¹¹affiliationtext: Tsinghua University, China

Multiple SLEs and Dyson Brownian motion:
transition density and Green’s function

Hao Wu [email protected]. Supported by Beijing Natural Science Foundation (JQ20001). Lu Yang [email protected]

Abstract

1 Introduction

In this article, we consider connections between multiple chordal SLEs and Dyson Brownian motion. The motivation to study multiple SLEs is to describe the scaling limit of multiple interfaces in polygons, see for instance [BBK05, Dub07, KP16]. The study on its connections with Dyson Brownian motion goes back to [Car03] in multiple radial case, and some recent research including [KK21, CM22, CLM23] has also proved properties of multiple chordal SLE driven by Dyson Brownian motion, which is a local formulation of multiple chordal SLE defined through multiple Loewner equations.

There is another global formulation of multiple chordal SLE defined as commuting Loewner chains and we first introduce its notions. We call $(\Omega;x_{1},\ldots,x_{2N})$ a polygon if $\Omega\subset\mathbb{C}$ is a simply connected domain such that $\partial\Omega$ is locally connected and $x_{1},\ldots,x_{2N}$ are $2N$ marked points lying on $\partial\Omega$ in counterclockwise order. We consider $N$ non-crossing simple curves in $\Omega$ such that each curve connects two points among $\{x_{1},\ldots,x_{2N}\}$ . These curves can have various planar connectivities. We describe such connectivities by link patterns $\alpha=\{\{a_{1},b_{1}\},\ldots,\{a_{N},b_{N}\}\}$ where $\{a_{1},b_{1},\ldots,a_{N},b_{N}\}=\{1,2,\ldots,2N\}$ , and denote by $\mathrm{LP}_{N}$ the set of all link patterns. Note that $\#\mathrm{LP}_{N}$ is the Catalan number $C_{N}=\frac{1}{N+1}\binom{2N}{N}$ .

When $N=1$ , we denote by $X_{\mathrm{simple}}(\Omega;x_{1},x_{2})$ the set of continuous simple unparameterized curves in $\Omega$ connecting $x_{1}$ and $x_{2}$ such that they only touch the boundary $\partial\Omega$ in $\{x_{1},x_{2}\}$ . We denote by $X(\Omega;x_{1},x_{2})$ the closure of $X_{\mathrm{simple}}(\Omega;x_{1},x_{2})$ in the metric topology of the set $X$ of all planar oriented curves (i.e., continuous mappings from $[0,1]$ to $\mathbb{C}$ modulo reparameterization) equipped with metric

\mathrm{dist}(\eta,\tilde{\eta}):=\inf_{\varphi,\tilde{\varphi}}\sup_{t\in[0,1]}|\eta(\varphi(t))-\tilde{\eta}(\tilde{\varphi}(t))|,\quad\eta,\tilde{\eta}\in X,

where the infimum is taken over all increasing homeomorphisms $\varphi,\tilde{\varphi}:[0,1]\mapsto[0,1]$ . Chordal SLE in $(\Omega;x_{1},x_{2})$ is a continuous random curve in $\Omega$ connecting $x_{1}$ and $x_{2}$ . Note that $\mathrm{SLE}_{\kappa}$ in $(\Omega;x_{1},x_{2})$ is a random curve in $X_{\mathrm{simple}}(\Omega;x_{1},x_{2})$ when $\kappa\leq 4$ and that $\mathrm{SLE}_{\kappa}$ in $(\Omega;x_{1},x_{2})$ is a random curve in $X(\Omega;x_{1},x_{2})$ when $\kappa\in(4,8)$ . In general, for $N\geq 2$ and $\alpha=\{\{a_{1},b_{1}\},\ldots,\{a_{N},b_{N}\}\}\in\mathrm{LP}_{N}$ , we denote by $X_{\alpha}(\Omega;x_{1},\ldots,x_{2N})$ the set of families $(\gamma_{1},\ldots,\gamma_{N})$ of curves such that, for each $s\in\{1,\ldots,N\}$ , we have $\gamma_{s}\in X(\Omega;x_{a_{s}},x_{b_{s}})$ and $\gamma_{s}$ does not disconnect any two points $x_{a},x_{b}$ with $\{a,b\}\in\alpha$ from each other.

Fix $\kappa\in(0,8)$ . For $N\geq 1$ and $\alpha\in\mathrm{LP}_{N}$ , there exists a unique probability measure on $(\gamma_{1},\ldots,\gamma_{N})\in X_{\alpha}(\Omega;x_{1},\ldots,x_{2N})$ such that, for each $j\in\{1,\ldots,N\}$ , the conditional law of $\gamma_{j}$ given $\{\gamma_{1},\ldots,\gamma_{N}\}\setminus\{\gamma_{j}\}$ is the chordal $\mathrm{SLE}_{\kappa}$ connecting $x_{a_{j}}$ and $x_{b_{j}}$ in the connected component of the domain $\Omega\setminus\bigcup_{k\neq j}\gamma_{k}$ having $x_{a_{j}}$ and $x_{b_{j}}$ on its boundary. For the existence and uniqueness of such measures, see [KL07], [PW19, Theorem 1.3] and [BPW21, Theorem 1.2] for $\kappa\in(0,4]$ , and [AHSY23, Theorem 1.2], [Zha23, Theorem 4.2] and [FLPW24, Theorem 1.21] for $\kappa\in(4,8)$ . We call this measure chordal $N$ - $\mathrm{SLE}_{\kappa}$ in polygon $(\Omega;x_{1},\ldots,x_{2N})$ associated to link pattern $\alpha$ .

In the companion paper [FWY24], we derived the connection between chordal $N$ - $\mathrm{SLE}_{\kappa}$ in the unit disc and Dyson circular ensemble for $\kappa\in(0,4]$ . In this article, we will prove corresponding conclusions for chordal $N$ - $\mathrm{SLE}_{\kappa}$ and Dyson Brownian motion, and extend the conclusions from $\kappa\in(0,4]$ to $\kappa\in(0,8)$ . The main conclusions are summarized as follows. We will parameterize multiple chordal SLEs by a “common time” and derive the transition density of the driving function in Theorem 1.1: it can be given by the transition density of Dyson Brownian motion and Green’s function (Definition 1.2). Using such connection and the asymptotic of the transition density of Dyson Brownian motion, we derive the asymptotic of the probability of a rare event and show that, conditional on this rare event, the driving function of multiple chordal SLEs converges to Dyson Brownian motion (Proposition 1.3). In particular, we take $\kappa=4$ as an example and connect our result to Gaussian free field level lines (Proposition 4.1). In this case, the driving function has an explicit SDE (4.2), Green’s function has an explicit formula (4.4), and the Dyson Brownian motion is of parameter $\beta=2$ . Finally, we give one application of our main result: we derive the asymptotic of the probability for chordal $N$ -SLE to hit a small neighborhood of a boundary point (Theorem 1.4).

Let us elaborate on how we extend our results from $\kappa\in(0,4]$ to $\kappa\in(0,8)$ . To prove the first main conclusion Theorem 1.1, we first relate global multiple SLEs to multiple SLE partition functions “pure partition functions” (see Section 2.1); second, we use these partition functions to construct $2N$ -time local martingales (see Section 3.2); and finally, we take the common time in these local martingales and derive the desired transition density. This proof works for all $\kappa\in(0,8)$ . To prove the second main conclusion Proposition 1.3 and Theorem 1.4, we need further analysis on pure partition functions in Section 2.2. Such analysis relies on the relation between Coulomb gas integrals and pure partition functions which is only available due to recent development in [FPW24, FLPW24]. Using these tools, we are also able to extend previous result [FWY24, Theorem 1.2] from $\kappa\in(0,4]$ to $\kappa\in(0,8)$ , see Remark 1.5.

1.1 Multiple SLEs and Dyson Brownian motion

We focus on the polygon $(\mathbb{H};x_{1},\ldots,x_{2N})$ where $\mathbb{H}$ is the upper-half plane and $x_{1}<\cdots<x_{2N}$ . We denote

\mathfrak{X}_{2N}:=\{(x_{1},\ldots,x_{2N})\in\mathbb{R}^{2N}:x_{1}<\cdots<x_{2N}\}.

We denote by $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ the law of chordal $N$ - $\mathrm{SLE}_{\kappa}$ in polygon $(\mathbb{H};x_{1},\ldots,x_{2N})$ associated to link pattern $\alpha=\{\{a_{1},b_{1}\},\ldots,\{a_{N},b_{N}\}\}\in\mathrm{LP}_{N}$ . Suppose $(\gamma_{1},\ldots,\gamma_{N})\sim\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ . Recall that $\gamma_{j}$ is a curve in $\mathbb{H}$ from $x_{a_{j}}$ to $x_{b_{j}}$ . We view $(\gamma_{1},\ldots,\gamma_{N})$ as a $2N$ -tuple of continuous non-self-crossing curves $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ : for $j\in\{1,\ldots,N\}$ , we define $\eta_{a_{j}}$ to be $\gamma_{j}$ and $\eta_{b_{j}}$ to be the time-reversal of $\gamma_{j}$ . In this way, $\eta_{j}$ is a continuous curve in $\mathbb{H}$ starting from $x_{j}$ for $j\in\{1,\ldots,2N\}$ . We introduce the common parameterization under which the $2N$ curves $(\eta_{1},\ldots,\eta_{2N})$ grow simultaneously and are parameterized by a single time parameter $t$ . For $\boldsymbol{t}=(t_{1},\ldots,t_{2N})\in\mathbb{R}_{+}^{2N}$ , let $H_{\boldsymbol{t}}$ be the unbounded connected component of $\mathbb{H}\setminus(\cup_{j}\eta_{j}([0,t_{j}]))$ . Let $g_{\boldsymbol{t}}:H_{\boldsymbol{t}}\to\mathbb{H}$ be the unique conformal transformation with

g_{\boldsymbol{t}}(z)=z+\frac{\aleph(\boldsymbol{t})}{z}+o(1/|z|),\quad z\to\infty.

For $a>0$ , we say that the collection of curves $(\eta_{j}([0,t_{j}]),1\leq j\leq 2N)$ have $a$ -common parameterization if for each $t<\tau:=\min_{1\leq j\leq 2N}t_{j}$ ,

\displaystyle\partial_{j}\aleph(t,t,\ldots,t)=a,\quad j\in\{1,2,\ldots,2N\}.

In particular, under $a$ -common parameterization, we denote $g_{t}:=g_{(t,\ldots,t)}$ and then

\displaystyle g_{t}(z)=z+\frac{2aNt}{z}+o(1/|z|),\quad z\to\infty.

Moreover, under $a$ -common parameterization, $g_{t}$ satisfies the following chordal Loewner equation

\displaystyle\partial_{t}g_{t}(z)=\sum_{j=1}^{2N}\frac{a}{g_{t}(z)-X_{t}^{j}},\quad\forall z\in\mathbb{H}\setminus\left(\cup_{j}\eta_{j}([0,t])\right),\quad\forall t<\tau.

We say that $\{(\eta_{j}(t),0\leq t<\tau)\}_{1\leq j\leq 2N}$ is the chordal Loewner chain with $a$ -common parameterization and with driving functions $\{(X_{t}^{j},0\leq t<\tau)\}_{1\leq j\leq 2N}$ .

For chordal $N$ - $\mathrm{SLE}_{\kappa}$ , we view it as a chordal Loewner chain $(\boldsymbol{\eta}(t)=(\eta_{1}(t),\ldots,\eta_{2N}(t)),t\geq 0)$ with $2/\kappa$ -common parameterization and with driving functions $(\boldsymbol{X}_{t}=(X^{1}_{t},\ldots,X^{2N}_{t}),t\geq 0)$ . Denote by $(\mathcal{F}_{t},t\geq 0)$ the filtration generated by $(\boldsymbol{\eta}(t),t\geq 0)$ , and define the lifetime $T$ as the first time $t$ that $X_{t}^{j}=X_{t}^{k}$ for some $j\neq k$ , then we have $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[T<\infty]=1$ . We will prove in Lemma 3.2 that, under $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ , the driving functions $(\boldsymbol{X}_{t},0\leq t<T)$ satisfies the SDE

\mathrm{d}X_{t}^{j}=\mathrm{d}B_{t}^{j}+(\partial_{j}\log\mathcal{Z}_{\alpha})(X_{t}^{1},\ldots,X_{t}^{2N})\mathrm{d}t+\sum_{k\neq j}\frac{2/\kappa}{X_{t}^{j}-X_{t}^{k}}\mathrm{d}t,\quad 1\leq j\leq 2N,

(1.1)

where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions and $\mathcal{Z}_{\alpha}$ is the pure partition function for chordal $N$ - $\mathrm{SLE}_{\kappa}$ defined in Section 2.3. The main goal of this article is to compare the solution to (1.1) with Dyson Brownian motion.

Theorem 1.1.

Fix $\kappa\in(0,8)$ and $\alpha\in\mathrm{LP}_{N}$ .

•

Denote by $p_{\alpha}(t;\cdot,\cdot)$ the transition density for the solution to (1.1).¹¹1By saying that $p_{\alpha}(t;\cdot,\cdot)$ is the transition density, we mean that for any $t>0$ and $\boldsymbol{x}\in\mathfrak{X}_{2N}$ , if $(\boldsymbol{X}_{t},0\leq t<T)$ starts from $\boldsymbol{x}$ , then for any bounded measurable function $f$ on $\mathfrak{X}_{2N}$ , $\mathbb{E}_{\alpha}^{(\boldsymbol{x})}\left[\mathbb{1}_{\{T>t\}}f(\boldsymbol{X}_{t})\right]=\int_{\mathfrak{X}_{2N}}p_{\alpha}(t;\boldsymbol{x},\boldsymbol{y})f(\boldsymbol{y})\mathrm{d}\boldsymbol{y}.$
•

Denote by $p_{*}(t;\cdot,\cdot)$ the transition density for Dyson Brownian motion with parameter $\beta=8/\kappa$ :

$\mathrm{d}X_{t}^{j}=\mathrm{d}B_{t}^{j}+\sum_{k\neq j}\frac{4/\kappa}{X_{t}^{j}-X_{t}^{k}}\mathrm{d}t,\quad 1\leq j\leq 2N,$ (1.2)

where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions.

Then we have

p_{\alpha}(t;\boldsymbol{x},\boldsymbol{y})=p_{*}(t;\boldsymbol{x},\boldsymbol{y})\frac{G_{\alpha}(\boldsymbol{x})}{G_{\alpha}(\boldsymbol{y})},\quad\text{for all }t\geq 0,\text{ and }\boldsymbol{x},\boldsymbol{y}\in\mathfrak{X}_{2N},

(1.3)

where $G_{\alpha}$ is the Green’s function in Definition 1.2.

Definition 1.2.

Fix $\kappa\in(0,8)$ and $\alpha\in\mathrm{LP}_{N}$ . For $(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ , we define Green’s function in the boundary for chordal $N$ - $\mathrm{SLE}_{\kappa}$ as

G_{\alpha}(x_{1},\ldots,x_{2N}):=\frac{\mathcal{Z}_{*}(x_{1},\ldots,x_{2N})}{\mathcal{Z}_{\alpha}(x_{1},\ldots,x_{2N})},

where $\mathcal{Z}_{*}$ is the partition function for $\mathrm{SLE}_{\kappa}(2,\ldots,2)$ :

\mathcal{Z}_{*}(x_{1},\ldots,x_{2N})=\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{2/\kappa},

(1.4)

and $\mathcal{Z}_{\alpha}$ is the pure partition function for chordal $N$ - $\mathrm{SLE}_{\kappa}$ defined in Section 2.3.

The relation (1.3) provides a tool to analyze multiple SLEs using Dyson Brownian motion. For instance, one may analyze multiple SLEs as $N\to\infty$ using (1.3) as the asymptotic of Dyson Brownian motion as $N\to\infty$ is known, see [dMS16, HK18, HS21]. One may analyze multiple SLEs as $\kappa\to 0+$ using (1.3) as the asymptotic of Dyson Brownian motion as $\beta\to\infty$ is known, this is related to large deviation principle for multiple SLEs, see [PW24, AHP24]. In the following, we analyze multiple SLEs as $t\to\infty$ using (1.3) as the asymptotic of Dyson Brownian motion as $t\to\infty$ is known [Rös98] (see Lemma 2.6) and we derive the asymptotic of the probability for multiple SLEs to hit a small neighborhood of a boundary point.

1.2 Estimates on multiple SLEs

Proposition 1.3.

Fix $\kappa\in(0,8)$ , $\alpha\in\mathrm{LP}_{N}$ and $\boldsymbol{x}\in\mathfrak{X}_{2N}$ . We have

\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[T>t]=\mathcal{I}_{*}^{-1}\mathcal{J}_{\alpha}G_{\alpha}(\boldsymbol{x})\left(\frac{1}{\sqrt{t}}\right)^{A_{2N}^{+}}\left(1+O\left(\frac{|\boldsymbol{x}|}{\sqrt{t}}\right)\right),\quad\text{as }t\to\infty,

(1.5)

where $\mathcal{I}_{*}$ and $\mathcal{J}_{\alpha}$ are normalization constants defined in (2.13) and (2.14), $G_{\alpha}$ is Green’s function in Definition 1.2, and $A_{2N}^{+}$ is the half-plane arm exponent for SLE:

A_{2N}^{+}=\frac{N(4N+4-\kappa)}{\kappa}.

(1.6)

Moreover,

•

For $t>0$ , denote by $\mu_{\alpha\,|\,t}^{(\boldsymbol{x})}$ the law of the solution to (1.1) started from $\boldsymbol{x}$ conditional on $\{T>t\}$ .
•

Denote by $\mu^{(\boldsymbol{x})}_{*}$ the law of Dyson Brownian motion (1.2) with parameter $\beta=8/\kappa$ started from $\boldsymbol{x}$ .

Denote by $(\mathcal{F}_{t},t\geq 0)$ the filtration generated by the Brownian motions in the equations (1.1) or (1.2). Then, for any $s>0$ , when both measures $\mu^{(\boldsymbol{x})}_{\alpha\,|\,t}$ and $\mu^{(\boldsymbol{x})}_{*}$ are restricted to $\mathcal{F}_{s}$ , the total variation distance between the two measures goes to zero as $t\to\infty$ :

\lim_{t\to\infty}\mathrm{dist}_{\mathrm{TV}}\left(\mu_{\alpha\,|\,t}^{(\boldsymbol{x})}[\cdot\,|\,_{\mathcal{F}_{s}}],\mu^{(\boldsymbol{x})}_{*}[\cdot\,|\,_{\mathcal{F}_{s}}]\right)=0.

(1.7)

Theorem 1.4.

Fix $\kappa\in(0,8)$ , $\boldsymbol{x}=(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ , $\alpha\in\mathrm{LP}_{N}$ and suppose $(\gamma_{1},\ldots,\gamma_{N})\sim\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ . Then we have²²2For two functions $F_{1}(\boldsymbol{x};R)$ and $F_{2}(\boldsymbol{x};R)$ , we write $F_{1}(\boldsymbol{x};R)\asymp F_{2}(\boldsymbol{x};R)$ as $R/|\boldsymbol{x}|\to\infty$ if there exist constants $C\in(1,\infty)$ and $R_{0}$ such that $C^{-1}\leq\frac{F_{1}(\boldsymbol{x};R)}{F_{2}(\boldsymbol{x};R)}\leq C$ for all $R\geq R_{0}|\boldsymbol{x}|$ .

\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[\gamma_{j}\cap\partial B(0,R)\neq\emptyset,1\leq j\leq N]\asymp G_{\alpha}(\boldsymbol{x})R^{-A_{2N}^{+}},\quad\text{as }R/|\boldsymbol{x}|\to\infty,

(1.8)

where $A_{2N}^{+}$ is half-plane arm exponent (1.6) and $G_{\alpha}$ is Green’s function in Definition 1.2, and the implicit constants depend only on $\kappa,N,\alpha$ .

The rest of this article is organized as follows. We give preliminaries on multiple SLEs and Dyson Brownian motion in Section 2. We prove Theorem 1.1 in Section 3. The main ingredients for the proof are $2N$ -time local martingales constructed in Section 3.2. Using Theorem 1.1 and the asymptotic of the transition density of Dyson Brownian motion, we complete the proof of Proposition 1.3 in Section 3.3. We prove Proposition 4.1 in Section 4. Finally, we prove Proposition 1.4 in Section 5 using Proposition 1.3. The conclusion in Proposition 1.4 is proved for $\kappa\in(0,8),N=1$ in [Law15] and is proved for $\kappa\in(0,8),N=2$ in [Zha21]. We will prove (1.8) in Section 5 for all $\kappa\in(0,8),N\geq 3,\alpha\in\mathrm{LP}_{N}$ .

Remark 1.5.

In the companion paper [FWY24], we derived the asymptotic of the probability for chordal $N$ - $\mathrm{SLE}_{\kappa}$ to hit a small neighborhood of an interior point for $\kappa\in(0,4]$ . Using recent development in [FPW24, FLPW24], we are able to extend it from $\kappa\in(0,4]$ to $\kappa\in(0,8)$ . More precisely, fix $\kappa\in(0,8)$ , consider the polygon $(\mathbb{U};x_{1},\ldots,x_{2N})$ where $\mathbb{U}$ is the unit disc. Let $(\gamma_{1},\ldots,\gamma_{N})$ be the chordal $N$ - $\mathrm{SLE}_{\kappa}$ in $(\mathbb{U};x_{1},\ldots,x_{2N})$ associated to link pattern $\alpha\in\mathrm{LP}_{N}$ and denote its law by $\mathbb{P}_{\alpha}^{(\mathbb{U};\boldsymbol{x})}$ . Then³³3For two functions $F_{1}(\boldsymbol{x};\epsilon)$ and $F_{2}(\boldsymbol{x};\epsilon)$ , we write $F_{1}(\boldsymbol{x};\epsilon)\sim F_{2}(\boldsymbol{x};\epsilon)$ as $\epsilon\to 0$ if the limit $\lim_{\epsilon\to 0}\frac{F_{1}(\boldsymbol{x};\epsilon)}{F_{2}(\boldsymbol{x};\epsilon)}$ exists and takes value in $(0,\infty)$ .

\mathbb{P}_{\alpha}^{(\mathbb{U};\boldsymbol{x})}\left[\gamma_{j}\cap B(0,\epsilon)\neq\emptyset,1\leq j\leq N\right]\sim\tilde{G}_{\alpha}(\boldsymbol{x})\epsilon^{A_{2N}},\quad\text{as }\epsilon\to 0,

(1.9)

where $A_{2N}$ is whole-plane arm exponent

A_{2N}=\frac{16N^{2}-(\kappa-4)^{2}}{8\kappa};

and $\tilde{G}_{\alpha}$ is Green’s function defined in [FWY24, Definition 1.3]. See discussion at the end of Section 5.

2 Preliminaries

We fix parameters

\kappa\in(0,8),\qquad a=\frac{2}{\kappa},\qquad h=\frac{6-\kappa}{2\kappa},\qquad c=\frac{(6-\kappa)(3\kappa-8)}{2\kappa}.

(2.1)

$\mathbb{H}$ -hulls and half-plane capacity

An $\mathbb{H}$ -hull is a relatively closed subset $K$ of $\mathbb{H}$ such that $K$ is bounded and $\mathbb{H}\setminus K$ is a simply connected domain. For an $\mathbb{H}$ -hull $K$ , there exists a unique conformal map $g_{K}:\mathbb{H}\setminus K\to\mathbb{H}$ with

g_{K}(z)=z+\frac{c_{K}}{z}+o(1/|z|),\quad z\to\infty

for some constant $c_{K}\geq 0$ . The constant $c_{K}$ is called the half-plane capacity of $K$ and denoted as $\text{hcap}(K)$ . We give some estimates about the relation between half-plane capacity and diameter of $\mathbb{H}$ -hulls, which will be used in the proof of Theorem 1.4.

Lemma 2.1.

Suppose $K\subset\mathbb{H}$ is an $\mathbb{H}$ -hull with half-plane capacity $\emph{hcap}(K)>0$ as described above.

•

If $K\cap\{|z|>r\}=\emptyset$ , then $\emph{hcap}(K)\leq r^{2}$ .
•

If $\emph{hcap}(K)<r^{2}/4$ , then $K\cap\{|z|>r\}=\emptyset$ .

Proof.

This is a standard estimate using the Koebe’s $1/4$ Theorem and Koebe’s distortion Theorem. See [Zha21, Proposition 2.4 and Lemma 6.1]. ∎

2.1 Pure partition functions

The description of the Loewner chain of chordal $N$ - $\mathrm{SLE}_{\kappa}$ involves the notion of pure paritition functions of multiple $\mathrm{SLE}_{\kappa}$ . These are the recursive collection $\{\mathcal{Z}_{\alpha}\colon\alpha\in\bigsqcup_{N\geq 0}\mathrm{LP}_{N}\}$ of functions $\mathcal{Z}_{\alpha}\colon\mathfrak{X}_{2N}\to\mathbb{R}$ uniquely determined by the following 4 properties:

•

BPZ equations: for all $j\in\{1,\ldots,2N\}$ ,

\displaystyle\left[\frac{1}{2}\partial_{j}^{2}+\sum_{k\neq j}\left(\frac{a}{x_{k}-x_{j}}\partial_{k}-\frac{ah}{(x_{k}-x_{j})^{2}}\right)\right]\mathcal{Z}_{\alpha}(x_{1},\ldots,x_{2N})=0.

(2.2)

•

Möbius covariance: for all Möbius maps $\varphi$ of the upper half-plane $\mathbb{H}$ such that $\varphi(x_{1})<\cdots<\varphi(x_{2N})$ , we have

\displaystyle\mathcal{Z}_{\alpha}(x_{1},\ldots,x_{2N})=\prod_{j=1}^{2N}\varphi^{\prime}(x_{j})^{h}\times\mathcal{Z}_{\alpha}(\varphi(x_{1}),\ldots,\varphi(x_{2N})).

•

Asymptotics: with $\mathcal{Z}_{\emptyset}\equiv 1$ for the empty link pattern $\emptyset\in\mathrm{LP}_{0}$ , the collection $\{\mathcal{Z}_{\alpha}\colon\alpha\in\mathrm{LP}_{N}\}$ satisfies the following recursive asymptotics property. Fix $N\geq 1$ and $j\in\{1,2,\ldots,2N-1\}$ . Then, we have

\displaystyle\lim_{x_{j},x_{j+1}\to\xi}\frac{\mathcal{Z}_{\alpha}(x_{1},\ldots,x_{2N})}{(x_{j+1}-x_{j})^{-2h}}=\begin{cases}\mathcal{Z}_{\alpha/\{j,j+1\}}(x_{1},\ldots,x_{j-1},x_{j+2},\ldots,x_{2N}),&\quad\text{if }\{j,j+1\}\in\alpha,\\ 0,&\quad\text{if }\{j,j+1\}\not\in\alpha,\end{cases}

where $\xi\in(x_{j-1},x_{j+2})$ (with the convention that $x_{0}=-\infty$ and $x_{2N+1}=+\infty$ ), and $\alpha/\{k,l\}$ denotes the link pattern in $\mathrm{LP}_{N-1}$ obtained by removing $\{k,l\}$ from $\alpha$ and then relabeling the remaining indices so that they are the first $2(N-1)$ positive integers.

•

The functions satisfy the following power-law bound: there exist constants $C\in(0,\infty)$ and $p\in(0,\infty)$ such that for all $(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ ,

\displaystyle|\mathcal{Z}_{\alpha}(x_{1},\ldots,x_{2N})|\leq C\prod_{1\leq j<k\leq 2N}|x_{k}-x_{j}|^{\mu_{kj}(p)},\quad\text{where }\mu_{kj}(p)=\begin{cases}p,&\text{if }|x_{k}-x_{j}|\geq 1,\\ -p,&\text{if }|x_{k}-x_{j}|<1.\end{cases}

(2.3)

The uniqueness when $\kappa\in(0,8)$ of such collection of functions were proved in [FK15a]. The existence is proved in [FK15b, KP16, PW19, Wu20, FLPW24]. Pure partition functions enjoy a refined power law bound which we summarize in the following two lemmas. They will be used later.

Lemma 2.2.

Fix $\kappa\in(0,6]$ , pure partition functions satisfy the following refined power law bound:

\displaystyle 0<\mathcal{Z}_{\alpha}(x_{1},\ldots,x_{2N})\leq\prod_{\{k,l\}\in\alpha}|x_{k}-x_{l}|^{-2h},\quad\text{for all }x_{1}<\cdots<x_{2N}.

(2.4)

Lemma 2.3.

Fix $\kappa\in(4,8)$ , pure partition functions satisfy the following refined power law bound: there exists constant $C\in(0,\infty)$ (depending only on $N$ and $\kappa$ ) such that for all $(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ ,

\displaystyle 0<\mathcal{Z}_{\alpha}(x_{1},\ldots,x_{2N})\leq C\prod_{1\leq j<k\leq 2N}|x_{k}-x_{j}|^{\mu_{kj}},\quad\text{where }\mu_{kj}=\begin{cases}10/\kappa,&\text{if }|x_{k}-x_{j}|\geq 1,\\ -6/\kappa,&\text{if }|x_{k}-x_{j}|<1.\end{cases}

(2.5)

The bound (2.4) is only proved for $\kappa\in(0,6]$ (see [Wu20, Theorem 1.7]) and we will prove the bound (2.5) for $\kappa\in(4,8)$ in Section 2.2. Note that the bound (2.5) is stronger than (2.3) but is weaker than (2.4). Nevertheless, it suffices for us to complete the proof of main conclusions.

2.2 Proof of Lemma 2.3

The proof of Lemma 2.3 relies on recent development [FPW24, FLPW24] on the relation between Coulomb gas integrals and pure partition functions, both of which are solutions to BPZ equations (2.2). We first introduce some necessary notations. Fix $\kappa\in(4,8)$ . For $N\geq 1$ , we define

\displaystyle\mathcal{G}^{(N)}(x_{1},\ldots,x_{2N}):=\left(\frac{2|\cos(4\pi/\kappa)|\Gamma(2-8/\kappa)}{\Gamma(1-4/\kappa)^{2}}\right)^{N}\int_{x_{1}}^{x_{2}}\cdots\int_{x_{2N-1}}^{x_{2N}}f(\boldsymbol{x};u_{1},\ldots,u_{N})\mathrm{d}u_{1}\cdots\mathrm{d}u_{N},

(2.6)

where $\mathrm{d}u_{q}$ is integrated from $x_{2q-1}$ to $x_{2q}$ for $1\leq q\leq N$ , and the integrand is

\displaystyle f(\boldsymbol{x};u_{1},\ldots,u_{N}):=\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{2/\kappa}\prod_{1\leq r<s\leq N}|u_{s}-u_{r}|^{8/\kappa}\prod_{1\leq j\leq 2N,1\leq q\leq N}|u_{q}-x_{j}|^{-4/\kappa}.

(2.7)

The function $\mathcal{G}^{(N)}$ is the Coulomb gas integral function with link pattern

\boldsymbol{\underline{\cap\cap}}=\{\{1,2\},\ldots,\{2N-1,2N\}\}\in\mathrm{LP}_{N},

see for instance [FLPW24, Definition 1.4]. We have the following two lemmas for $\mathcal{G}^{(N)}$ .

Lemma 2.4.

Fix $\kappa\in(4,8)$ . For all $(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ ,

\mathcal{G}^{(N)}(x_{1},\ldots,x_{2N})=\sum_{\alpha\in\mathrm{LP}_{N}}\mathcal{M}_{\nu}(\alpha,\boldsymbol{\underline{\cap\cap}})\mathcal{Z}_{\alpha}(x_{1},\ldots,x_{2N}),

(2.8)

where $\nu=2|\cos(4\pi/\kappa)|$ and $\mathcal{M}_{\nu}(\alpha,\boldsymbol{\underline{\cap\cap}})=\nu^{\ell(\alpha,\boldsymbol{\underline{\cap\cap}})}$ and $\ell(\alpha,\boldsymbol{\underline{\cap\cap}})$ denotes the number of loops in the meander formed from the two link patterns $\alpha$ and $\boldsymbol{\underline{\cap\cap}}$ .

Proof.

See [FPW24, Proposition 1.10] and [FLPW24, Proposition 1.11]. ∎

Lemma 2.5.

Fix $\kappa\in(4,8)$ . There exists constant $C\in(0,\infty)$ (depending only on $N$ and $\kappa$ ) such that for all $(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ ,

0<\mathcal{G}^{(N)}(x_{1},\ldots,x_{2N})\leq C\prod_{1\leq j<k\leq 2N}|x_{k}-x_{j}|^{\mu_{kj}},\quad\text{where }\mu_{kj}=\begin{cases}10/\kappa,&\text{if }|x_{k}-x_{j}|\geq 1,\\ -6/\kappa,&\text{if }|x_{k}-x_{j}|<1.\end{cases}

(2.9)

Proof.

The second term in the integrand (2.7) can be bounded from above: for $1\leq r<s\leq N$ ,

|u_{s}-u_{r}|\leq|x_{2s}-x_{2r-1}|.

The third term in the integrand (2.7) can be bounded from above: for $1\leq q\leq N$ ,

\prod_{1\leq j\leq 2N}|u_{q}-x_{j}|^{-4/\kappa}\leq\prod_{1\leq j<2q-1}|x_{2q-1}-x_{j}|^{-4/\kappa}\times|u_{q}-x_{2q-1}|^{-4/\kappa}|u_{q}-x_{2q}|^{-4/\kappa}\times\prod_{2q<j\leq 2N}|x_{2q}-x_{j}|^{-4/\kappa}.

Note that, using change of variables $v=\frac{u_{q}-x_{2q-1}}{x_{2q}-x_{2q-1}}$ ,

	$\displaystyle\int_{x_{2q-1}}^{x_{2q}}\|u_{q}-x_{2q-1}\|^{-4/\kappa}\|u_{q}-x_{2q}\|^{-4/\kappa}\mathrm{d}u_{q}=$	$\displaystyle(x_{2q}-x_{2q-1})^{1-8/\kappa}\int_{0}^{1}v^{-4/\kappa}(1-v)^{-4/\kappa}\mathrm{d}v$
	$\displaystyle=$	$\displaystyle(x_{2q}-x_{2q-1})^{1-8/\kappa}\frac{\Gamma(1-4/\kappa)^{2}}{\Gamma(2-8/\kappa)}.$

Plugging these three observations into (2.6), there exists constant $C\in(0,\infty)$ (depending only on $N$ and $\kappa$ ),

	$\displaystyle\mathcal{G}^{(N)}(x_{1},\ldots,x_{2N})\leq$	$\displaystyle C\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{2/\kappa}\times\prod_{1\leq r<s\leq N}\|x_{2s}-x_{2r-1}\|^{8/\kappa}$		(2.10)
		$\displaystyle\times\prod_{1\leq q\leq N}\prod_{1\leq j<2q-1}\|x_{2q-1}-x_{j}\|^{-4/\kappa}\times\|x_{2q}-x_{2q-1}\|^{1-8/\kappa}\times\prod_{2q<j\leq 2N}\|x_{2q}-x_{j}\|^{-4/\kappa}.$

In the right-hand side of (2.10),

•

the accumulated exponent for the term $|x_{2q}-x_{2q-1}|$ with $1\leq q\leq N$ is given by

$2/\kappa+1-8/\kappa=1-6/\kappa;$
•

the accumulated exponent for the term $|x_{2q}-x_{2r-1}|$ with $1\leq r<q\leq N$ is given by

$2/\kappa+8/\kappa=10/\kappa;$
•

the accumulated exponent for the term $|x_{2q+1}-x_{2q}|$ with $1\leq q\leq N-1$ is given by

$2/\kappa-4/\kappa=-2/\kappa;$
•

the accumulated exponent for the term $|x_{2j+1}-x_{2q}|$ with $1\leq q<j\leq N-1$ is given by

$2/\kappa-4/\kappa-4/\kappa=-6/\kappa;$
•

the accumulated exponent for the term $|x_{2r}-x_{2q}|$ with $1\leq q<r\leq N$ is given by

$2/\kappa-4/\kappa=-2/\kappa;$
•

the accumulated exponent for the term $|x_{2r-1}-x_{2q-1}|$ with $1\leq q<r\leq N$ is given by

$2/\kappa-4/\kappa=-2/\kappa.$

Combining these observations, we obtain (2.9) as desired. ∎

Proof of Lemma 2.3.

From (2.8), for any $\alpha\in\mathrm{LP}_{N}$ , we have $\mathcal{G}^{(N)}\geq\mathcal{M}_{\nu}(\alpha,\boldsymbol{\underline{\cap\cap}})\mathcal{Z}_{\alpha}$ . Combining with (2.9), we obtain (2.5) as desired. ∎

2.3 Chordal $N$ - $\mathrm{SLE}_{\kappa}$

The law of chordal $N$ - $\mathrm{SLE}$ can be described by the pure partition functions defined in Section 2.1. To this end, we first introduce usual parameterization. We fix $N\geq 1$ and let $\boldsymbol{x}=(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ . Let $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ be $2N$ -tuple of continuous non-self-crossing curves $\eta_{j}:[0,T_{j}]\to\overline{\mathbb{H}}$ such that $\eta_{j}(0)=x_{j}$ . For $j\in\{1,\ldots,2N\}$ , let $H_{t}^{j}$ be the unbounded connected component of $\mathbb{H}\setminus\eta_{j}([0,t])$ . Let $g_{t}^{j}:H_{t}^{j}\to\mathbb{H}$ be the unique conformal transformation with $\lim_{z\to\infty}|g_{t}^{j}(z)-z|=0$ . We say that $\eta_{j}=(\eta_{j}(t),0\leq t<T_{j})$ has $a$ -usual parameterization if for any $t<T_{j}$ ,

\displaystyle g_{t}^{j}(z)=z+\frac{at}{z}+o(1/|z|),\quad\text{ as }z\to\infty.

Under this parameterization, the half-plane capacity of $\eta_{j}[0,t]$ is $at$ and $g_{t}^{j}$ satisfies the following chordal Loewner equation

\displaystyle\partial_{t}{g}_{t}^{j}(z)=\frac{a}{g^{j}_{t}(z)-W_{t}^{j}},\quad\forall z\in H_{t}^{j},\quad\forall t<T_{j}.

We say that $(\eta_{j}(t),0\leq t<T_{j})$ is the chordal Loewner chain with $a$ -usual parameterization and with driving function $(W_{t}^{j},0\leq t<T_{j})$ .

Let $(\gamma_{1},\ldots,\gamma_{N})\sim\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ be the chordal $N$ - $\mathrm{SLE}_{\kappa}$ in polygon $(\mathbb{H};x_{1},\ldots,x_{2N})$ associated to link pattern $\alpha=\{\{a_{1},b_{1}\},\ldots,\{a_{N},b_{N}\}\}\in\mathrm{LP}_{N}$ . We view it as a $2N$ -tuple of continuous non-self-crossing curves $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ as described in Section 1.1: for $j\in\{1,\ldots,N\}$ , we define $\eta_{a_{j}}$ to be $\gamma_{j}$ and $\eta_{b_{j}}$ to be the time-reversal of $\gamma_{j}$ . In this way, $\eta_{j}$ is a continuous curve in $\mathbb{H}$ starting from $x_{j}$ for $j\in\{1,\ldots,2N\}$ . For $j\in\{1,\ldots,2N\}$ , if we parameterize $\eta_{j}$ with $a$ -usual parameterization, then its driving function $(W^{j}_{t},t\geq 0)$ satisfies (see [PW19, Proposition 4.10] for $\kappa\in(0,4]$ and [FLPW24, Theorem 1.21] for $\kappa\in(4,8)$ )

\begin{cases}\mathrm{d}W_{t}^{j}=\mathrm{d}B_{t}^{j}+(\partial_{j}\log\mathcal{Z}_{\alpha})(V^{1}_{t},\ldots,V^{j-1}_{t},W_{t}^{j},V^{j+1}_{t},\ldots,V^{2N}_{t})\mathrm{d}t,\quad W_{0}^{j}=x_{j};\\ \mathrm{d}V_{t}^{k}=\frac{a}{V_{t}^{k}-W_{t}^{j}}\mathrm{d}t,\quad V_{0}^{k}=x_{k},\quad k\in\{1,\ldots,j-1,j+1,\ldots,2N\};\end{cases}

(2.11)

where $(B^{j}_{t},t\geq 0)$ is a standard Brownian motion.

2.4 Dyson Brownian motion

In this section, we recall a classical result for Dyson Brownian motion. For each $v>0$ and $\boldsymbol{x}=(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ , define

f_{v}(\boldsymbol{x})=\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{v}.

Note that $\mathcal{Z}_{*}(\boldsymbol{x}):=\mathcal{Z}_{*}(x_{1},\ldots,x_{2N})$ in (1.4) is exactly $f_{a}(\boldsymbol{x})$ .

Lemma 2.6.

Fix $\kappa\in(0,8)$ and $a=2/\kappa$ . Denote by $p_{*}(t;\boldsymbol{x},\cdot)$ the transition density for Dyson Brownian motion (1.2) with parameter $\beta=8/\kappa$ . Then, for any $\boldsymbol{x},\boldsymbol{y}\in\mathfrak{X}_{2N}$ ,

p_{*}(t;\boldsymbol{x},\boldsymbol{y})=\mathcal{I}_{*}^{-1}\left(\frac{1}{\sqrt{t}}\right)^{B_{N}}f_{4a}(\boldsymbol{y})\exp\left(-\frac{|\boldsymbol{y}|^{2}}{2t}\right)\left(1+O\left(\frac{|\boldsymbol{x}|}{\sqrt{t}}\right)\right),

(2.12)

where

B_{N}=\frac{2N(8N-4+\kappa)}{\kappa},

and $\mathcal{I}_{*}\in(0,\infty)$ is a normalization constant

\mathcal{I}_{*}=\int_{\mathfrak{X}_{2N}}f_{4a}(\boldsymbol{x})e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}\mathrm{d}\boldsymbol{x}.

(2.13)

Proof.

The explicit formula of transition density $p_{*}(t;\boldsymbol{x},\boldsymbol{y})$ can be obtained from [Rös98, Equation (4.5)]:

p_{*}(t;\boldsymbol{x},\boldsymbol{y})=\mathcal{I}_{*}^{-1}\left(\frac{1}{\sqrt{t}}\right)^{B_{N}}f_{4a}(\boldsymbol{y})\exp\left(-\frac{|\boldsymbol{x}|^{2}+|\boldsymbol{y}|^{2}}{2t}\right)J_{2a}^{A}\left(\frac{\boldsymbol{x}}{\sqrt{t}},\frac{\boldsymbol{y}}{\sqrt{t}}\right),\quad\boldsymbol{x},\boldsymbol{y}\in\mathfrak{X}_{2N},

where $J_{2a}^{A}$ is multivariate Bessel function of type $A$ with multiplicity $k=2a$ and is analytic on $\mathbb{R}^{2N}\times\mathbb{R}^{2N}$ with $J_{2a}^{A}(\boldsymbol{0},\boldsymbol{y})=1$ (see [Rös98, Section 2] or [Voi19, Section 1]). Letting $|\boldsymbol{x}|/\sqrt{t}\to 0$ , we obtain (2.12) and complete the proof. ∎

The following lemma will be used in the proof of Proposition 1.3.

Lemma 2.7.

The constant

\mathcal{J}_{\alpha}=\int_{\mathfrak{X}_{2N}}f_{4a}(\boldsymbol{x})G_{\alpha}(\boldsymbol{x})^{-1}e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}\mathrm{d}\boldsymbol{x}

(2.14)

is finite, where $G_{\alpha}$ is the Green’s function in Definition 1.2.

Proof.

Note that

0<f_{4a}(\boldsymbol{x})G_{\alpha}(\boldsymbol{x})^{-1}e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}=f_{4a}(\boldsymbol{x})\mathcal{Z}_{\alpha}(\boldsymbol{x})\mathcal{Z}_{*}(\boldsymbol{x})^{-1}e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}=f_{3a}(\boldsymbol{x})\mathcal{Z}_{\alpha}(\boldsymbol{x})e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}.

From the power-law bound (2.4) and (2.5) and the relation (2.1), we know that $f_{3a}(\boldsymbol{x})\mathcal{Z}_{\alpha}(\boldsymbol{x})e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}$ is integrable on $\mathfrak{X}_{2N}$ . Thus $\mathcal{J}_{\alpha}$ is a finite constant. ∎

3 Proof of Theorem 1.1 and Proposition 1.3

3.1 Transition density and proof of Theorem 1.1

Fix $N\geq 1$ and $\boldsymbol{x}=(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ . To derive transition density $p_{\alpha}(t;\cdot,\cdot)$ for driving functions of chordal $N$ - $\mathrm{SLE}_{\kappa}$ , we derive Radon-Nikodym derivatives between three measures:

•

Recall that $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ denotes the law of chordal $N$ - $\mathrm{SLE}_{\kappa}$ in polygon $(\mathbb{H};x_{1},\ldots,x_{2N})$ associated to link pattern $\alpha\in\mathrm{LP}_{N}$ . We view it as a measure on $2N$ -tuples of curves $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ .
•

Denote by $\mathbb{P}_{*}^{(\boldsymbol{x})}$ the law of $2N$ -tuples of curves $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ in polygon $(\mathbb{H};x_{1},\ldots,x_{2N})$ driven by Dyson Brownian motion (1.2).
•

Denote by $\mathbb{P}$ the law on $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ where $(\eta_{1},\ldots,\eta_{2N})$ are independent chordal $\mathrm{SLE}_{\kappa}$ in $\mathbb{H}$ started from $(x_{1},\ldots,x_{2N})$ respectively.

We fix the same parameters as in (2.1). We parameterize these curves with $a$ -common parameterization and recall from Section 2.3 that $(W_{t}^{j},t\geq 0)$ is the driving function of $(\eta^{j}(t),t\geq 0)$ and from Section 1.1 that $\{(X_{t}^{j},t\geq 0)\}_{1\leq j\leq 2N}$ is driving functions of $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ as a chordal Loewner chain with $a$ -common parameterization. Under $a$ -common parameterization, recall that $(\mathcal{F}_{t},t\geq 0)$ is the filtration generated by $(\boldsymbol{\eta}(t),t\geq 0)$ and $T$ is the lifetime of driving functions $(\boldsymbol{X}_{t},t\geq 0)$ , and we define the following normalized conformal maps for $t<T$ :

•

$g_{t}^{j}$ is the conformal map from the unbounded connected component of $\mathbb{H}\setminus\eta_{j}([0,t])$ onto $\mathbb{H}$ with $\lim_{z\to\infty}|g_{t}^{j}(z)-z|=0$ ;
•

$g_{t}$ is the conformal map from the unbounded connected component of $\mathbb{H}\setminus\cup_{j}\eta_{j}([0,t])$ onto $\mathbb{H}$ with $g_{t}(z)=z+2aNt/z+o(1/|z|)$ as $z\to\infty$ ;
•

$g_{t,j}$ is the conformal map from the unbounded connected component of $\mathbb{H}\setminus\cup_{k\neq j}g_{t}^{j}(\eta_{k}([0,t]))$ onto $\mathbb{H}$ with $\lim_{z\to\infty}|g_{t,j}(z)-z|=0$ .

Then we have $g_{t}=g_{t,j}\circ g_{t}^{j}$ for $1\leq j\leq 2N$ . For a conformal map $g$ , we denote the Schwarzian derivative of $g$ by

Sg:=\left(\frac{g^{\prime\prime}}{g^{\prime}}\right)^{\prime}-\frac{1}{2}\left(\frac{g^{\prime\prime}}{g^{\prime}}\right)^{2}=\frac{g^{\prime\prime\prime}}{g^{\prime}}-\frac{3}{2}\left(\frac{g^{\prime\prime}}{g^{\prime}}\right)^{2}.

Lemma 3.1.

The Radon-Nikodym derivative between $\mathbb{P}_{*}^{(\boldsymbol{x})}$ and $\mathbb{P}$ when both measures are restricted to $\mathcal{F}_{t}$ and $\{T>t\}$ is given by

\frac{\mathrm{d}\mathbb{P}_{*}^{(\boldsymbol{x})}[\cdot|_{\mathcal{F}_{t}}]}{\mathrm{d}\mathbb{P}[\cdot|_{\mathcal{F}_{t}\cap\{T>t\}}]}=\frac{M^{*}_{t}}{M^{*}_{0}},\quad t>0,

where

M^{*}_{t}=\prod_{j=1}^{2N}g_{t,j}^{\prime}(W_{t}^{j})^{h}\exp\left(-\frac{ac}{12}\int_{0}^{t}Sg_{s,j}(W_{s}^{j})\mathrm{d}s\right)\times\mathcal{Z}_{*}(X^{1}_{t},\ldots,X^{2N}_{t}),

(3.1)

and $\mathcal{Z}_{*}$ is the partition function (1.4).

Lemma 3.2.

The Radon-Nikodym derivative between $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ and $\mathbb{P}$ when both measures are restricted to $\mathcal{F}_{t}$ and $\{T>t\}$ is given by

\frac{\mathrm{d}\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[\cdot|_{\mathcal{F}_{t}\cap\{T>t\}}]}{\mathrm{d}\mathbb{P}[\cdot|_{\mathcal{F}_{t}\cap\{T>t\}}]}=\frac{M^{\alpha}_{t}}{M^{\alpha}_{0}},\quad t>0,

where

M_{t}^{\alpha}=\prod_{j=1}^{2N}g_{t,j}^{\prime}(W_{t}^{j})^{h}\exp\left(-\frac{ac}{12}\int_{0}^{t}Sg_{s,j}(W_{s}^{j})\mathrm{d}s\right)\times\mathcal{Z}_{\alpha}(X^{1}_{t},\ldots,X^{2N}_{t}),

(3.2)

and $\mathcal{Z}_{\alpha}$ is the pure partition function in Section 2.3. Moreover, under $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ , the driving functions $(\boldsymbol{X}_{t},0\leq t<T)$ satisfies the SDE (1.1) where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions.

We will prove Lemmas 3.1 and 3.2 in Section 3.2 where we introduce local martingales with $2N$ time parameters. Assuming these lemmas, we are able to derive the transition density for driving functions of chordal $N$ - $\mathrm{SLE}_{\kappa}$ .

Proof of Theorem 1.1.

For each $t>0$ , we obtain from Lemmas 3.1 and 3.2 the Radon-Nikodym derivative of $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ against $\mathbb{P}_{*}^{(\boldsymbol{x})}$ when both measures are restricted to $\mathcal{F}_{t}$ and $\{T>t\}$ :

\frac{\mathrm{d}\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[\cdot|_{\mathcal{F}_{t}\cap\{T>t\}}]}{\mathrm{d}\mathbb{P}_{*}^{(\boldsymbol{x})}[\cdot|_{\mathcal{F}_{t}}]}=\frac{M^{\alpha}_{t}/M^{\alpha}_{0}}{M^{*}_{t}/M^{*}_{0}}=\frac{\mathcal{Z}_{\alpha}(\boldsymbol{X}_{t})/\mathcal{Z}_{\alpha}(\boldsymbol{x})}{\mathcal{Z}_{*}(\boldsymbol{X}_{t})/\mathcal{Z}_{*}(\boldsymbol{x})}=\frac{G_{\alpha}(\boldsymbol{x})}{G_{\alpha}(\boldsymbol{X}_{t})}.

Since $(\boldsymbol{X}_{t},t\geq 0)$ is a Markov process with transition density $p_{*}(t;\boldsymbol{x},\boldsymbol{y})$ under $\mathbb{P}_{*}^{(\boldsymbol{x})}$ , the Radon-Nikodym derivative gives the transition density $p_{\alpha}(t;\boldsymbol{x},\boldsymbol{y})$ as desired. ∎

3.2 $2N$ -time local martingales

We will prove Lemmas 3.1 and 3.2 in this section. To this end, we introduce local martingales with $2N$ time parameters. Consider $2N$ chordal curves $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ in the polygon $(\mathbb{H};x_{1},\ldots,x_{2N})$ . Suppose each $\eta_{j}$ is parameterized with $a$ -usual parameterization and we get $2N$ time parameters $\boldsymbol{t}=(t_{1},\ldots,t_{2N})\in\mathbb{R}_{+}^{2N}$ . We define the following normalized conformal maps:

•

$g_{t_{j}}^{j}$ is the conformal map from the unbounded connected component of $\mathbb{H}\setminus\eta_{j}([0,t_{j}])$ onto $\mathbb{H}$ with $g_{t_{j}}^{j}(z)=z+at_{j}/z+o(1/|z|)$ as $z\to\infty$ , $1\leq j\leq 2N$ .
•

$g_{\boldsymbol{t}}$ is the conformal map from the unbounded connected component of $\mathbb{H}\setminus\bigcup_{j=1}^{2N}\eta_{j}([0,t_{j}])$ onto $\mathbb{H}$ with $g_{\boldsymbol{t}}(z)=z+\aleph(\boldsymbol{t})/z+o(1/|z|)$ as $z\to\infty$ .
•

$g_{\boldsymbol{t},j}$ is the conformal map from the unbounded connecetd component of $\mathbb{H}\setminus\bigcup_{k\neq j}g_{t_{j}}^{j}(\eta_{k}([0,t_{k}]))$ onto $\mathbb{H}$ with $\lim_{z\to\infty}|g_{\boldsymbol{t},j}(z)-z|=0$ , $1\leq j\leq 2N$ .

Then $g_{\boldsymbol{t}}=g_{\boldsymbol{t},j}\circ g_{t_{j}}^{j}$ for $1\leq j\leq 2N$ . Denote by $(W^{j}_{t_{j}},t_{j}\geq 0)$ the driving function of $\eta_{j}$ as a chordal Loewner chain with $a$ -usual parameterization. Let $\boldsymbol{X}_{\boldsymbol{t}}=(X^{1}_{\boldsymbol{t}},\ldots,X^{2N}_{\boldsymbol{t}})$ with $X^{j}_{\boldsymbol{t}}=g_{\boldsymbol{t},j}(W_{t_{j}}^{j})$ , $1\leq j\leq 2N$ .

Lemma 3.3.

M^{*}_{\boldsymbol{t}}=\prod_{j=1}^{2N}g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{h}\exp\left(-\frac{ac}{12}\int_{0}^{t_{j}}Sg_{\boldsymbol{t},j}(W_{t_{j}}^{j})\mathrm{d}t_{j}\right)\times\mathcal{Z}_{*}(X^{1}_{\boldsymbol{t}},\ldots,X^{2N}_{\boldsymbol{t}})

(3.3)

is a $2N$ -time-parameter local martingale with respect to $\mathbb{P}$ .

Proof.

Note that $\{W^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions under $\mathbb{P}$ . Since $X_{\boldsymbol{t}}^{j}=g_{\boldsymbol{t},j}(W_{t_{j}}^{j})$ for $1\leq j\leq 2N$ , by Itô’s formula and a standard calculation, we have

\mathrm{d}X_{\boldsymbol{t}}^{j}=g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})\mathrm{d}W_{t_{j}}^{j}-hg_{\boldsymbol{t},j}^{\prime\prime}(W_{t_{j}}^{j})\mathrm{d}t_{j}+a\sum_{k\neq j}\frac{1}{X_{\boldsymbol{t}}^{j}-X_{\boldsymbol{t}}^{k}}g_{\boldsymbol{t},k}^{\prime}(W_{t_{k}}^{k})^{2}\mathrm{d}t_{k},\quad 1\leq j\leq 2N.

Note that $M_{\boldsymbol{t}}^{*}$ defined in (3.3) has a decomposition $M_{\boldsymbol{t}}^{*}=N_{\boldsymbol{t}}\times N_{\boldsymbol{t}}^{*}$ , where

N_{\boldsymbol{t}}=\prod_{j=1}^{2N}g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{h}\exp\left(-\frac{ac}{12}\int_{0}^{t_{j}}Sg_{\boldsymbol{t},j}(W_{t_{j}}^{j})\mathrm{d}t_{j}\right)\times\exp\left(ah\sum_{j=1}^{2N}\int_{0}^{t_{j}}\sum_{k\neq j}\frac{g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{2}}{(X_{\boldsymbol{t}}^{j}-X_{\boldsymbol{t}}^{k})^{2}}\mathrm{d}t_{j}\right),

(3.4)

and

N_{\boldsymbol{t}}^{*}=\mathcal{Z}_{*}(X^{1}_{\boldsymbol{t}},\ldots,X^{2N}_{\boldsymbol{t}})\times\exp\left(-ah\sum_{j=1}^{2N}\int_{0}^{t_{j}}\sum_{k\neq j}\frac{g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{2}}{(X_{\boldsymbol{t}}^{j}-X_{\boldsymbol{t}}^{k})^{2}}\mathrm{d}t_{j}\right).

It is proved by direct calculation that $N_{\boldsymbol{t}}$ is a local martingale with respect to $\mathbb{P}$ satisfying

\mathrm{d}N_{\boldsymbol{t}}=N_{\boldsymbol{t}}\sum_{j=1}^{2N}h\frac{g_{\boldsymbol{t},j}^{\prime\prime}(W_{t_{j}}^{j})}{g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})}\mathrm{d}W_{t_{j}}^{j}.

Denote by $\mathbb{Q}$ the probability measure obtained by tilting $\mathbb{P}$ by $N_{\boldsymbol{t}}$ . By Girsanov’s theorem, under $\mathbb{Q}$ , we have

\mathrm{d}X_{\boldsymbol{t}}^{j}=g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})\mathrm{d}B_{t_{j}}^{j}+a\sum_{k\neq j}\frac{1}{X^{j}_{\boldsymbol{t}}-X^{k}_{\boldsymbol{t}}}g_{\boldsymbol{t},k}^{\prime}(W_{t_{k}}^{k})^{2}\mathrm{d}t_{k},\quad 1\leq j\leq 2N,

where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions.

It remains to prove that $N_{\boldsymbol{t}}^{*}$ is a local martingale with respect to $\mathbb{Q}$ . By Itô’s formula, we have

\mathrm{d}N_{\boldsymbol{t}}^{*}=N_{\boldsymbol{t}}^{*}\sum_{j=1}^{2N}\sum_{k\neq j}\frac{a}{X_{\boldsymbol{t}}^{j}-X_{\boldsymbol{t}}^{k}}g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})\mathrm{d}B_{t_{j}}^{j}.

This completes the proof. ∎

Proof of Lemma 3.1.

Denote by $\hat{\mathbb{P}}$ the probability measure obtained by tilting $\mathbb{P}$ by $M_{\boldsymbol{t}}^{*}$ , then under $\hat{\mathbb{P}}$ , for $1\leq j\leq 2N$ ,

\mathrm{d}X_{\boldsymbol{t}}^{j}=g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})\mathrm{d}B_{t_{j}}^{j}+a\sum_{k\neq j}\frac{1}{X^{j}_{\boldsymbol{t}}-X^{k}_{\boldsymbol{t}}}g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{2}\mathrm{d}t_{j}+a\sum_{k\neq j}\frac{1}{X^{j}_{\boldsymbol{t}}-X^{k}_{\boldsymbol{t}}}g_{\boldsymbol{t},k}^{\prime}(W_{t_{k}}^{k})^{2}\mathrm{d}t_{k},

(3.5)

where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent Brownian motions under $\hat{\mathbb{P}}$ . Under $a$ -common parameterization, we have $\mathrm{d}t_{j}=g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{-2}\mathrm{d}t$ for $1\leq j\leq 2N$ (see [HL21, Lemma 3.2] for similar proof), then the SDE (3.5) becomes SDE (1.2), and the martingale (3.3) becomes (3.1). This implies $\hat{\mathbb{P}}=\mathbb{P}_{*}^{(\boldsymbol{x})}$ and completes the proof. ∎

Lemma 3.4.

M^{\alpha}_{\boldsymbol{t}}=\prod_{j=1}^{2N}g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{h}\exp\left(-\frac{ac}{12}\int_{0}^{t_{j}}Sg_{\boldsymbol{t},j}(W_{t_{j}}^{j})\mathrm{d}t_{j}\right)\times\mathcal{Z}_{\alpha}(X^{1}_{\boldsymbol{t}},\ldots,X^{2N}_{\boldsymbol{t}})

(3.6)

is a $2N$ -time-parameter local martingale with respect to $\mathbb{P}$ .

Proof.

Note that $M_{\boldsymbol{t}}^{*}$ defined in (3.6) has a decomposition $M_{\boldsymbol{t}}^{\alpha}=N_{\boldsymbol{t}}\times N_{\boldsymbol{t}}^{\alpha}$ , where $N_{\boldsymbol{t}}$ is defined in (3.4) and

N_{\boldsymbol{t}}^{\alpha}=\mathcal{Z}_{\alpha}(X^{1}_{\boldsymbol{t}},\ldots,X^{2N}_{\boldsymbol{t}})\times\exp\left(-ah\sum_{j=1}^{2N}\int_{0}^{t_{j}}\sum_{k\neq j}\frac{g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{2}}{(X_{\boldsymbol{t}}^{j}-X_{\boldsymbol{t}}^{k})^{2}}\mathrm{d}t_{j}\right).

Recall from the proof of Lemma 3.3 that $N_{\boldsymbol{t}}$ is a local martingale with respect to $\mathbb{P}$ , and $\mathbb{Q}$ is the probability measure obtained by tilting $\mathbb{P}$ by $N_{\boldsymbol{t}}$ . It remains to prove that $N_{\boldsymbol{t}}^{\alpha}$ is a local martingale with respect to $\mathbb{Q}$ . By Itô’s formula and the PDE (2.2) satisfied by $\mathcal{Z}_{\alpha}$ , we have

\mathrm{d}N_{\boldsymbol{t}}^{\alpha}=N_{\boldsymbol{t}}^{\alpha}\sum_{j=1}^{2N}(\partial_{j}\log\mathcal{Z}_{\alpha})(X_{\boldsymbol{t}}^{1},\ldots,X_{\boldsymbol{t}}^{2N})g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})\mathrm{d}B_{t_{j}}^{j}.

This completes the proof. ∎

Proof of Lemma 3.2.

Denote by $\hat{\mathbb{P}}$ the probability measure obtained by tilting $\mathbb{P}$ by $M^{\alpha}_{\boldsymbol{t}}$ in (3.6), then under $\hat{\mathbb{P}}$ , for $1\leq j\leq 2N$ ,

\mathrm{d}X^{j}_{\boldsymbol{t}}=g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})\mathrm{d}B_{t_{j}}^{j}+(\partial_{j}\log\mathcal{Z}_{\alpha})(X^{1}_{\boldsymbol{t}},\ldots,X^{2N}_{\boldsymbol{t}})g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{2}\mathrm{d}t_{j}+a\sum_{k\neq j}\frac{1}{X^{j}_{\boldsymbol{t}}-X^{k}_{\boldsymbol{t}}}g_{\boldsymbol{t},k}^{\prime}(W_{t_{k}}^{k})^{2}\mathrm{d}t_{k},

(3.7)

where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions under $\hat{\mathbb{P}}$ . Suppose that $\{\eta_{j}([0,T_{j}])\}_{1\leq j\leq 2N}$ are restrictions of $\{\eta_{j}\}_{1\leq j\leq 2N}$ such that, for each $1\leq j\leq 2N$ , the curve $\eta_{j}([0,T_{j}])$ does not disconnect any $\eta_{k}([0,T_{k}])$ with $k\neq j$ from $\infty$ . We will argue that $\hat{\mathbb{P}}$ is the same as $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ when restricted to $\{\eta_{j}([0,T_{j}])\}_{1\leq j\leq 2N}$ .

On the one hand, by the domain Markov property of chordal $N$ - $\mathrm{SLE}_{\kappa}$ , $(\eta_{1},\ldots,\eta_{2N})$ commute with each other in the following sense. Let $\tau_{j}$ be any stopping time before $T_{j}$ for $1\leq j\leq 2N$ . For each $\eta_{j}$ , conditionally on $\bigcup_{k\neq j}\eta_{k}([0,\tau_{k}])$ , let $g$ be the conformal map from the unbounded connected component of $\mathbb{H}\setminus\bigcup_{k\neq j}\eta_{k}([0,\tau_{k}])$ onto $\mathbb{H}$ with $\lim_{z\to\infty}|g(z)-z|=0$ , then the $g$ -image of $\eta_{j}$ is a chordal Loewner chain in $\mathbb{H}$ started from $g(x_{j})$ with $a$ -usual parameterization and with driving function $(W^{j}_{t},t\geq 0)$ satisfying SDE (2.11) up to a time change.

On the other hand, we know from (3.7) that under the measure $\hat{\mathbb{P}}$ and conditionally on $\bigcup_{k\neq j}\eta_{k}([0,\tau_{k}])$ , $(g(\eta_{j}(t_{j})),t_{j}\geq 0)$ is a chordal Loewner chain in $\mathbb{H}$ started from $g(x_{j})$ and driven by $(X_{t_{j}}^{j},t_{j}\geq 0)$ satisfying the SDE

\mathrm{d}X_{t_{j}}^{j}=g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})\mathrm{d}B_{t_{j}}^{j}+(\partial_{j}\log\mathcal{Z}_{\alpha})(X^{1}_{\boldsymbol{t}},\ldots,X^{2N}_{\boldsymbol{t}})g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{2}\mathrm{d}t_{j},

where $\boldsymbol{t}=(\tau_{1},\ldots,\tau_{j-1},t_{j},\tau_{j+1},\ldots,\tau_{2N})$ , $t_{j}\geq 0$ . This is the same as SDE (2.11) after a time change, which implies that the chordal Loewner curves $(\eta_{1},\ldots,\eta_{2N})$ under $\hat{\mathbb{P}}$ locally commute with each other in the sense of the last paragraph. Therefore, the joint law $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ of chordal $N$ - $\mathrm{SLE}_{\kappa}$ $(\eta_{1},\ldots,\eta_{2N})$ coincides with $\hat{\mathbb{P}}$ when restricted to $\{\eta_{j}([0,T_{j}])\}_{1\leq j\leq 2N}$ .

Finally, we parameterize these $2N$ -tuple of curves with $a$ -common parameterization by making the time change $\mathrm{d}t_{j}=g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{-2}\mathrm{d}t$ , then the $2N$ -time-parameter local martingale (3.6) becomes $M_{t}^{\alpha}$ in (3.2), and the SDE (3.7) becomes SDE (1.1). Note also that, on the event $\{T>t\}$ , the collection $\{\eta_{j}([0,t])\}_{1\leq j\leq 2N}$ has the property that the curve $\eta_{j}([0,t])$ does not disconnect any $\eta_{k}([0,t])$ with $k\neq j$ from $\infty$ . This completes the proof. ∎

3.3 Proof of Proposition 1.3

Recall that $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ denotes the law of chordal $N$ - $\mathrm{SLE}_{\kappa}$ in polygon $(\mathbb{H};x_{1},\ldots,x_{2N})$ associated to link pattern $\alpha\in\mathrm{LP}_{N}$ . We view it as a $2N$ -tuple of continuous non-self-crossing curves $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ as described in Section 1.1: for $j\in\{1,\ldots,N\}$ , we define $\eta_{a_{j}}$ to be $\gamma_{j}$ and $\eta_{b_{j}}$ to be the time-reversal of $\gamma_{j}$ . In this way, $\eta_{j}$ is a continuous non-self-crossing curve in $\mathbb{H}$ starting from $x_{j}$ for $j\in\{1,\ldots,2N\}$ . We parameterize $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ with $a$ -common parameterization. We denote by $(\mathcal{F}_{t},t\geq 0)$ the filtration generated by $(\boldsymbol{\eta}(t),t\geq 0)$ and by $T$ the lifetime of its driving function $(\boldsymbol{X}_{t},t\geq 0)$ . Under $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ , the lifetime $T$ is finite almost surely. We will prove Proposition 1.3 in this section.

Proof of (1.5).

For each $\boldsymbol{x}=(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ , we have

$\displaystyle\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[T>t]$	$\displaystyle=\int_{\mathfrak{X}_{2N}}p_{\alpha}(t;\boldsymbol{x},\boldsymbol{y})\mathrm{d}\boldsymbol{y}$
	$\displaystyle=\int_{\mathfrak{X}_{2N}}p_{*}(t;\boldsymbol{x},\boldsymbol{y})\frac{G_{\alpha}(\boldsymbol{x})}{G_{\alpha}(\boldsymbol{y})}\mathrm{d}\boldsymbol{y}$	(due to (1.3))
	$\displaystyle=\int_{\mathfrak{X}_{2N}}\mathcal{I}_{*}^{-1}\left(\frac{1}{\sqrt{t}}\right)^{B_{N}}f_{4a}(\boldsymbol{y})\exp\left(-\frac{\|\boldsymbol{y}\|^{2}}{2t}\right)\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right)\frac{G_{\alpha}(\boldsymbol{x})}{G_{\alpha}(\boldsymbol{y})}\mathrm{d}\boldsymbol{y}$	(due to (2.12))
	$\displaystyle=\mathcal{I}_{*}^{-1}G_{\alpha}(\boldsymbol{x})\left(\frac{1}{\sqrt{t}}\right)^{B_{N}}\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right)\times\int_{\mathfrak{X}_{2N}}f_{4a}(\boldsymbol{y})G_{\alpha}(\boldsymbol{y})^{-1}\exp\left(-\frac{\|\boldsymbol{y}\|^{2}}{2t}\right)\mathrm{d}\boldsymbol{y}$
	$\displaystyle=\mathcal{I}_{*}^{-1}G_{\alpha}(\boldsymbol{x})\left(\frac{1}{\sqrt{t}}\right)^{B_{N}}\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right)\times\mathcal{J}_{\alpha}\times\left(\frac{1}{\sqrt{t}}\right)^{-B_{N}^{\prime}}$

where $\mathcal{J}_{\alpha}$ is the constant in (2.14) and the exponent $B_{N}^{\prime}$ is obtained by change of variable in integration:

B_{N}^{\prime}=\frac{N(12N-12+3\kappa)}{\kappa}.

Note that

B_{N}-B_{N}^{\prime}=\frac{2N(8N-4+\kappa)}{\kappa}-\frac{N(12N-12+3\kappa)}{\kappa}=\frac{N(4N+4-\kappa)}{\kappa}=A_{2N}^{+}.

This completes the proof. ∎

Using the estimates in (1.5), we are able to show that the law $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ conditional on the event $\{T>t\}$ will converge to $\mathbb{P}_{*}^{(\boldsymbol{x})}$ . This implies the convergence of driving functions.

Corollary 3.5.

Consider the following two measures:

•

For $t>0$ , denote by $\mathbb{P}_{\alpha\,|\,t}^{(\boldsymbol{x})}$ the measure $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ conditional on the event $\{T>t\}$ .
•

Recall that $\mathbb{P}_{*}^{(\boldsymbol{x})}$ denotes the law of $2N$ -tuples of curves $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ in polygon $(\mathbb{H};x_{1},\ldots,x_{2N})$ driven by Dyson Brownian motion (1.2).

Then for any $s>0$ , when both measures $\mathbb{P}_{\alpha\,|\,t}^{(\boldsymbol{x})}$ and $\mathbb{P}_{*}^{(\boldsymbol{x})}$ are restricted to $\mathcal{F}_{s}$ , the total variation distance between the two measures goes to zero as $t\to\infty$ :

\lim_{t\to\infty}\mathrm{dist}_{\mathrm{TV}}\left(\mathbb{P}_{\alpha\,|\,t}^{(\boldsymbol{x})}[\cdot\,|\,_{\mathcal{F}_{s}}],\mathbb{P}_{*}^{(\boldsymbol{x})}[\cdot\,|\,_{\mathcal{F}_{s}}]\right)=0.

Proof.

For any $t>s>0$ , from the domain Markov property of chordal $N$ - $\mathrm{SLE}_{\kappa}$ and Proposition 1.3, we have

\frac{\mathrm{d}\mathbb{P}_{\alpha\,|\,t}^{(\boldsymbol{x})}\left[\cdot|_{\mathcal{F}_{s}}\right]}{\mathrm{d}\mathbb{P}_{\alpha}^{(\boldsymbol{x})}\left[\cdot|_{\mathcal{F}_{s}\cap\{T>s\}}\right]}=\frac{\mathbb{P}_{\alpha}^{(\boldsymbol{X}_{s})}[T>t-s]}{\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[T>t]}=\left(\frac{t-s}{t}\right)^{-A_{2N}^{+}/2}\frac{G_{\alpha}(X^{1}_{s},\ldots,X^{2N}_{s})}{G_{\alpha}(x_{1},\ldots,x_{2N})}\left(1+O\left(\frac{|\boldsymbol{X}_{s}|}{\sqrt{t}}\right)\right).

Combining this with

\frac{\mathrm{d}\mathbb{P}_{\alpha}^{(\boldsymbol{x})}\left[\cdot|_{\mathcal{F}_{s}\cap\{T>s\}}\right]}{\mathrm{d}\mathbb{P}_{*}^{(\boldsymbol{x})}\left[\cdot|_{\mathcal{F}_{s}}\right]}=\frac{G_{\alpha}(x_{1},\ldots,x_{2N})}{G_{\alpha}(X^{1}_{s},\ldots,X^{2N}_{s})},

we conclude that

\frac{\mathrm{d}\mathbb{P}_{\alpha\,|\,t}^{(\boldsymbol{x})}\left[\cdot|_{\mathcal{F}_{s}}\right]}{\mathrm{d}\mathbb{P}_{*}^{(\boldsymbol{x})}\left[\cdot|_{\mathcal{F}_{s}}\right]}=\left(1-\frac{s}{t}\right)^{-A_{2N}^{+}/2}\left(1+O\left(\frac{|\boldsymbol{X}_{s}|}{\sqrt{t}}\right)\right),

which implies that

\lim_{t\to\infty}\mathbb{E}_{*}^{(\boldsymbol{x})}\left[\left(1-\frac{\mathrm{d}\mathbb{P}_{\alpha\,|\,t}^{(\boldsymbol{x})}\left[\cdot|_{\mathcal{F}_{s}}\right]}{\mathrm{d}\mathbb{P}_{*}^{(\boldsymbol{x})}\left[\cdot|_{\mathcal{F}_{s}}\right]}\right)^{+}\right]=0.

This gives the convergence in total-variation distance and completes the proof. ∎

Proof of Proposition 1.3.

The estimate in (1.5) is already proved. The convergence (1.7) follows from Corollary 3.5 because $\mu_{\alpha\,|\,t}^{(\boldsymbol{x})}$ is the induced measure on the driving function $(\boldsymbol{X}_{t},t\geq 0)$ under $\mathbb{P}_{\alpha\,|\,t}^{(\boldsymbol{x})}$ and $\mu_{*}^{(\boldsymbol{x})}$ is the induced measure on the driving function under $\mathbb{P}_{*}^{(\boldsymbol{x})}$ . ∎

4 Gaussian free field level lines and Dyson Brownian motion $\beta=2$

We will give a very brief summary on Gaussian free field (GFF) and its level lines, more details can be found in [She07, SS13, WW17]. Suppose $D\subset\mathbb{C}$ is a domain. The Sobolev space $H_{0}^{1}(D)$ is the Hilbert space closure of $C_{0}^{\infty}(D)$ with respect to the Dirichlet inner product $(f,g)_{\nabla}=\frac{1}{2\pi}\int\nabla f\cdot\nabla g\;\mathrm{d}x$ . The zero-boundary GFF $\Gamma$ on $D$ is given by $\Gamma=\sum_{n}X_{n}f_{n}$ where $\{X_{n}\}$ is a sequence of i.i.d. normal Gaussian random variables and $\{f_{n}\}$ is an orthonormal basis for $H_{0}^{1}(D)$ . Such sum does not converge in $H_{0}^{1}(D)$ , but it does converge in an appropriate space of distributions. The GFF with boundary data $\Gamma_{0}$ is the sum of the zero-boundary GFF on $D$ and the function in $D$ , still denoted by $\Gamma_{0}$ , which is harmonic in $D$ and is equal to $\Gamma_{0}$ on $\partial D$ . It is shown in [SS13] that variants of $\mathrm{SLE}_{4}$ process can be coupled with GFF as “level lines”. In this section, we focus on the coupling between chordal $N$ - $\mathrm{SLE}_{4}$ and GFF.

We fix $\lambda=\pi/2$ and fix $\boldsymbol{x}=(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ . We consider GFF $\Gamma$ in $\mathbb{H}$ with the following boundary data:

\lambda\text{ on }(x_{2j-1},x_{2j})\text{ for }1\leq j\leq N,\quad-\lambda\text{ on }(x_{2j},x_{2j+1})\text{ for }0\leq j\leq N,

(4.1)

with the convention that $x_{0}=-\infty$ and $x_{2N+1}=\infty$ . For $1\leq j\leq N$ , let $\eta_{2j-1}$ be the level line of $\Gamma$ starting from $x_{2j-1}$ and let $\eta_{2j}$ be the level line of $-\Gamma$ starting from $x_{2j}$ . We denote the law of $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ by $\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}$ . We view it as a chordal Loewner chain with $1/2$ -common parameterization and with driving functions $(\boldsymbol{X}_{t}=(X_{t}^{1},\ldots,X_{t}^{2N}),t\geq 0)$ . Denote by $(\mathcal{F}_{t},t\geq 0)$ the filtration generated by $(\boldsymbol{\eta}(t),t\geq 0)$ , and the lifetime $T$ as the first time $t$ that $X_{t}^{j}=X_{t}^{k}$ for some $j\neq k$ . We will prove in Lemma 4.2 that, under $\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}$ , the driving functions $(\boldsymbol{X}_{t},0\leq t<T)$ satisfies the SDE

\mathrm{d}X_{t}^{j}=\mathrm{d}B_{t}^{j}+\sum_{k\neq j}\frac{\frac{1}{2}\left((-1)^{j-k}+1\right)}{X_{t}^{j}-X_{t}^{k}}\mathrm{d}t,\quad 1\leq j\leq 2N,

(4.2)

where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions. The main conclusion of this section is the following.

Proposition 4.1.

Consider level lines $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ of the GFF with boundary data (4.1).

•

Denote by $q(t;\cdot,\cdot)$ the transition density for the solution to (4.2).
•

Denote by $q_{*}(t;\cdot,\cdot)$ the transition density for Dyson Brownian motion with parameter $\beta=2$ :

$\mathrm{d}X_{t}^{j}=\mathrm{d}B_{t}^{j}+\sum_{k\neq j}\frac{1}{X_{t}^{j}-X_{t}^{k}}\mathrm{d}t,\quad 1\leq j\leq 2N,$

where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions.

Then we have

q(t;\boldsymbol{x},\boldsymbol{y})=q_{*}(t;\boldsymbol{x},\boldsymbol{y})\frac{G(\boldsymbol{x})}{G(\boldsymbol{y})},\quad\text{for all }t\geq 0,\text{ and }\boldsymbol{x},\boldsymbol{y}\in\mathfrak{X}_{2N},

(4.3)

where

G(x_{1},\ldots,x_{2N})=\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{\frac{1}{2}(1-(-1)^{j-k})}.

(4.4)

Moreover,

\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}[T>t]=\mathcal{I}_{N}^{-1}\mathcal{J}_{N}G(\boldsymbol{x})t^{-N^{2}/2}\left(1+O\left(\frac{|\boldsymbol{x}|}{\sqrt{t}}\right)\right),\quad\text{as }t\to\infty,

(4.5)

where $\mathcal{I}_{N}$ and $\mathcal{J}_{N}$ are constants:

\mathcal{I}_{N}=(2\pi)^{N}\prod_{n=1}^{2N-1}n!,\qquad\mathcal{J}_{N}=\int_{\mathfrak{X}_{2N}}\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{\frac{1}{2}(3+(-1)^{j-k})}\times e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}\mathrm{d}\boldsymbol{x}.

The law of the solution to (4.2) conditional on $\{T>t\}$ converges to the law of Dyson Brownian motion with parameter $\beta=2$ in total variation distance as in (1.7).

The rest of this section is devoted to proving Proposition 4.1. We denote the law of $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ by $\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}$ . For $j\in\{1,\ldots,2N\}$ , if we parameterize $\eta_{j}$ with $1/2$ -usual parameterization, then its driving function $(W_{t}^{j},t\geq 0)$ satisfies (see e.g. [MS16, Theorem 1.1])

\begin{cases}\mathrm{d}W_{t}^{j}=\mathrm{d}B_{t}^{j}+(\partial_{j}\log\mathcal{Z}_{\mathrm{GFF}}^{(N)})(V^{1}_{t},\ldots,V^{j-1}_{t},W_{t}^{j},V^{j+1}_{t},\ldots,V^{2N}_{t})\mathrm{d}t,\quad W_{0}^{j}=x_{j};\\ \mathrm{d}V_{t}^{k}=\frac{1/2}{V_{t}^{k}-W_{t}^{j}}\mathrm{d}t,\quad V_{0}^{k}=x_{k},\quad k\in\{1,\ldots,j-1,j+1,\ldots,2N\};\end{cases}

where $(B^{j}_{t},t\geq 0)$ is a standard Brownian motion and $\mathcal{Z}_{\mathrm{GFF}}^{(N)}$ is given by

\mathcal{Z}_{\mathrm{GFF}}^{(N)}(x_{1},\ldots,x_{2N})=\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{\frac{1}{2}(-1)^{j-k}}.

(4.6)

If we parameterize $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ with $1/2$ -common parameterization and use the same notations as in Section 3.1 for $\kappa=4$ , we will obtain the following lemma.

Lemma 4.2.

Recall that $\mathbb{P}$ denotes the law on $\boldsymbol{\eta}=(\eta_{1},\ldots,\eta_{2N})$ where $(\eta_{1},\ldots,\eta_{2N})$ are independent chordal $\mathrm{SLE}_{4}$ in $\mathbb{H}$ started from $(x_{1},\ldots,x_{2N})$ respectively. The Radon-Nikodym derivative between $\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}$ and $\mathbb{P}$ when both measures are restricted to $\mathcal{F}_{t}$ and $\{T>t\}$ is given by

\frac{\mathrm{d}\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}[\cdot|_{\mathcal{F}_{t}\cap\{T>t\}}]}{\mathrm{d}\mathbb{P}[\cdot|_{\mathcal{F}_{t}\cap\{T>t\}}]}=\frac{M_{t}}{M_{0}},\quad t>0,

where

M_{t}=\prod_{j=1}^{2N}g_{t,j}^{\prime}(W_{t}^{j})^{1/4}\exp\left(-\frac{1}{24}\int_{0}^{t}Sg_{s,j}(W_{s}^{j})\mathrm{d}s\right)\times\mathcal{Z}_{\mathrm{GFF}}^{(N)}(X^{1}_{t},\ldots,X^{2N}_{t}),

and $\mathcal{Z}_{\mathrm{GFF}}^{(N)}$ is the partition function (4.6). Moreover, under $\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}$ , the driving functions $(\boldsymbol{X}_{t},0\leq t<T)$ satisfies the SDE (4.2) where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions.

Proof.

We fix $\kappa=4$ . From [PW19, Theorem 1.4], we know that

\mathcal{Z}_{\mathrm{GFF}}^{(N)}(\boldsymbol{x})=\sum_{\alpha\in\mathrm{LP}_{N}}\mathcal{Z}_{\alpha}(\boldsymbol{x}),

(4.7)

and that

\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}=\sum_{\alpha}\frac{\mathcal{Z}_{\alpha}(\boldsymbol{x})}{\mathcal{Z}_{\mathrm{GFF}}^{(N)}(\boldsymbol{x})}\mathbb{P}_{\alpha}^{(\boldsymbol{x})}.

(4.8)

Therefore,

$\displaystyle\frac{\mathrm{d}\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}[\cdot\,\|\,_{\mathcal{F}_{t}\cap\{T>t\}}]}{\mathrm{d}\mathbb{P}[\cdot\,\|\,_{\mathcal{F}_{t}\cap\{T>t\}}]}=$	$\displaystyle\sum_{\alpha\in\mathrm{LP}_{N}}\frac{\mathcal{Z}_{\alpha}(\boldsymbol{x})}{\mathcal{Z}_{\mathrm{GFF}}^{(N)}(\boldsymbol{x})}\times\frac{\mathrm{d}\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[\cdot\,\|\,_{\mathcal{F}_{t}\cap\{T>t\}}]}{\mathrm{d}\mathbb{P}[\cdot\,\|\,_{\mathcal{F}_{t}\cap\{T>t\}}]}$	(due to (4.8))
$\displaystyle=$	$\displaystyle\sum_{\alpha\in\mathrm{LP}_{N}}\frac{\mathcal{Z}_{\alpha}(\boldsymbol{x})}{\mathcal{Z}_{\mathrm{GFF}}^{(N)}(\boldsymbol{x})}\times\frac{M_{t}^{\alpha}}{M_{0}^{\alpha}}$	(due to Lemma 3.2)
$\displaystyle=$	$\displaystyle\frac{M_{t}}{M_{0}}.$	(due to (4.7))

Note that $\mathcal{Z}_{\mathrm{GFF}}^{(N)}$ is a combination of $\mathcal{Z}_{\alpha}$ as in (4.7) and hence it also satisfies the PDE (2.2), and that under $\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ , the driving functions $(\boldsymbol{X}_{t},0\leq t<T)$ satisfies the SDE (1.1) with $\kappa=4$ . Then similar calculation using Itô’s formula implies that under $\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}$ , $(\boldsymbol{X}_{t},0\leq t<T)$ satisfies the SDE

\mathrm{d}X_{t}^{j}=\mathrm{d}B_{t}^{j}+(\partial_{j}\log\mathcal{Z}_{\mathrm{GFF}}^{(N)})(X_{t}^{1},\ldots,X_{t}^{2N})\mathrm{d}t+\sum_{k\neq j}\frac{1/2}{X_{t}^{j}-X_{t}^{k}}\mathrm{d}t,\quad 1\leq j\leq 2N,

where $\{B^{j}\}_{1\leq j\leq 2N}$ are independent standard Brownian motions. From the explicit expression (4.6), this is the same as SDE (4.2) and we complete the proof. ∎

Proof of Proposition 4.1.

We fix $\kappa=4$ . For each $t>0$ , we obtain from Lemma 3.1 with $\kappa=4$ and Lemma 4.2, the Radon-Nikodym derivative of $\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}$ against $\mathbb{P}_{*}^{(\boldsymbol{x})}$ when both measures are restricted to $\mathcal{F}_{t}$ and $\{T>t\}$ :

\frac{\mathrm{d}\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}[\cdot|_{\mathcal{F}_{t}\cap\{T>t\}}]}{\mathrm{d}\mathbb{P}_{*}^{(\boldsymbol{x})}[\cdot|_{\mathcal{F}_{t}}]}=\frac{M_{t}/M_{0}}{M^{*}_{t}/M^{*}_{0}}=\frac{\mathcal{Z}_{\mathrm{GFF}}^{(N)}(\boldsymbol{X}_{t})/\mathcal{Z}_{\mathrm{GFF}}^{(N)}(\boldsymbol{x})}{\mathcal{Z}_{*}(\boldsymbol{X}_{t})/\mathcal{Z}_{*}(\boldsymbol{x})}=\frac{G(\boldsymbol{x})}{G(\boldsymbol{X}_{t})},

where

G(\boldsymbol{x})=\mathcal{Z}_{*}(\boldsymbol{x})/\mathcal{Z}_{\mathrm{GFF}}^{(N)}(\boldsymbol{x})=\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{\frac{1}{2}(1-(-1)^{j-k})}.

Since $(\boldsymbol{X}_{t},t\geq 0)$ is a Markov process with transition density $q_{*}(t;\boldsymbol{x},\boldsymbol{y})$ under $\mathbb{P}_{*}^{(\boldsymbol{x})}$ , the Radon-Nikodym derivative gives the transition density $q(t;\boldsymbol{x},\boldsymbol{y})$ as desired in (4.3).

Lemma 2.6 gives the asymptotic of transition density for Dyson Brownian motion with parameter $\beta=2$ :

\displaystyle q_{*}(t;\boldsymbol{x},\boldsymbol{y})=\mathcal{I}_{N}^{-1}t^{-2N^{2}}f_{2}(\boldsymbol{y})\exp\left(-\frac{|\boldsymbol{y}|^{2}}{2t}\right)\left(1+O\left(\frac{|\boldsymbol{x}|}{\sqrt{t}}\right)\right),\quad\text{as }t\to\infty,

where (see [Meh04, Eq. (17.6.7)])

\displaystyle\mathcal{I}_{N}=\int_{\mathfrak{X}_{2N}}f_{2}(\boldsymbol{x})e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}\mathrm{d}\boldsymbol{x}=\frac{1}{(2N)!}\int_{\mathbb{R}^{2N}}|f_{2}(\boldsymbol{x})|e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}\mathrm{d}\boldsymbol{x}=(2\pi)^{N}\prod_{n=1}^{2N-1}n!.

Then we obtain

	$\displaystyle\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}[T>t]$	$\displaystyle=\int_{\mathfrak{X}_{2N}}q(t;\boldsymbol{x},\boldsymbol{y})\mathrm{d}\boldsymbol{y}=\int_{\mathfrak{X}_{2N}}q_{*}(t;\boldsymbol{x},\boldsymbol{y})\frac{G(\boldsymbol{x})}{G(\boldsymbol{y})}\mathrm{d}\boldsymbol{y}$
		$\displaystyle=\mathcal{I}_{N}^{-1}G(\boldsymbol{x})t^{-2N^{2}}\int_{\mathfrak{X}_{2N}}\frac{f_{2}(\boldsymbol{y})}{G(\boldsymbol{y})}\exp\left(-\frac{\|\boldsymbol{y}\|^{2}}{2t}\right)\mathrm{d}\boldsymbol{y}\times\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right)$
		$\displaystyle=\mathcal{I}_{N}^{-1}G(\boldsymbol{x})t^{-2N^{2}}\times\mathcal{J}_{N}t^{3N^{2}/2}\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right)$
		$\displaystyle=\mathcal{I}_{N}^{-1}\mathcal{J}_{N}G(\boldsymbol{x})t^{-N^{2}/2}\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right),\quad\text{as }t\to\infty,$

where

\displaystyle\mathcal{J}_{N}=\int_{\mathfrak{X}_{2N}}\frac{f_{2}(\boldsymbol{x})}{G(\boldsymbol{x})}e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}\mathrm{d}\boldsymbol{x}=\int_{\mathfrak{X}_{2N}}\prod_{1\leq j<k\leq 2N}(x_{k}-x_{j})^{\frac{1}{2}(3+(-1)^{j-k})}e^{-\frac{1}{2}|\boldsymbol{x}|^{2}}\mathrm{d}\boldsymbol{x}.

This completes the proof of (4.5).

The same analysis in the proof of Corollary 3.5 implies the convergence as $t\to\infty$ of $\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}$ conditional on $\{T>t\}$ to $\mathbb{P}_{*}^{(\boldsymbol{x})}$ for $\kappa=4$ . As their induced measures on driving function, the law of the solution to (4.2) conditional on $\{T>t\}$ converges to the law of Dyson Brownian motion with parameter $\beta=2$ . ∎

5 Proof of Theorem 1.4

Fix $\kappa\in(0,8)$ , $\boldsymbol{x}=(x_{1},\ldots,x_{2N})\in\mathfrak{X}_{2N}$ , $\alpha\in\mathrm{LP}_{N}$ and suppose $(\gamma_{1},\ldots,\gamma_{N})\sim\mathbb{P}_{\alpha}^{(\boldsymbol{x})}$ is a chordal $N$ - $\mathrm{SLE}_{\kappa}$ in $(\mathbb{H};x_{1},\ldots,x_{2N})$ associated to link pattern $\alpha$ .

Proof of Theorem 1.4.

First, we regard $(\gamma_{1},\ldots,\gamma_{N})$ as a $2N$ -tuple of curves $(\eta_{1},\ldots,\eta_{2N})$ as described in Section 1.1 and derive the relation between different time parameterizations of the curves. Suppose that for each $1\leq j\leq 2N$ , $\eta_{j}$ is parameterized with $a$ -usual parameterization. Then we get a $2N$ -time parameter $\boldsymbol{t}=(t_{1},\ldots,t_{2N})\in\mathbb{R}_{+}^{2N}$ . Recall the normalized conformal maps $g_{t_{j}}^{j}$ , $g_{\boldsymbol{t}}$ , $g_{\boldsymbol{t},j}$ in Section 3.2. A standard calculation gives that

\mathrm{d}\aleph(\boldsymbol{t})=a\sum_{j=1}^{2N}g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{2}\mathrm{d}t_{j}.

For each $1\leq j\leq 2N$ , we have $g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})\in(0,1]$ , and for $\boldsymbol{t}=(0,\ldots,0,t_{j},0,\ldots,0)\in\mathbb{R}_{+}^{2N}$ , we have $\aleph(\boldsymbol{t})=at_{j}$ . Therefore, for $\boldsymbol{t}=(t_{1},\ldots,t_{2N})\in\mathbb{R}_{+}^{2N}$ ,

a\max_{1\leq j\leq 2N}t_{j}\leq\aleph(\boldsymbol{t})\leq a(t_{1}+\cdots+t_{2N}).

There exists a time change $\boldsymbol{t}=\boldsymbol{t}(t)=(t_{1}(t),\ldots,t_{2N}(t))$ with $t_{j}^{\prime}(t)=g_{\boldsymbol{t},j}^{\prime}(W_{t_{j}}^{j})^{-2}$ for $1\leq j\leq 2N$ , such that $(\boldsymbol{\eta}(t)=(\eta_{1}(t_{1}(t)),\ldots,\eta_{2N}(t_{2N}(t))),0\leq t<T)$ is parameterized with $a$ -common parameterization. In this case, we have $\aleph(t)=2aNt$ , which gives the relation

\max_{1\leq j\leq 2N}t_{j}\leq 2Nt\leq t_{1}+\cdots+t_{2N},

and from the time change, we have $t_{j}(t)\geq t$ for $1\leq j\leq 2N$ . We may conclude the relation as

t\leq t_{j}(t)\leq 2Nt,\quad\text{for all }1\leq j\leq 2N.

(5.1)

Second, we compare the events $\{\gamma_{j}\cap\partial B(0,R)\neq\emptyset,1\leq j\leq N\}$ and $\{T>t\}$ for some proper $t>0$ . Precisely, we will prove the relation

\left\{T>\frac{R^{2}}{a}\right\}\subset\{\gamma_{j}\cap\partial B(0,R)\neq\emptyset,1\leq j\leq N\}\subset\left\{T>\frac{R^{2}}{8aN}\right\}.

(5.2)

Suppose that for each $1\leq j\leq 2N$ , $\eta_{j}$ is parameterized with $a$ -usual parameterization, and we define $\tau_{j}=\min(\inf\{t_{j}:\eta_{j}(0,t_{j}]\cap\partial B(0,R)\neq\emptyset\},t_{j}(T))$ .

•

On the event $\{T>R^{2}/a\}$ , we know from (5.1) that $t_{j}(T)\geq T>R^{2}/a$ for any $1\leq j\leq 2N$ . Then we have $\text{hcap}(\eta(0,t_{j}])>R^{2}$ , and by Lemma 2.1, $\eta_{j}(0,t_{j}]\cap\partial B(0,R)\neq\emptyset$ for any $1\leq j\leq 2N$ . This implies $\{\gamma_{j}\cap\partial B(0,R)\neq\emptyset,1\leq j\leq N\}$ .
•

On the event $\{\gamma_{j}\cap\partial B(0,R)\neq\emptyset,1\leq j\leq N\}$ , for each $1\leq j\leq N$ , we have $\tau_{a_{j}}\wedge\tau_{b_{j}}<\infty$ , and by Lemma 2.1, $\tau_{a_{j}}\wedge\tau_{b_{j}}>R^{2}/4a$ . Note that $\tau_{j}\leq t_{j}(T)$ implies the non-collision of driving functions for $\{\eta_{j}(0,\tau_{j}),1\leq j\leq 2N\}$ , then by (5.1), the lifetime $T$ of $(\eta_{1},\ldots,\eta_{2N})$ has a lower bound

$T>\frac{1}{2N}\max_{1\leq j\leq 2N}\tau_{j}>\frac{R^{2}}{8aN}.$

This completes the proof of (5.2).

Finally, we prove (1.8):

$\displaystyle\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[\gamma_{j}\cap\partial B(0,R)\neq\emptyset,1\leq j\leq N]\leq$	$\displaystyle\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[T>R^{2}/(8aN)]$	(due to (5.2))
$\displaystyle=$	$\displaystyle\mathcal{I}_{*}^{-1}\mathcal{J}_{\alpha}G_{\alpha}(\boldsymbol{x})\left(\frac{\sqrt{8aN}}{R}\right)^{A_{2N}^{+}}\left(1+O\left(\frac{\|\boldsymbol{x}\|}{R}\right)\right)$	(due to (1.5))
$\displaystyle\asymp$	$\displaystyle G_{\alpha}(\boldsymbol{x})R^{-A_{2N}^{+}};$
$\displaystyle\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[\gamma_{j}\cap\partial B(0,R)\neq\emptyset,1\leq j\leq N]\geq$	$\displaystyle\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[T>R^{2}/a]$	(due to (5.2))
$\displaystyle=$	$\displaystyle\mathcal{I}_{*}^{-1}\mathcal{J}_{\alpha}G_{\alpha}(\boldsymbol{x})\left(\frac{\sqrt{a}}{R}\right)^{A_{2N}^{+}}\left(1+O\left(\frac{\|\boldsymbol{x}\|}{R}\right)\right)$	(due to (1.5))
$\displaystyle\asymp$	$\displaystyle G_{\alpha}(\boldsymbol{x})R^{-A_{2N}^{+}};$

where the implicit constants depend only on $\kappa,N,\alpha$ . This completes the proof. ∎

We expect that a stronger conclusion holds:⁴⁴4For two functions $F_{1}(\boldsymbol{x};R)$ and $F_{2}(\boldsymbol{x};R)$ , we write $F_{1}(\boldsymbol{x};R)\sim F_{2}(\boldsymbol{x};R)$ as $R\to\infty$ if the limit $\lim_{R\to\infty}\frac{F_{1}(\boldsymbol{x};R)}{F_{2}(\boldsymbol{x};R)}$ exists and takes value in $(0,\infty)$ .

\mathbb{P}_{\alpha}^{(\boldsymbol{x})}[\gamma_{j}\cap\partial B(0,R)\neq\emptyset,1\leq j\leq N]\sim G_{\alpha}(\boldsymbol{x})R^{-A_{2N}^{+}},\quad\text{as }R\to\infty,

(5.3)

This stronger estimate (5.3) is proved for $\kappa\in(0,8),N=1$ in [Law15, Theorem 1] and is proved for $\kappa\in(0,8),N=2$ in [Zha21, Theorem 1.1]. We are not able to prove (5.3) due to technical difficulty.

Finally, let us briefly explain how we extend [FWY24, Theorem 1.2] from $\kappa\in(0,4]$ to $\kappa\in(0,8)$ as in Remark 1.5. To prove (1.9), we need the extension to $\kappa\in(4,8)$ of the following three basic inputs:

•

Radon-Nikodym derivative of the law $\mathbb{P}_{\alpha}^{(\mathbb{U};\boldsymbol{x})}$ of chordal $N$ - $\mathrm{SLE}_{\kappa}$ against the law of radial Loewner chains driven by $2N$ -radial Bessel process with parameter $2a$ .
•

Transition density for $2N$ -radial Bessel process and its convergence to invariant density as $t\to\infty$ ([HL21, Proposition 5.5]).
•

The quasi-invariant density for the argument process under $\mathbb{P}_{\alpha}^{(\mathbb{U};\boldsymbol{x})}$ .

The first input is an analogue of Lemmas 3.1 and 3.2 in this article. Note that the extension to $\kappa\in(4,8)$ of SDE (2.11) for driving function due to recent development in [FPW24, FLPW24] can be modified to the radial case, by using Möbius covariance of pure partition functions and technicals of coordinate changes for $\mathrm{SLE}$ (see [SW05]) respectively. The second input of $2N$ -radial Bessel process is an analogue of Lemma 2.6, which is true when parameter $2a\geq 1/2$ (i.e., $\kappa\in(0,8]$ ). The third input is an analogue of Proposition 1.3, and can be now extended to $\kappa\in(4,8)$ due to Lemma 2.3. Based on the extension of these main ingredients, the proof of [FWY24, Theorem 1.2] can be extended to $\kappa\in(4,8)$ using the same analysis.

Acknowledgements.

We thank Zhonggen Su who suggested this topic to H. W. and provided helpful discussions. We thank Vadim Gorin for helpful discussion and in particular, for pointing out useful references in Lemma 2.6.

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	$\displaystyle\mathbb{P}_{\mathrm{GFF}}^{(\boldsymbol{x})}[T>t]$	$\displaystyle=\int_{\mathfrak{X}_{2N}}q(t;\boldsymbol{x},\boldsymbol{y})\mathrm{d}\boldsymbol{y}=\int_{\mathfrak{X}_{2N}}q_{*}(t;\boldsymbol{x},\boldsymbol{y})\frac{G(\boldsymbol{x})}{G(\boldsymbol{y})}\mathrm{d}\boldsymbol{y}$
		$\displaystyle=\mathcal{I}_{N}^{-1}G(\boldsymbol{x})t^{-2N^{2}}\int_{\mathfrak{X}_{2N}}\frac{f_{2}(\boldsymbol{y})}{G(\boldsymbol{y})}\exp\left(-\frac{\|\boldsymbol{y}\|^{2}}{2t}\right)\mathrm{d}\boldsymbol{y}\times\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right)$
		$\displaystyle=\mathcal{I}_{N}^{-1}G(\boldsymbol{x})t^{-2N^{2}}\times\mathcal{J}_{N}t^{3N^{2}/2}\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right)$
		$\displaystyle=\mathcal{I}_{N}^{-1}\mathcal{J}_{N}G(\boldsymbol{x})t^{-N^{2}/2}\left(1+O\left(\frac{\|\boldsymbol{x}\|}{\sqrt{t}}\right)\right),\quad\text{as }t\to\infty,$

Multiple SLEs and Dyson Brownian motion: transition density and Green’s function

Abstract

1 Introduction

1.1 Multiple SLEs and Dyson Brownian motion

Theorem 1.1.

Definition 1.2.

1.2 Estimates on multiple SLEs

Proposition 1.3.

Theorem 1.4.

Remark 1.5.

2 Preliminaries

ℍ\mathbb{H}-hulls and half-plane capacity

Lemma 2.1.

Proof.

2.1 Pure partition functions

Lemma 2.2.

Lemma 2.3.

2.2 Proof of Lemma 2.3

Lemma 2.4.

Proof.

Lemma 2.5.

Proof.

Proof of Lemma 2.3.

2.3 Chordal NN-SLEκ\mathrm{SLE}_{\kappa}

2.4 Dyson Brownian motion

Lemma 2.6.

Proof.

Lemma 2.7.

Proof.

3 Proof of Theorem 1.1 and Proposition 1.3

3.1 Transition density and proof of Theorem 1.1

Lemma 3.1.

Lemma 3.2.

Proof of Theorem 1.1.

3.2 2​N2N-time local martingales

Lemma 3.3.

Proof.

Proof of Lemma 3.1.

Lemma 3.4.

Proof.

Proof of Lemma 3.2.

3.3 Proof of Proposition 1.3

Proof of (1.5).

Corollary 3.5.

Proof.

Proof of Proposition 1.3.

4 Gaussian free field level lines and Dyson Brownian motion β=2\beta=2

Proposition 4.1.

Lemma 4.2.

Proof.

Proof of Proposition 4.1.

5 Proof of Theorem 1.4

Proof of Theorem 1.4.

Acknowledgements.

References

Multiple SLEs and Dyson Brownian motion:
transition density and Green’s function

$\mathbb{H}$ -hulls and half-plane capacity

2.3 Chordal $N$ - $\mathrm{SLE}_{\kappa}$

3.2 $2N$ -time local martingales

4 Gaussian free field level lines and Dyson Brownian motion $\beta=2$