpha_{1}}$. If $D_{1}\neq D_{2}$, then for
$Q^{D_{1}}_{\alpha_{1}}$-a.s. every $\eta_{1}$,
$Q^{U\setminus\eta_{1}}_{\alpha_{2}}=Q^{D_{2}}_{\alpha_{2}}$. Thus, by (3.1),
$Q^{U}_{\alpha}=Q^{D_{1}}_{\alpha_{1}}\times Q^{D_{2}}_{\alpha_{2}}$.
Similarly, $Q^{U}_{\sigma(\alpha)}=Q^{D_{2}}_{\alpha_{2}}\times
Q^{D_{1}}_{\alpha_{1}}$. So $\sigma(Q^{U}_{\alpha})=Q^{U}_{\sigma(\alpha)}$.

Now consider the case $D_{1}=D_{2}$. We may assume $U=D_{1}$ since other
components of $U$ are irrelevant. By reversibility of chordal SLEκ and the
fact that $H^{U}_{(a,b)}=H^{U}_{(b,a)}$, we may relabeling the points of
$\alpha_{j}=\\{a_{j},b_{j}\\}$, $j=1,2$, such that $a_{1},b_{1},b_{2},a_{2}$
are ordered clockwise or counterclockwise on $\partial U$. Let
$H^{U}_{\alpha}=|Q^{U}_{\alpha}|$. By (3.1) and (2.1),

$H^{U}_{\alpha}=H^{U}_{\alpha_{1}}\int\mu^{U}_{\alpha_{1}}(d\eta_{1})H^{U\setminus\eta_{1}}_{\alpha_{2}}.$
(3.5)

Let $f$ be a conformal map from $U$ onto $\mathbb{H}$ such that $f(a_{1})=0$
and $f(b_{1})=\infty$. Let $x=f(a_{2})$ and $y=f(b_{2})$. By conformal
covariance,

$\int\mu^{U}_{\alpha_{1}}(d\eta_{1})H^{U\setminus\eta_{1}}_{\alpha_{2}}=|f^{\prime}(a_{2})|^{\operatorname{b}}|f^{\prime}(b_{2})|^{\operatorname{b}}\int\mu^{\mathbb{H}}_{(0,\infty)}(d\gamma)H^{\mathbb{H}\setminus\gamma}_{(x,y)}.$
(3.6)

Now $r:=x/y\in(0,1)$ is the cross ratio of $\alpha$ in $U$. By [15, Lemma 3.5
and Proposition 3.6], if $\gamma$ is SLEκ in $\mathbb{H}$ from $0$ to
$\infty$, there is an associated uniformly integrable martingale
$(M_{t})_{t\geq 0}$ satisfying

$M_{0}=|x-y|^{-2\operatorname{b}}r^{\frac{2}{\kappa}}F(r),\quad
M_{\infty}=H^{\mathbb{H}\setminus\gamma}_{(x,y)},$

where
$F:=_{2}F_{1}(\frac{4}{\kappa},1-\frac{4}{\kappa},\frac{8}{\kappa},\cdot)$ is
a hypergeometric function. Let $G(x)=x^{\frac{2}{\kappa}}F(x)$. Then

$\int\mu^{\mathbb{H}}_{(0,\infty)}(d\gamma)H^{\mathbb{H}\setminus\gamma}_{(x,y)}=\mathbb{E}[M_{\infty}]=M_{0}=H^{\mathbb{H}}_{(x,y)}G(r),$

which combined with (3.5,3.6) and the conformal covariance of
$H^{U}_{\alpha_{j}}$, $j=1,2$, implies that
$H^{U}_{\alpha}=G(r)H^{U}_{\alpha_{1}}H^{U}_{\alpha_{2}}\in(0,\infty)$. Thus,
$Q^{U}_{\alpha}$ is finite. Since the cross ratio of $\alpha$ and
$\sigma(\alpha)$ in $U$ agree, we get $H^{U}_{\alpha}=H^{U}_{\sigma(\alpha)}$.

By [15, Proposition 6.10], for any simply connected domain $D$ and
$\beta\in\operatorname{LP}^{D}_{2}$, the $2$-SLEκ law associated with $\beta$,
denoted by $\mu^{D}_{\beta}$, exists uniquely and satisfies conformal
invariance. This immediately implies that
$\sigma(\mu^{U}_{\alpha})=\mu^{U}_{\sigma(\alpha)}$ and
$\mu^{U}_{\alpha}=f^{-1}(\mu^{\mathbb{H}}_{((0,\infty),(x,y))})$. That
proposition and [15, Proposition 3.6] together imply that the first marginal
of $\mu^{\mathbb{H}}_{((0,\infty),(x,y))}$ is absolutely continuous w.r.t.
$\mu^{\mathbb{H}}_{(0,\infty)}$ with Radon-Nikodym derivative
$M_{\infty}/M_{0}=H^{\mathbb{H}\setminus\gamma}_{(x,y)}/(H^{\mathbb{H}}_{(x,y)}G(r))$.
Thus, by (2.5),

$\mu^{\mathbb{H}}_{((0,\infty),(x,y))}(d(\gamma_{1},\gamma_{2}))=H^{\mathbb{H}\setminus\gamma_{1}}_{(x,y)}/(H^{\mathbb{H}}_{(x,y)}G(r))\cdot\mu^{\mathbb{H}}_{(0,\infty)}(d\gamma_{1}){\otimes}\mu^{\mathbb{H}\setminus\gamma_{1}}_{(x,y)}(d\gamma_{2}).$

By this formula and the conformal invariance of SLEκ and $2$-SLEκ,

$\mu^{U}_{\alpha}(d(\eta_{1},\eta_{2}))=H^{\mathbb{H}\setminus
f(\eta_{1})}_{(x,y)}/(H^{\mathbb{H}}_{(x,y)}G(r))\cdot\mu^{U}_{\alpha_{1}}(d\eta_{1}){\otimes}\mu^{U\setminus\eta_{1}}_{\alpha_{2}}(d\eta_{2}).$

By the construction of $Q^{U}_{\alpha}$ and (2.3),

$Q^{U}_{\alpha}(d(\eta_{1},\eta_{2}))=H^{U}_{\alpha_{1}}H^{U\setminus\eta_{1}}_{\alpha_{2}}\cdot\mu^{U}_{\alpha_{1}}(d\eta_{1}){\otimes}\mu^{U\setminus\eta_{1}}_{\alpha_{2}}(d\eta_{2}).$

Comparing the above two displayed formulas and using
$H^{U\setminus\eta_{1}}_{\alpha_{2}}/{H^{\mathbb{H}\setminus
f(\eta_{1})}_{(x,y)}}=H^{U}_{\alpha_{2}}/H^{\mathbb{H}}_{(x,y)}$ and
$H^{U}_{\alpha}=G(r)H^{U}_{\alpha_{1}}H^{U}_{\alpha_{2}}$, we get
$Q^{U}_{\alpha}=H^{U}_{\alpha}\mu^{U}_{\alpha}$, which together with
$H^{U}_{\alpha}=H^{U}_{\sigma(\alpha)}$ and
$\sigma(\mu^{U}_{\alpha})=\mu^{U}_{\sigma(\alpha)}$ implies
$\sigma(Q^{U}_{\alpha})=Q^{U}_{\sigma(\alpha)}$. ∎

######  Lemma 3.6.

For any $\alpha\in\widehat{\operatorname{LP}}^{U}_{N}$ and $\sigma\in S_{N}$,
$\sigma(Q^{U}_{\alpha})=Q^{U}_{\sigma(\alpha)}$.

###### Proof.

By Lemma 3.5, the statement holds for $N=1,2$. Suppose $N\geq 3$. By (3.4),

$Q^{U}_{\alpha}=Q^{U}_{\alpha^{N-2}}(d(\eta_{1},\dots,\eta_{N-2})){\otimes}Q^{U\setminus\bigcup_{j=1}^{N-2}\eta_{j}}_{(\alpha_{N-1},\alpha_{N})}(d(\eta_{N-1},\eta_{N})).$

By Lemma 3.5, for $\sigma_{N}:=(N-1,N)\in S_{N}$,
$\sigma_{N}(Q^{U}_{\alpha})=Q^{U}_{\sigma_{N}(\alpha)}$. This result applied
to $\alpha^{k}$ and $\sigma_{k}=(k-1,k)\in S_{k}$ together with Lemma 3.5
implies that $\sigma_{k}(Q^{U}_{\alpha^{k}})=Q^{U}_{\sigma_{k}(\alpha^{k})}$,
$2\leq k\leq N$. We view $\sigma_{k}$ as an element of $S_{m}$ for any $k\leq
m\leq N$. By (3.4) and induction, we get
$\sigma_{k}(Q^{U}_{\alpha^{m}})=Q^{U}_{\sigma_{k}(\alpha^{m})}$ for $k\leq
m\leq N$. Taking $m=N$, we get
$\sigma_{k}(Q^{U}_{\alpha})=Q^{U}_{\sigma_{k}(\alpha)}$. The proof now
finishes since $\\{\sigma_{k}:2\leq k\leq N\\}$ generates the group $S_{N}$. ∎

######  Lemma 3.7.

For $\alpha\in\operatorname{LP}^{U}_{N}$, $Q^{U}_{\alpha}$ is an $N$-SLEκ
measure in $U$ with link pattern $\alpha$.

###### Proof.

Fix $j\in\mathbb{N}_{N}$. We need to show that,

$Q^{U}_{\alpha}=\sigma_{[j,N]}(P_{\widehat{j}}(Q^{U}_{\alpha})(d(\eta_{1},\dots,\widehat{\eta}_{j},\dots,\eta_{N}))){\otimes}\mu^{U\setminus\bigcup_{k\neq
j}\eta_{k}}_{\alpha_{j}}(d\eta_{j}).$ (3.7)

That (3.7) holds for $j=N$ follows from (3.1) and (2.4). By Lemma 3.6,
${\sigma}_{[j,N]}^{-1}(Q^{U}_{\alpha})=Q^{U}_{\sigma_{[j,N]}^{-1}(\alpha)}$.
Since
$\sigma_{[j,N]}^{-1}(\alpha)=(\alpha_{1},\dots,\alpha_{j-1},\alpha_{j+1},\dots,\alpha_{N},\alpha_{j})$
and (3.7) holds for $j=N$,

${\sigma}_{[j,N]}^{-1}(Q^{U}_{\alpha})=P_{\widehat{N}}({\sigma}_{[j,N]}^{-1}(Q^{U}_{\alpha}))(d(\eta_{1},\dots,\widehat{\eta}_{j},\dots,\eta_{N})){\otimes}\mu^{U\setminus\bigcup_{k\neq
j}\eta_{k}}_{\alpha_{j}}(d\eta_{j})$
$=P_{\widehat{j}}(Q^{U}_{\alpha})(d(\eta_{1},\dots,\widehat{\eta}_{j},\dots,\eta_{N})){\otimes}\mu^{U\setminus\bigcup_{k\neq
j}\eta_{k}}_{\alpha_{j}}(d\eta_{j}),$

where the last “$=$” follows from (3.2). This immediately implies (3.7). ∎

Combining Lemmas 3.7, 3.3 and 3.4, we obtain the existence theorem.

######  Theorem 3.8 (Existence).

Let $\kappa\in(0,6]$. Then for any $\alpha\in\operatorname{LP}^{U}_{N}$,
$Q^{U}_{\alpha}/|Q^{U}_{\alpha}|$ is an $N$-SLEκ law in $U$ with link pattern
$\alpha$.

Apparently, the existence statement holds for any $\kappa\in(6,8)$ such that
$Q^{U}_{\alpha}$ is finite.

##  4 Uniqueness

In this section we prove the uniqueness of $N$-SLEκ law in a simply connected
domain with any given link pattern. We will use the Markov chain theory in
[10]. By Proposition 10.1.1 and Theorem 10.4.9 of the book, if a Markov chain
$\Phi$ is irreducible and admits an invariant probability measure $\mu$, then
all ($\sigma$-finite) invariant measures of $\Phi$ are some constant times
$\mu$. By Section 4.2.1 of the book, $\Phi$ is called ($\phi$-)irreducible if
there is a nontrivial $\sigma$-finite measure $\phi$ such that if $A$
satisfies $\phi(A)>0$, then for any $x$ in the space, $L(x,A)>0$, which is
equivalent to $\sum_{m=1}^{\infty}P^{m}(x,A)>0$. Such $\phi$ is called an
irreducibility measure for $\Phi$.

Fix a simply connected domain $D$ with locally connected boundary and
$\alpha\in\operatorname{LP}^{D}_{N}$. For $j\in\mathbb{N}_{N}$, let $X_{j}$ be
the space of unparametrized curves in $\overline{D}$ which connect points of
$\alpha_{j}$. Let $\underline{X}$ be the space of
$(\eta_{1},\dots,\eta_{N})\in\prod_{j=1}^{N}X_{j}$ such that for any
$j\in\mathbb{N}_{N}$,
$\alpha_{j}\in\operatorname{LP}^{D\setminus\bigcup_{k\neq j}\eta_{k}}_{1}$.
For any $\underline{A}\subset\underline{X}$,
$\underline{\eta}\in\underline{X}$, and $j\in\mathbb{N}_{N}$, let
$\pi^{\underline{\eta}}_{j}(\underline{A})$ denote the set of $\gamma\in
X_{j}$ such that if the $j$-th component of $\underline{\eta}$ were replaced
by $\gamma$, then the modified $\underline{\eta}$ would lie in
$\underline{A}$.

Define a Markov chain $\Phi$ on $\underline{X}$, whose transition kernel is
$P=\frac{1}{N}\sum_{j=1}^{N}P_{j}$, where

$P_{j}(\underline{\eta},\underline{A})=P_{j}((\eta_{1},\dots,\eta_{N}),\underline{A})=\mu^{D\setminus\bigcup_{k\neq{j}}\eta_{k}}_{\alpha_{j}}(\pi^{\underline{\eta}}_{j}(\underline{A})).$

In plain words, given $\Phi_{0}=\underline{\eta}^{0}$, the next step
$\Phi_{1}=\underline{\eta}^{1}$ is given by $\eta^{1}_{k}=\eta^{0}_{k}$ for
$k\neq\bf j$ and $\eta^{1}_{\bf j}=\gamma$, where $\bf j$ is chosen from
$\mathbb{N}_{N}$ uniformly randomly, and $\gamma$ is sampled according to the
SLEκ law $\mu^{D\setminus\bigcup_{k\neq{\bf j}}\eta_{k}^{0}}_{\alpha_{\bf
j}}$. It is obvious that if $\mu$ is an $N$-SLEκ measure in $D$, then it is
also an invariant measure for $\Phi$.

######  Lemma 4.1.

$\Phi$ is irreducible.

###### Proof.

We use the notion of tubes in the proof of Lemma 3.3, and find mutually
disjoint tubes ${T}_{j}$, $j\in\mathbb{N}_{N}$, in $D$ such that $T_{j}$
connects $\alpha_{j}$. Let ${\cal T}_{j}=\\{\eta\in X_{j}:\eta\subset
T_{j}\\}$, and $\phi_{j}=\mathbbm{1}_{{{T}_{j}}}\mu^{D}_{\alpha_{j}}$, i.e.,
the law of the chordal SLEκ in $D$ connecting $\alpha_{j}$ restricted to the
event that the curve stays in ${T}_{j}$. By Proposition 2.1, each $\phi_{j}$
is a nontrivial finite measure on $X_{j}$. Let $\phi=\prod_{j=1}^{N}\phi_{j}$.
We are going to show that $\phi$ is an irreducibility measure for $\Phi$.

Suppose $\underline{\cal A}\subset\underline{X}$ satisfies that
$\phi(\underline{\cal A})>0$. Let
$\underline{\eta}=(\eta_{1},\dots,\eta_{N})\in\underline{X}$. We need to show
that $P^{m}(\underline{\eta},\underline{\cal A})>0$ for some $m\in\mathbb{N}$.
It suffices to construct measurable $\underline{\cal
A}^{0},\dots,\underline{\cal A}^{m}\subset\underline{X}$ with $\underline{\cal
A}^{0}=\\{\underline{\eta}\\}$ and $\underline{\cal A}^{m}=\underline{\cal A}$
such that for any $\underline{\gamma}\in\underline{\cal A}^{s-1}$, $1\leq
s\leq m$, $P(\underline{\gamma},\underline{\cal A}^{s})>0$, which holds if
there is $j_{s}\in\mathbb{N}_{N}$ such that
$P_{j_{s}}(\underline{\gamma},\underline{\cal A}^{s})>0$.

First, suppose $\underline{\cal A}=\underline{\cal T}:=\prod_{j=1}^{N}{{\cal
T}_{j}}$. By reordering the components of $\alpha$ and using Lemma 3.6, we may
assume that for any $1\leq j\leq N$, there is an open boundary arc $I_{j}$ of
$D$ with endpoints $\alpha_{j}$ such that ${\cal
I}_{j}:=\\{k\in\mathbb{N}_{N}:\alpha_{k}\subset
I_{j}\\}\subset\mathbb{N}_{j-1}$. We have the following fact.

  * (F)

Let $r\in\mathbb{N}_{N}$. Let $A,B$ be closed subset of $\overline{D}$ such
that $A\cup{I_{r}}$ is connected, $A\cap({\partial D\setminus
I_{r}})=\emptyset$, $B\cup({\partial D\setminus I_{r}})$ is connected,
$B\cap\overline{I_{r}}=\emptyset$, and $A\cap B=\emptyset$. Then there is a
tube connecting $\alpha_{r}$ in $D$, which is disjoint from $A\cup B$.

Applying (F) to $r=1$, $A=\emptyset$ and $B=\bigcup_{j=2}^{N}(\eta_{j}\cup
T_{j})$, we construct a tube $S_{1}$ connecting $\alpha_{1}$, which is
disjoint from all $\eta_{j},T_{j}$, $2\leq j\leq N$. Suppose for some $m\leq
N-1$, we have constructed mutually disjoint tubes $S_{j}$ connecting
$\alpha_{j}$ for $1\leq j\leq m$, which are disjoint from all $\eta_{k},T_{k}$
for $k\geq m+1$. Applying (F) to $r=m+1$, $A=\bigcup_{j\in{\cal
I}_{m+1}}S_{j}$ and $B=\bigcup_{j\in\mathbb{N}_{m}\setminus{\cal
I}_{m+1}}S_{j}\cup\bigcup_{j=m+2}^{N}(\eta_{j}\cup T_{j})$, we construct a
tube $S_{m+1}$ connecting $\alpha_{m+1}$, which is disjoint from $S_{j}$ for
$1\leq j\leq m$ and $\alpha_{j}\cup T_{j}$ for $m+2\leq j\leq N$. By induction
we get mutually disjoint tubes $S_{j}$ connecting $\alpha_{j}$, $1\leq j\leq
N$, such that $S_{j}$ is disjoint from $\eta_{k}$ and $T_{k}$ whenever $j<k$.
Let ${\cal S}_{j}=\\{\eta\in X_{j}:\eta\subset S_{j}\\}$. For $1\leq s\leq N$,
we define $\underline{\cal A}^{s}=\prod_{j=1}^{s}{\cal
S}_{j}\times\prod_{j={s+1}}^{N}\\{\eta_{j}\\}$. Note that $\underline{\cal
A}^{0}=\\{\underline{\eta}\\}$ and $\underline{\cal
A}^{N}=\prod_{j=1}^{N}{\cal S}_{j}=:\underline{\cal S}$. Let
$s\in\mathbb{N}_{N}$ and
$\underline{\gamma}=(\gamma_{1},\dots,\gamma_{N})\in{\cal A}^{s-1}$. Then
$\gamma_{j}\subset S_{j}$ for $1\leq j\leq s$ and $\gamma_{j}=\eta_{j}$ for
$s+1\leq j\leq N$. Since $S_{s}$ is disjoint from $S_{j}$, $j<s$, and
$\eta_{j}$, $j>s$, it is a tube in $D\setminus\bigcup_{j\neq s}\gamma_{j}$
connecting $\alpha_{j}$. Thus, by Proposition 2.1,
$P_{s}(\underline{\gamma},\underline{\cal
A}^{s})=\mu^{D\setminus\bigcup_{j\neq
s}\gamma_{j}}_{\alpha_{s}}(\pi^{\underline{\gamma}}_{s}(\underline{\cal
A}^{s}))=\mu^{D\setminus\bigcup_{j\neq s}\gamma_{j}}_{\alpha_{s}}({\cal
S}_{s})>0$.

For $1\leq r\leq N$, we let $\underline{\cal A}^{N+r}=\prod_{j=1}^{N-r}{\cal
S}_{j}\times\prod_{j=N+1-r}^{N}{\cal T}_{j}$. Note that $\underline{\cal
A}^{2N}=\underline{\cal T}$. Let $r\in\mathbb{N}_{N}$ and
$\underline{\gamma}=(\gamma_{1},\dots,\gamma_{N})\in\underline{\cal
A}^{N+r-1}$. Since $T_{N+1-r}$ is disjoint from $T_{j}$, which contains
$\gamma_{j}$, for $j>N+1-r$ and $S_{j}$, which also contains $\gamma_{j}$, for
$j<N+1-r$, it is a tube in $D\setminus\bigcup_{j\neq s}\gamma_{j}$ connecting
$\alpha_{N+1-r}$. Thus, by Proposition 2.1,

$P_{N+1-r}(\underline{\gamma},\underline{\cal
A}^{N+r})=\mu^{D\setminus\bigcup_{j\neq
N+1-r}\gamma_{j}}_{\alpha_{N+1-r}}(\pi^{\underline{\gamma}}_{N+1-r}(\underline{\cal
A}^{N+r}))=\mu^{D\setminus\bigcup_{j\neq
N+1-r}\gamma_{j}}_{\alpha_{N+1-r}}({\cal T}_{N+1-r})>0.$

So the sequence $\underline{\cal A}^{s}$, $0\leq s\leq 2N$, satisfy the
desired properties for $\underline{\cal A}=\underline{\cal T}$.

Now we consider the general case. Since $\phi$ is supported by
$\underline{\cal T}$, we may assume $\underline{\cal A}\subset\underline{\cal
T}$. It suffices to further construct $\underline{\cal A}^{2N+t}$, $1\leq
t\leq N$, with $\underline{\cal A}^{3N}=\underline{\cal A}$, such that for any
$\underline{\gamma}\in\underline{\cal A}^{2N+t-1}$, $1\leq t\leq N$,
$P_{t}(\underline{\gamma},\underline{\cal A}^{2N+t})>0$. We define these
$\underline{\cal A}^{2N+t}$’s backward inductively such that $\underline{\cal
A}^{3N}=\underline{\cal A}$ and

$\underline{\cal A}^{2N+t-1}=\\{\underline{\gamma}\in\underline{\cal
T}:\phi_{t}(\pi^{\underline{\gamma}}_{t}(\underline{\cal A}^{2N+t}))>0\\}$
(4.1)

for $2\leq t\leq N$. Since $\phi=\prod_{j=1}^{N}\phi_{j}$ and
$\phi(\underline{\cal A}^{3N})>0$, by Fubini theorem and backward induction,
each $\underline{\cal A}^{2N+t}$ has the form ${\cal
C}_{t}\times\prod_{j=t+1}^{N}{\cal T}_{t}$ for some ${\cal
C}_{t}\subset\prod_{j=1}^{t}{\cal T}_{j}$ with $\prod_{j=1}^{t}\phi_{j}({\cal
C}_{t})>0$. From $\underline{\cal A}^{2N+1}={\cal
C}_{1}\times\prod_{j=2}^{N}{\cal T}_{j}$ and $\phi_{1}({\cal C})>0$ we see
that (4.1) also holds for $t=1$.

Let $1\leq t\leq N$ and $\underline{\gamma}\in\underline{\cal A}^{2N+t-1}$. By
(4.1), $\mu^{D}_{\alpha_{t}}(\pi^{\underline{\gamma}}_{t}(\underline{\cal
A}^{2N+t}))=\phi_{t}(\pi^{\underline{\gamma}}_{t}(\underline{\cal
A}^{2N+t}))>0$. Since $T_{t}$ is a tube in $D\setminus\cup_{j\neq
t}\gamma_{j}$ connecting $\alpha_{t}$, by Proposition 2.1,

$P_{t}(\underline{\gamma},\underline{\cal
A}^{2N+t})=\mu^{D\setminus\bigcup_{j\neq
t}\gamma_{j}}_{\alpha_{t}}(\pi^{\underline{\gamma}}_{t}(\underline{\cal
A}^{2N+t}))=\mu^{D_{t}(\underline{\gamma})}_{\alpha_{t}}(\pi^{\underline{\gamma}}_{t}(\underline{\cal
A}^{2N+t}))>0.$

Thus, $\underline{\cal A}^{s}$, $0\leq s\leq 3N$, satisfy the desired
properties for the general $\underline{\cal A}$. ∎

The irreducible Markov chain theory then implies the uniqueness theorem.

######  Theorem 4.2 (Uniqueness).

Let $\kappa\in(0,8)$. Let $D$ be a simply connected domain with locally
connected boundary and $\alpha\in\operatorname{LP}^{D}_{N}$. If there exists
an $N$-SLEκ law $\mu$ in $D$ with link pattern $\alpha$, then $Q^{D}_{\alpha}$
is finite, and its normalization $Q^{D}_{\alpha}/|Q^{D}_{\alpha}|$ equals
$\mu$.

##  5 Partition function

For $\alpha\in\operatorname{LP}^{U}_{N}$, we call
$H^{U}_{\alpha}:=|Q^{U}_{\alpha}|$ a multiple SLEκ partition function. When
$N=1,2$, the function is finite and has a closed formula. For the rest of the
section we assume $\kappa\leq 6$, in which case $H^{U}_{\alpha}$ is known to
be finite by Lemma 3.4. By (2.9,3.1), $H^{U}_{\alpha}$ satisfies the following
conformal covariance. Suppose $f$ is a conformal map on $U$, which extends to
neighborhoods of points in $\alpha$. Then

$H^{U}_{\alpha}=\prod_{x\in\alpha}|f^{\prime}(x)|^{\operatorname{b}}H^{f(U)}_{f(\alpha)}.$
(5.1)

Thus, it suffices to consider the case that $U=\mathbb{H}$. In that case,
$\operatorname{LP}_{N}:=\operatorname{LP}^{\mathbb{H}}_{N}$ is an open subset
of $\mathbb{R}^{2N}$. By a slight abuse of notation, we also call the function
$H(\alpha):=H^{\mathbb{H}}_{\alpha}$ defined on $\operatorname{LP}_{N}$ a
multiple ($N$-)SLEκ partition function.

Our definition follows the theme of [6], where the partition function is
defined as the total mass of a measure. It is different from the multiple SLE
partition function introduced in [4, 3], in which the function is assumed to
satisfy a system of $2N$ second-order linear PDE (cf. [7, Formula (1.2)]),
which guarantees the existence of local multiple SLE.

A local multiple SLEκ consists of $2N$ SLEκ-type curves emanating from the
points of $\alpha$ and stopped before leaving neighborhoods of these points,
which satisfy some commutation relation. These curves may or may not be
extended to form an $N$-SLEκ associated with $\alpha$. Some authors call the
usual multiple SLE global multiple SLE, and call a partition function multiple
SLE pure partition function whenever it generates a local multiple SLE that is
a part of a global multiple SLE with a fixed link pattern.

By (5.1), $H$ satisfies the conformal covariance for multiple SLE partition
function (cf. [7, Formula (1.3)]). We will show that $H$ satisfies the
asymptotics for pure partition function (cf. [15, Formula (1.3)]) in Lemma
5.5.

For $N\geq 2$ and $j\in\mathbb{N}_{N}$, let $P_{\widehat{j}}$ denote the map
$\underline{x}\mapsto(x_{1},\dots,\widehat{x_{2j-1}},\widehat{x_{2j}},\dots,x_{2N})$
from $\mathbb{R}^{2N}$ to $\mathbb{R}^{2(N-1)}$. This is consistent with our
definition of
$P_{\widehat{j}}:\operatorname{LP}_{N}\to\operatorname{LP}_{N-1}$.

Let $D$ be a simply connected domain and $\alpha\in\operatorname{LP}^{D}_{N}$.
For two prime ends $a\neq b$ of $D$, which are different from points in
$\alpha$, we define

$I^{D;(a,b)}_{\alpha}=\int\mu^{D}_{\alpha}(d\eta)H^{U\setminus\eta}_{\alpha}.$
(5.2)

The conformal covariance of $H^{U}_{\alpha}$ implies the following conformal
covariance of $I^{D;(a,b)}_{\alpha}$. If $f$ satisfies the conditions of the
$f$ in (5.1), then

$I^{U;(a,b)}_{\alpha}=\prod_{x\in\alpha}|f^{\prime}(x)|^{\operatorname{b}}H^{f(U);(f(a),f(b))}_{f(\alpha)}.$
(5.3)

By (3.1) and Lemma 3.6, if $N\geq 2$, then for any $j\in\mathbb{N}_{N}$,

$H^{D}_{\alpha}=H^{D}_{\alpha_{j}}I^{D;\alpha_{j}}_{P_{\widehat{j}}(\alpha)}.$
(5.4)

######  Lemma 5.1.

Let $N\geq 2$ and $j\in\mathbb{N}_{N}$. Let $\underline{y}\in\mathbb{R}^{2N}$
be such that $y_{2j}=y_{2j-1}\neq y_{k}$ for any
$k\in\mathbb{N}_{2N}\setminus\\{2j,2j-1\\}$, and
$P_{\widehat{j}}(\underline{y})\in\operatorname{LP}_{N-1}$. Then

$\lim_{\operatorname{LP}_{N}\ni\underline{x}\to\underline{y}}|x_{2j}-x_{2j-1}|^{2\operatorname{b}}H(\underline{x})\to
H(P_{\widehat{j}}(\underline{y})).$ (5.5)

###### Proof.

By (5.2,5.4),

$|x_{2j}-x_{2j-1}|^{2\operatorname{b}}H(\underline{x})=\int\mu^{\mathbb{H}}_{(x_{2j},x_{2j-1})}H^{\mathbb{H}\setminus\eta}_{P_{\widehat{j}}(\underline{x})}.$
(5.6)

As $\underline{x}\to\underline{y}$, $x_{2j},x_{2j-1}\to y_{2j}$, so the SLEκ
curve in $\mathbb{H}$ from $x_{2j}$ to $x_{2j-1}$ shrinks to the single point
$y_{2j}$, which implies that the integrand in (5.6) tends to the RHS of (5.5).
So we get (5.5) using dominated convergence theorem and monotonicity of $H$. ∎

######  Lemma 5.2.

For any $N\in\mathbb{N}$, $H$ is continuous on ${\operatorname{LP}}_{N}$.

###### Proof.

The continuity of $H$ when $N=1$ is obvious because
$H(a,b)=|a-b|^{-2\operatorname{b}}$. Suppose $H$ is continuous on
$\operatorname{LP}_{k}$ for any $k\leq N$. We now show that $H$ is continuous
on ${\operatorname{LP}}_{N+1}$. Suppose
$\underline{x}^{n}\to\underline{x}^{0}$ in $\operatorname{LP}_{N+1}$. We need
to show that $H(\underline{x}^{n})\to H(\underline{x}^{0})$. We may and will
assume that $x^{n}_{1}=0$ and $x^{n}_{2}=1$ for all
$n\in\mathbb{N}\cup\\{0\\}$ by applying (2.9) to linear functions.

By (5.2,5.4),

$H(\underline{x}^{n})=\int\mu^{\mathbb{H}}_{(0,1)}(d\eta)H^{\mathbb{H}\setminus\eta}_{P_{\widehat{1}}(\underline{x}^{n})},\quad
n\in\mathbb{N}\cup\\{0\\}.$

For any fixed $\eta$,
$H^{\mathbb{H}\setminus\eta}_{P_{\widehat{1}}(\underline{x}^{n})}$ equals the
product of the partition functions in each component of
$\mathbb{H}\setminus\eta$ whose boundary contains points of
$\underline{x}^{n}$. By applying the continuity of $H$ on
$\operatorname{LP}_{k}$ for $k\leq N$ and the conformal covariance of $H$ to
each component of $\mathbb{H}\setminus\eta$, we get
$H^{\mathbb{H}\setminus\eta}_{P_{\widehat{1}}(\underline{x}^{n})}\to
H^{\mathbb{H}\setminus\eta}_{P_{\widehat{1}}(\underline{x}^{0})}$. By
monotonicity,
$H^{\mathbb{H}\setminus\eta}_{P_{\widehat{1}}(\underline{x}^{n})}\leq
H(P_{\widehat{1}}(\underline{x}^{n}))\to
H(P_{\widehat{1}}(\underline{x}^{0}))$. So we get $H(\underline{x}^{n})\to
H(\underline{x}^{0})$ by dominated convergence theorem. The proof is completed
by induction. ∎

######  Theorem 5.3.

Let $\underline{u}=(u_{1},\dots,u_{2N})\in\operatorname{LP}_{N}$, $N\geq 2$.
Let $B_{t}$ be a standard Brownian motion. Let
$\\{a,b\\}=\\{2j_{0}-1,2j_{0}\\}$ for some $j_{0}\in\mathbb{N}_{N}$. Let
$W_{t}$ and $V_{t}$ solve the system of SDE/ODE

$\left\\{\begin{array}[]{ll}dW_{t}=\frac{\kappa-6}{W_{t}-V_{t}}\,dt+\sqrt{\kappa}dB_{t},&W_{0}=u_{a};\\\
dV_{t}=\frac{2}{V_{t}-W_{t}}\,dt,&V_{0}=u_{b}.\end{array}\right.$ (5.7)

Let $g_{t}$ be the chordal Loewner maps driven by $W_{t}$, and let $\tau$ be
the first time that $g_{t}(u_{j})$ is not defined for some
$j\in\mathbb{N}_{2N}\setminus\\{a\\}$. Let $U^{j}_{t}=g_{t}(u_{j})$ for $j\neq
a$, and $U^{a}_{t}=W_{t}$. Note that $U^{b}_{t}=V_{t}$. Then

$M_{t}:=\Big{(}\prod_{j\not\in\\{a,b\\}}|g_{t}^{\prime}(u_{j})|\Big{)}^{\operatorname{b}}H(W_{t},V_{t})^{-1}H(U^{1}_{t},\dots,U^{2N}_{t}),\quad
0\leq t<\tau,$ (5.8)

is a bounded local martingale.

###### Proof.

If $N=1$, $M_{t}$ is constant $1$, and the conclusion is obvious. Suppose
$N\geq 2$. By symmetry we may assume that $a=1$ and $b=2$. Let $K_{t}$ and
$\gamma(t)$, $0\leq t<\tau$, be the chordal Loewner hulls and curve driven by
$W_{t}$. By SLE coordinate changes ([14]), $\gamma$ is a part of a chordal
SLEκ curve, say $\eta$, in $\mathbb{H}$ from $x_{1}$ to $x_{2}$. Let
$Z=H^{\mathbb{H}\setminus\eta}_{(u_{3},\dots,u_{2N})}$. Let $T$ be any
stopping time. Let $\eta_{T}$ and $\eta^{T}$ be respectively the parts of
$\eta$ before and after $T$. By DMP of SLEκ, conditionally on ${\cal F}_{T}$
and the event $E_{T}:=\\{T<\tau\\}$, $\eta^{T}$ is an SLEκ in
$\mathbb{H}\setminus K_{T}$ from $\eta(T)$ to $u_{2}$. Since
$Z=H^{(\mathbb{H}\setminus\eta_{T})\setminus\eta^{T}}_{(u_{3},\dots,u_{2N})}=H^{(\mathbb{H}\setminus
K_{T})\setminus\eta^{T}}_{(u_{3},\dots,u_{2N})}$ on $E_{T}$, we get
$\mathbb{E}[Z|{\cal F}_{T},E_{T}]=I^{\mathbb{H}\setminus
K_{t};(\eta(T),u_{2})}_{(u_{3},\dots,u_{2N})}$. Since $g_{T}$ maps
$\mathbb{H}\setminus K_{T}$ conformally onto $\mathbb{H}$, and sends $\eta(T)$
and $u_{2}$ respectively to $W_{t}$ and $V_{T}$, by (5.3,5.4), on the event
$E_{T}$,

$M_{T}=\Big{(}\prod_{j\not\in\\{1,2\\}}|g_{t}^{\prime}(u_{j})|\Big{)}^{\operatorname{b}}I^{\mathbb{H};(W_{t},V_{t})}_{(U^{3}_{t},\dots,U^{2N}_{t})}=I^{\mathbb{H}\setminus
K_{t};(\eta(T),u_{2})}_{(u_{3},\dots,u_{2N})}=\mathbb{E}[Z|{\cal F}_{T}]$

on the event $E_{T}=\\{T<\tau\\}$. Since this holds for any stopping time $T$,
and $Z\leq H(u_{3},\dots,u_{2N})$ by monotonicity, we get the conclusion. ∎

Once we know that $H$ is smooth, Itô’s calculus together with (2.6,2.7)
implies that $H$ satisfies the PDE for multiple SLE partition function (cf.
[7, Formula (1.2)]):

${\cal D}_{r}H:=\Big{[}\frac{\kappa}{2}\partial_{r}^{2}+\sum_{j\neq
r}\Big{(}\frac{2}{x_{j}}\partial_{j}-\frac{2\operatorname{b}}{x_{j}^{2}}\Big{)}\Big{]}H=0,\quad
1\leq r\leq N,$ (5.9)

For $\kappa\in(0,6]$, if $(\eta_{1},\dots,\eta_{N})$ is an $N$-SLEκ in $D$
with link pattern $\alpha$, then for every $j\in\mathbb{N}_{N}$, the law of
$\eta_{j}$ is absolutely continuous w.r.t. $\mu^{D}_{\alpha_{j}}$ with Radon-
Nikodym derivative
$H^{D\setminus\eta}_{P_{\widehat{j}}(\alpha)}/H^{D}_{\alpha}$. To see this for
$j=1$, we use (3.1) to conclude that
$Q^{D}_{\alpha}=Q^{D}_{\alpha_{1}}(d\eta_{1})\otimes
Q^{D\setminus\eta_{1}}_{P_{\widehat{1}}(\alpha)}(d(\eta_{2},\dots,\eta_{N}))$,
and then apply (2.2). The statement for other $j$ follows from symmetry.
Unlike the case $\kappa\in(0,4]$, if $\kappa>4$, for any $j\neq
k\in\mathbb{N}_{N}$, the joint law of $(\eta_{j},\eta_{k})$ is not absolutely
continuous w.r.t. $\mu^{D}_{\alpha_{j}}\times\mu^{D}_{\alpha_{k}}$.

The partition function may also be used to describe the law of local multiple
SLE. Let $(\eta_{1},\dots,\eta_{N})$ be multiple SLEκ in $U$ with link pattern
$\alpha$. Suppose $\alpha_{j}=(a_{j},b_{j})$, and let $U^{a}_{j},U^{b}_{j}$ be
neighborhoods of $a_{j},b_{j}$ in $D$, $j\in\mathbb{N}$, mutually disjoint
closures. Let $\eta^{a}_{j}$ (resp. $\eta^{b}_{j}$) be the part of $\eta_{j}$
started from $a_{j}$ (resp. $b_{j}$) and stopped on exiting $U^{a}_{j}$ (resp.
$U^{b}_{j}$). Then the joint law of
$(\eta^{a}_{1},\dots,\eta^{a}_{N};\eta^{b}_{1},\dots,\eta^{b}_{N})$ is
absolutely continuous w.r.t. the product of the laws of SLEκ in $D$ from
$a_{j}$ to $b_{j}$ stopped when exiting $U^{a}_{j}$ for $1\leq j\leq N$ and
the laws of SLEκ in $D$ from $b_{j}$ to $a_{j}$ stopped when exiting
$U^{b}_{j}$ for $1\leq j\leq N$, and the Radon-Nikodym derivative could be
expressed in terms of multiple SLEκ partition function and Brownian loop
measure.

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