The topology of $SLE_{\kappa}$ is random for $\kappa>4$

Here we discuss a few $SLE$ basics as well as how one defines the more general $SLE_{\kappa}(\bar{\rho})$ processes. Recall that we write ${\mathbb{H}}:=\{z\in{\mathbb{C}}:\mathfrak{Im}(z)>0\}.$ If $K$ is a bounded closed subset of ${\mathbb{H}}$ such that ${\mathbb{H}}\setminus K$ is simply connected, then we call $K$ a hull in ${\mathbb{H}}$ w.r.t. $\infty$. For such $K$, there is a unique $g_{K}$ that maps ${\mathbb{H}}\setminus K$ conformally onto ${\mathbb{H}}$ such that $g_{K}(z)=z+\dfrac{a}{z}+O\left(\dfrac{1}{z^{2}}\right)$ as $z\to\infty$. The quantity $a$ is known as the half plane capacity of $K$, and is denoted by $\text{hcap}{K}$. It can be shown that $a\geq 0$. The map $g_{K}$ is said to satisfy the hydrodynamic normalization at infinity. For a real interval $I$, let $\mathcal{C}(I)$ denote the real-valued continuous functions on I. Suppose $U\in\mathcal{C}([0,T])$ for some $T\in(0,\infty]$. The chordal Loewner equation driven by $U$ is as follows:

For $0\leq t<T$, let $K_{t}$ and $g_{t}$ be the chordal Loewner hulls and maps, respectively, driven by $U_{t}$. Suppose that for every $t\in[0,T)$,

exists, and $\eta_{t},0\leq t<T$, is a continuous curve. Then for every $t\in[0,T),K_{t}$ is the complement of the unbounded component of ${\mathbb{H}}\setminus\eta((0,t])$. We call $\eta$ the chordal Loewner trace driven by $U_{t}$. In general, however, such a curve may not exist depending on the choice of driving function.

An $SLE_{\kappa}$ in ${\mathbb{H}}$ from $0$ to $\infty$ is defined by the random family of conformal maps $g_{t}$ obtained by solving the Loewner ODE driven by Brownian motion. In particular, we let $U_{t}=\sqrt{\kappa}B_{t}$, where $B_{t}$ is a standard Brownian motion. An $SLE_{\kappa}$ connecting boundary points $x$ and $y$ of an arbitrary simply connected Jordan domain can be constructed as the image of an $SLE_{\kappa}$ on ${\mathbb{H}}$ under a conformal transformation $\Psi\colon{\mathbb{H}}\to D$ sending $0$ to $x$ and $\infty$ to $y$. $SLE$ curves are characterized by scale invariance and the domain Markov property, and are viewed modulo reparametrization. The almost sure continuity of the curves of these processes has also been shown in [13].

$SLE(\kappa;\bar{\rho})$, which is often written as $SLE_{\kappa}(\bar{\rho}_{L};\bar{\rho}_{R})$, is the stochastic process one obtains by solving (1) with a modification on the driving process $U_{t}$, which we now discuss. It is a natural generalization of $SLE_{\kappa}$ in which one keeps track of additional marked points which are called force points. Fix $\bar{x_{L}}=(x_{l,L}<\dots<x_{1,L}\leq 0)$ and $\bar{x}_{R}=(0\leq x_{1,R}<\dots<x_{r,R})$. We associate with each $x_{i,q}$ for $q\in{L,R}$ a weight $\rho_{i,q}\in{\mathbb{R}}$. An $SLE_{\kappa}(\bar{\rho}_{L};\bar{\rho}_{R})$ process with force points $(\bar{x}_{L};\bar{x}_{R})$ is the measure on continuously growing compact hulls $K_{t}$ generated by the Loewner chain with $U_{t}$ replaced by the solution to the system of SDEs given by

The existence and uniqueness of solutions to (1) is discussed in [6]. In particular, it is shown that there is a unique solution to (1) until the first time $t$ that $U_{t}=V_{t}^{j,q}$ where $\sum\limits_{i=1}^{j}\rho^{i,q}\leq-2$ for $q\in\{L,R\}$ (we call this time the continuation threshold). In particular,if $\sum\limits_{i=1}^{j}\rho^{i,q}>-2$ for all $1\leq j\leq|\bar{\rho}^{q}|$ for $q\in\{L,R\}$, then (1) has a unique solution for all times $t$. This even holds when one or both of the $x^{1,q}$ are zero. Note that in [8] it is proved that $SLE_{\kappa}(\bar{\rho})$ is generated by a curve and is transient. In this paper, we need only consider the case of two force points, though the main ingredients for the proofs are stated in more generality.

For $\kappa>4$, there is also significant interest in the hulls that are generated by the $SLE_{\kappa}$ curves. Duplantier conjectured in [4, 5] the duality between $SLE_{\kappa}$ and $SLE_{16/\kappa}$, which says that the boundary of an $SLE_{\kappa}$ hull behaves like an $SLE_{16/\kappa}$ curve, for $\kappa>4$. Many versions of this duality have also been shown in [16, 17, 3, 8, 9].

Lemma 4.9 in [8] asserts that, for $\kappa>4$, the outer boundary $\eta^{\prime}$ of an $SLE_{\kappa}$ curve is an $SLE_{\kappa}(\bar{\rho})$ process. The $SLE$ curves are realized as flow lines of the Gaussian free field (i.e $SLE_{\kappa}(\bar{\rho})$ curves coupled with the Gaussian free field in ${\mathbb{H}}$), with the outer boundaries described as counterflow lines (in which the coupling is done with the negation of the Gaussian field). Though we do not need this machinery as presented in [8] and [11], it serves as an excellent framework for proving some general properties of $SLE_{\kappa}(\bar{\rho})$, some of which we rely on to prove the main results. We state one such fact as follows:

Intuitively, Lemma 2.1 tells us that an $SLE_{\kappa}(\bar{\rho}_{L};\bar{\rho}_{R})$ process has a positive chance to stay close to any fixed deterministic curve for a positive amount of time.

Consider the left and right boundaries of the $SLE$ curve $\eta$, which are boundary-touching $SLE_{\frac{16}{\kappa}}(\rho)$ curves, with force points starting at $0$. In fact, the left boundary of $SLE_{\kappa}$ turns out to be $SLE_{16/\kappa}(\frac{16}{\kappa}-4;\frac{8}{\kappa}-2)$ and by symmetry, the right boundary is $SLE_{16/\kappa}(\frac{8}{\kappa}-2;\frac{16}{\kappa}-4)$. This can be deduced from Proposition 7.31 in [8]. These curves are shown in the figure below. The open region between the left and right boundaries has countably many connected components, which are separated by the intersection points of the left and right boundaries, i.e., the cut points of $\eta$. These connected components have a total ordering, and come in four types:

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  Type 0: Neither the left nor the right boundary of the component intersects the real line.

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  Type 1: Only the right boundary intersects the real line.

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  Type 2: Only the left boundary intersects the real line.

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  Type 3: The left and right boundaries both intersect the real line.

Note that $\eta$ is a continuous curve that travels between the positive and negative real axes between any two consecutive components of type 3. This shows that the components of type 3 form a discrete set, to which we may assign a labeling by the integers - written as

uniquely, modulo index shift. For concreteness, we choose the indexing for the sequence so that $U_{0}$ is the first type 3 bubble which has Euclidean diameter at least 1. We remark here that our construction relies on a few tail triviality arguments, and so we require the following:

Between pairs of consecutive type 3 bubbles, $U_{i}$ and $U_{i+1}$, we may either observe a type 1 or 2 bubble, or we may not. Let $E_{i}$ be the event that there is a type 1 or type 2 bubble between $U_{i}$ and $U_{i+1}$, and define

the bi-infinite sequence of 0’s and 1’s consisting of the indicators of the $E_{i}$’s.

Here, we require a more subtle argument that relies on a less obvious statistic of observation. In this section, we fix $\kappa\geq 8$ and recall $SLE_{\kappa}$, in this instance, is plane filling. Let $\eta$ be an instance of $SLE_{\kappa}$ in $\overline{{\mathbb{H}}}$. We are interested in the successive crossing times (about the origin) of the curve $\eta$, i.e., the times at which $\eta$ hits the real line again, just after having hit it on the opposite side of the origin. We look at one such crossing time, and consider the part of the $SLE$ within this, observing the times it goes back and forth between the boundaries of the crossing excursion. As pictured below in Fig. 3, these left and right crossings (within the curve) define a sequence of marked points $\{X_{k}\}$ along the boundary, which accumulate only at the tip of the curve. Via the corresponding Loewner map $g_{t}^{\eta}$, we may conformally map this configuration as shown in Fig. 3, so that the tip goes to $0$, and we abtain a sequence of marked points along the left boundary. Notice these marked points are determined by the past, so we can condition on their locations, and the future will still be an $SLE$ by the Markov property.

A bit more care is needed in defining these quantities. Let $\tau(t)$ be the last time before $t$ such that $\eta(t)\in{\mathbb{R}}$. Define the sets

and set $\mathcal{S}=\bar{T}_{-}\cap\bar{T}_{+}$. Notice that $\mathcal{S}$ is a discrete set since $\eta$ is continuous, and so it cannot cross back and forth between $(-\infty,0)$ and $(0,\infty)$ infinitely many times during any compact time interval contained in $(0,\infty)$. Thus, we may index the elements of $\mathcal{S}$ as a countable sequence of well defined crossing times $\{\tau_{j}\}$.

Notice that these are not necessarily stopping times (which poses a problem in applying the strong Markov property), but this can be addressed by adopting some notation from the previous section as follows. Let $\eta_{j}:=\eta|_{[\tau_{j-1},\tau_{j}]}$, which is the $j$^th left-right crossing around $0$ that we observe. For $r>0$, let $J_{r}^{(n)}$ be the $n$^th smallest $j$ for which the Euclidean diameter of $\eta_{j}$ is at least $r$. It is not difficult to see that the set of times $\{\tau_{J_{r}^{(n)}}\}$ is indeed a set of stopping times. To see this, let $t>0$. If one sees $\eta|_{[0,t]}$, then one can determine the set $\{\tau_{j}:j\leq t\}$ (but not necessarily its indexing). This follows from the definition of the times $\{\tau_{j}\}$ as the intersection points of $T_{-}$ and $T_{+}$, as shown previously. Hence $\eta|_{[0,t]}$ determines the set of excursions $\{\eta_{j}:\tau_{j}\leq t\}$. We have $\tau_{J_{r}^{(n)}}\leq t$ if and only if this set of excursions includes at least $n$ elements which have Euclidean diameter at least $r$. Hence $\{\tau_{J_{r}^{(n)}}\leq t\}$ is determined by $\eta|_{[0,t]}$, which holds for any choice of $t$.

We fix some $r$ and some $n$, and set $J:=J_{r}^{(n)}$. Within $\eta_{J}$, i.e. between the outer boundaries of this crossing, we can keep track of the times at which $\{\eta_{t};t<\tau_{J}\}$ sequentially hits these boundaries. More precisely, we let $L_{J}$ be the outer boundary of $\eta[0,\tau_{J}]$. We define our sequence of crossing times inductively as follows:

and so on. The sequences $\{\sigma_{J,k}\}_{k\geq 1}$ and $\{\tilde{\sigma}_{J,k}\}_{k\geq 1}$ define two discrete sets of times that our curve successively hits the outer boundaries $L_{J}$ and $L_{J-1}$ respectively. We assume without loss of generality that the $J$^th excursion goes from left to right. By considering only the outer boundary $L_{J}$ (as a priori $\tau_{J}$ is a well-defined stopping time), we can construct a sequence of marked points $\{X_{J,k}\}_{k\geq 1}$ along the negative real axis, via the (shifted) Loewner map which sends $\eta(\tau_{J})$ to $0$. That is to say, $X_{J,k}:=g_{\tau_{J}}(\eta(\sigma_{J,k}))-U_{\tau_{J}}$. As we are considering a fixed $J$, we may write $X_{J,k}:=X_{k}$ for ease.

We let $N_{k}=\#\left\{\text{crossing endpoints of the future curve which lie in the interval }[X_{k+1},X_{k}]\right\}.$ In other words, we are looking at $g_{\tau_{J}}(\eta|_{[\tau_{J},\infty)})-U_{\tau_{J}}$ as it does these left-right crossings, conditioned on the past, and for each interval we are keeping track of how many endpoints it contains. We wish to show that for every sequence of deterministic integers $\{n_{k}\}_{k\in{\mathbb{N}}}$, we have that

It suffices to show that there are arbitrarily large $k$ such that ${\mathbb{P}}[N_{k}=n_{k}]$ is bounded away from $1$. Indeed, the event $\{N_{k}=n_{k}\text{ for infinitely many }k\}$ is a tail event for the Brownian motion driving the $SLE$, and the Blumenthal $0-1$ law implies that this has probability $0$ or $1$. Thus, being bounded away from $1$ guarantees that we have (4). We do this in cases as follows:

Case 1: Assume $\exists$ arbitrarily large $k$ such that $n_{k}\neq 0$. We claim that $\exists\,q>0$ such that

To see this, we consider the segment of the curve, just after the $(n-1)^{th}$ crossing is completed. Let $\mathcal{T}_{n}=\{n^{th}\,\text{time we have a crossing in the interval}\,[X_{k},X_{k+1}]\,\}$. Thus $\mathcal{T}_{n}$ is a stopping time, and conditioned on what we have seen up until this time, the future of the curve is still $SLE$.

The goal is to have an upper bound on the probability that there are exactly $n$ crossings, and we do so by comparing the harmonic measure (from $\infty$) of the interval $[X_{k+1},\eta(\mathcal{T}_{n-1})]$, to that of the outer boundary of the curve $\eta[0,\mathcal{T}_{n-1}]$(and more precisely, this is the harmonic measure from $\infty$ in ${\mathbb{H}}\setminus\eta[0,\mathcal{T}_{n-1}]$) . These quantities are denoted $a$ and $b$ respectively, as shown in Figure 3.

The proof relies on the following intuitive argument which we formalize later: If $a$ is larger than $b$, then with positive probability we observe $2$ further crossings, hence $n+1$ total crossings. If $a$ is smaller than $b$, then, with positive probability, we expect the interval $[X_{k+1},\eta(\mathcal{T}_{n-1})]$ to be covered before we observe the next crossing. In other words, there is always a positive chance that we observe either $n-1$ crossings or $n+1$ crossings, and so

We return to the notation introduced in Fig. 4, and we consider the the behavior of the $SLE$ curve given the relative quantities $a$ and $b$. In particular, we require the following two key lemmas to prove the original claim:

Case 2: $n_{k}=0$ for all but finitely many $k$.

This condition implies that the $SLE$ travels a positive distance of time without any left-right crossings, which happens with probability $0$. This shows that for any fixed deterministic sequence $\{n_{k}\}_{k\in{\mathbb{N}}}$ with only finitely many non-zero elements, we have that