On regularity of SLE_8 curves

Regularity of SLE in $(t,κ)$ and refined GRR estimates

Schramm-Loewner evolution (SLE$_κ$) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $\sqrtκ$ times Brownian motion. This yields a (half-plane) valued random field $γ= γ(t, κ; ω)$. (Hölder) regularity of in $γ(\cdot,κ;ω$), a.k.a. SLE trace, has been considered by many authors, starting with Rohde-Schramm (2005). Subsequently, Johansson Viklund, Rohde, and Wong (2014) showed a.s. Hölder continuity of this random field for $κ< 8(2-\sqrt{3})$. In this paper, we improve their result to joint Hölder continuity up to $κ< 8/3$. Moreover, we show that th…

Hausdorff dimension of the SLE curve intersected with the real line

We establish an upper bound on the asymptotic probability of an SLE(kappa) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < kappa < 8. As a consequence we are able to prove that the SLE curve intersected with the real line has Hausdorff dimension 2-8/kappa, almost surely.

The topology of $SLE_κ$ is random for $κ>4$

We study the topology of $SLE$ curves for $κ> 4$. More precisely, we show that, a.s., there is no homeomorphism $Φ: \overline{\mathbb{H}} \rightarrow \overline{\mathbb{H}}$, taking the range of one independent $SLE$ curve to another for $κ\in (4,8)$. Furthermore, we extend the result to $κ\geq 8$ by showing that there is no homeomorphism taking one $SLE$ curve to another, when viewed as curves modulo parametrization.

Restriction Properties of Annulus SLE

For $κ\in(0,4]$, a family of annulus SLE$(κ;Λ)$ processes were introduced in [14] to prove the reversibility of whole-plane SLE$(κ)$. In this paper we prove that those annulus SLE$(κ;Λ)$ processes satisfy a restriction property, which is similar to that for chordal SLE$(κ)$. Using this property, we construct $n\ge 2$ curves crossing an annulus such that, when any $n-1$ curves are given, the last curve is a chordal SLE$(κ)$ trace.

Up-to-constants bounds on the two-point Green's function for SLE curves

The Green's function for the chordal Schramm-Loewner evolution $SLE_κ$ for $0 < κ< 8$, gives the normalized probability of getting near points. We give up-to-constant bounds for the two-point Green's function.

On the $\log$-regularity of SLE$_4$

We prove that the welding homeomomorphism of SLE$_4$ is almost surely $\log$-regular. In a previous version of this work, we had erroneously deduced its removability from this property. Nevertheless, the $\log$-regularity does provide some information and could lead to future developments.

SLE_4统一化映射的正则性及SLE_8轨迹

Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace

研究SLE_4统一化映射的连续模，证明其为( log δ^{-1} )^{-1/3 + o(1)}，并展示SLE_4不满足Jones-Smirnov条件。同时证明SLE_8的连续模为( log δ^{-1} )^{-1/4 + o(1)}，验证Alvisio和Lawler的猜想。

Decomposition of Schramm-Loewner evolution along its curve

We show that, for $κ\in(0,8)$, the integral of the laws of two-sided radial SLE$_κ$ curves through different interior points against a measure with SLE$_κ$ Green function density is the law of a chordal SLE$_κ$ curve, biased by the path's natural length. We also show that, for $κ>0$, the integral of the laws of extended SLE$_κ(-8)$ curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE$_κ$ curve, biased by the path's capacity length restricted in that set. Another result is that, for $κ\in(4,8)$, if one i…

Some partial results on the convergence of loop-erased random walk to SLE(2) in the natural parametrization

We outline a strategy for showing convergence of loop-erased random walk on the Z^2 square lattice to SLE(2), in the supremum norm topology that takes the time parametrization of the curves into account. The discrete curves are parametrized so that the walker moves at a constant speed determined by the lattice spacing, and the SLE(2) curve has the recently introduced natural time parametrization. Our strategy can be seen as an extension of the one used by Lawler, Schramm, and Werner to prove convergence modulo time parametrization. The crucial extra step is showing that the expected occupation…

Reversed radial SLE and the Brownian loop measure

The Brownian loop measure is a conformally invariant measure on loops in the plane that arises when studying the Schramm-Loewner evolution (SLE). When an SLE curve in a domain evolves from an interior point, it is natural to consider the loops that hit the curve and leave the domain, but their measure is infinite. We show that there is a related normalized quantity that is finite and invariant under Möbius transformations of the plane. We estimate this quantity when the curve is small and the domain simply connected. We then use this estimate to prove a formula for the Radon-Nikodym derivative…

Green's function for chordal SLE curves

For a chordal SLE$_κ$ ($κ\in(0,8)$) curve in a domain $D$, the $n$-point Green's function valued at distinct points $z_1,\dots,z_n\in D$ is defined to be $$G(z_1,\dots,z_n)=\lim_{r_1,\dots,r_n\downarrow 0} \prod_{k=1}^n r_k^{d-2} \mathbb{P}[\mbox{dist}(γ,z_k)<r_k,1\le k\le n],$$ where $d=1+\fracκ{8}$ is the Hausdorff dimension of SLE$_κ$, provided that the limit converges. In this paper, we will show that such Green's functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green's functions as well. Finally, we give up-to-…

Multipoint Estimates for Radial and Whole-plane SLE

We prove upper bounds for the probability that a radial SLE$_κ$ curve, $κ\in(0,8)$, comes within specified radii of $n$ different points in the unit disc. Using this estimate, we then prove a similar upper bound for a whole-plane SLE$_κ$ curve. We then use these estimates to show that the lower Minkowski content of both the radial and whole-plane SLE$_κ$ traces restricted in a bounded region have finite moments of any order.

A Dimension Spectrum for SLE Boundary Collisions

We consider chordal SLE(kappa) curves for kappa > 4, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension min {2-8/kappa,1}. We study the random sets of points at which the curve collides with the real line at a specified "angle" and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov's theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.

Escape probability and transience for SLE

We give estimates for the probability that a chordal, radial or two-sided radial SLE$_κ$ curve retreats far from its terminal point after coming close to it, for $κ\leq 4$. The estimates are uniform over all initial segments of the curve, and are sharp up to a universal constant.

Two-curve Green's function for $2$-SLE: the boundary case

We prove that for $κ\in(0,8)$, if $(η_1,η_2)$ is a $2$-SLE$_κ$ pair in a simply connected domain $D$ with an analytic boundary point $z_0$, then $\lim_{r\to 0^+}r^{-α} \mathbb{P}[\mbox{dist}(z_0,η_j)<r,j=1,2]$ converges to a positive number for some $α>0$, which is called the two-curve Green's function. The exponent $α$ equals $\frac{12}κ-1$ or $2(\frac{12}κ-1)$ depending on whether $z_0$ is one of the endpoints of $η_1$ and $η_2$. We also find the convergence rate and the exact formula of the Green's function up to a multiplicative constant. To derive these results, we construct two-dimension…

Two-curve Green's function for $2$-SLE: the interior case

A $2$-SLE$_κ$ ($κ\in(0,8)$) is a pair of random curves $(η_1,η_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_κ$ curve in a complement domain. In this paper we prove that for any $z_0\in D$, the limit $\lim_{r\to 0^+}r^{-α_0} \mathbb{P}[\mbox{dist}(z_0,η_j)<r,j=1,2]$, where $α_0=\frac{(12-κ)(κ+4)}{8κ}$, exists. Such limit is called a two-curve Green's function. We find the convergence rate and the exact formula of the Green's function in terms of a hypergeometric function up to a multiplicative const…

Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve

In this paper we consider the natural random walk on a planar graph and scale it by a small positive number $δ$. Given a simply connected domain $D$ and its two boundary points $a$ and $b$, we start the scaled walk at a vertex of the graph nearby $a$ and condition it on its exiting $D$ through a vertex nearby $b$, and prove that the loop erasure of the conditioned walk converges, as $δ\downarrow 0$, to the chordal SLE$_{2}$ that connects $a$ and $b$ in $D$, provided that an invariance principle is valid for both the random walk and the dual walk of it.

SLE的精细正则性

Refined regularity of SLE

本文证明了容量参数化下SLE轨迹的精细变差与Hölder型正则性，简化SLE₈相关证明步骤并得到对应模，还对κ≥8的一致化映射证明了时间一致的正则性估计，结果通过正向Loewner微分方程分析得出。

Uniqueness of signature for simple curves

We propose a topological approach to the problem of determining a curve from its iterated integrals. In particular, we prove that a family of terms in the signature series of a two dimensional closed curve with finite p variation, 1\leq p<2, are in fact moments of its winding number. This relation allows us to prove that the signature series of a class of simple non-smooth curves uniquely determine the curves. This implies that outside a Chordal SLE_κ null set, where 0<κ\leq 4, the signature series of curves uniquely determine the curves. Our calculations also enable us to express the Fourier …

On the Continuity of SLE(κ) in κ

We prove that for almost every Brownian motion sample, the corresponding SLE(κ) curves parameterized by capacity exist and change continuously in the supremum norm when κvaries in the interval [0,κ_0), where κ_0=8(2-\sqrt{3})=2.143... We estimate the κ-dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with κ.

A support theorem for SLE curves

For all $κ> 0$, we show that the support of SLE$_κ$ curves is the closure in the sup-norm of the set of Loewner curves driven by nice (e.g. smooth) functions. It follows that the support is the closure of the set of simple curves starting at $0$.

Commutation relations for SLE

Schramm-Loewner Evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e. several random curves). In particular, one derives infinitesimal commutation conditions, discuss some elementary solutions, study integrability conditions following from commutation and show how to lift these infinitesimal relations to global relations in sim…

SLE curves and natural parametrization

Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_κ$ curves is well defined for all $κ<8$. Our proof uses a two-interior-point local martingale.

SLE$_κ(ρ)$ bubble measures

For $κ>0$ and $ρ>-2$, we construct a $σ$-finite measure, called a rooted SLE$_κ(ρ)$ bubble measure, on the space of curves in the upper half plane $\mathbb H$ started and ended at the same boundary point, which satisfies some SLE$_κ(ρ)$-related domain Markov property, and is the weak limit of SLE$_κ(ρ)$ curves in $\mathbb H$ with the two endpoints both tending to the root. For $κ\in(0,8)$ and $ρ\in ((-2)\vee(\fracκ2-4),\fracκ2-2)$, we derive decomposition theorems for the rooted SLE$_κ(ρ)$ bubble with respect to the Minkowski content measure of the intersection of the rooted SLE$_κ(ρ)$ bubble …

Almost sure multifractal spectrum for the tip of an SLE curve

The tip multifractal spectrum of a two-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm-Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability one, and we give applications to the scaling of harmonic measure at the tip.

Dimension and natural parametrization for SLE curves

Some possible definitions for the natural parametrization of SLE (Schramm-Loewner evolution) paths are proposed in terms of various limits. One of the definitions is used to give a new proof of the Hausdorff dimension of SLE paths.

The dimension of the SLE curves

Let $γ$ be the curve generating a Schramm--Loewner Evolution (SLE) process, with parameter $κ\geq0$. We prove that, with probability one, the Hausdorff dimension of $γ$ is equal to $\operatorname {Min}(2,1+κ/8)$.

Cadlag curves of SLE driven by Levy processes

Schramm Loewner Evolutions (SLE) are random increasing hulls defined through the Loewner equation driven by Brownian motion. It is known that the increasing hulls are generated by continuous curves. When the driving process is of the form \sqrtκ B+θ^{1/α}S for a Brownian motion B and a symmetric α-stable process S with κnot equal to 4 and 8, we prove that the corresponding increasing hulls are generated by Cadlag curves.

Convergence of Ising interfaces to Schramm's SLE curves

We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schramm's SLE curves with parameter $κ=3$ and $κ=16/3$ respectively.

The natural parametrization for the Schramm-Loewner evolution

The Schramm-Loewner evolution (SLE_κ) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ< 8, an instance of SLE_κis a random planar curve with almost sure Hausdorff dimension d = 1 + κ/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume. For κ< 8, we use a Doob-Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE_κthat transforms like a d-dimensional volume measure under conformal maps. We prove that this param…

Monte Carlo comparisons of the self-avoiding walk and SLE as parameterized curves

The scaling limit of the two-dimensional self-avoiding walk (SAW) is believed to be given by the Schramm-Loewner evolution (SLE) with the parameter kappa equal to 8/3. The scaling limit of the SAW has a natural parameterization and SLE has a standard parameterization using the half-plane capacity. These two parameterizations do not correspond with one another. To make the scaling limit of the SAW and SLE agree as parameterized curves, we must reparameterize one of them. We present Monte Carlo results that show that if we reparameterize the SAW using the half-plane capacity, then it agrees well…

Boundary proximity of SLE

This paper examines how close the chordal $\SLE_κ$ curve gets to the real line asymptotically far away from its starting point. In particular, when $κ\in(0,4)$, it is shown that if $β>β_κ:=1/(8/κ-2)$, then the intersection of the $\SLE_κ$ curve with the graph of the function $y=x/(\log x)^β$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $β=β_κ$. The critical $\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\log\log x)^α}$, $x>e^e$, in an unbounded set if $α\le 1$, but not if $α>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient in…

Hypergeometric SLE: Conformal Markov Characterization and Applications

This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property and symmetry single out a two-parameter family of random curves---Hypergeometric SLE---denoted by hSLE$_κ(ν)$ for $κ\in (0,4]$ and $ν<κ-6$. The proof relies crucially on Dubédat's commutation relation [Dub07] and a uniqueness result proved in [MS16b]. The classification indicates that hypergeometric SLE is the only possible scaling limit of the interfaces in critical lattice models (conjectured or…

Dimension transformation formula for conformal maps into the complement of an SLE curve

We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of $\mathbb R$ and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an SLE$_κ$ curve for $κ\not =4$. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes-Vargas (2011) and the KPZ formula of Gwynne-Holden-Miller (2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an SLE$_κ$ curve for $κ\in (0,4)\c…

Schramm's formula and the Green's function for multiple SLE

We construct martingale observables for systems of multiple SLE curves by applying screening techniques within the CFT framework recently developed by Kang and Makarov, extended to admit multiple SLEs. We illustrate this approach by rigorously establishing explicit expressions for the Green's function and Schramm's formula in the case of two curves growing towards infinity. In the special case when the curves are `fused' and start at the same point, some of the formulas we prove were predicted in the physics literature.

Convergence of an algorithm simulating Loewner curves

The development of Schramm--Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde \cite{MR05}, is to sample Brownian motion at discrete times, interpolate appropriately in between and solve explicitly the Loewner equation with this approximation. This algorithm always produces piecewise smooth non self-intersecting curves whereas SLE$_κ$ has been proven to be simple for $κ\in[0,4]$, self-touching for $κ\in(4,8)$ and space-filling for $κ\geq 8$. In this p…

$κ\in (4,8)$ 时全局 2-SLE 的分解及其在临界 FK-Ising 模型中的应用

Decomposition of global 2-SLE for $κ\in (4,8)$ and an application for critical FK-Ising model

该研究针对 $\kappa\in(4,8)$ 的拓扑矩形全局 2-SLE，推导了随机击中点分布、两曲线不交概率的渐近，并将其应用于临界 FK-Ising 模型，得到两条不相交开路存在概率的渐近结果。

Multiple SLE type scaling limits: from local to global

We consider collections of $N$ chordal random curves obtained from a critical lattice model on a planar graph, in the limit when a fine-mesh graph approximates a simply-connected domain. We define and study candidates for such limits in terms of conformally invariant collections of random curves, generated via iterated Loewner equations. These curves are a natural ``domain Markov extension'' of the earlier introduced local multiple SLE initial segments to global multiple SLE curves. For realizing them as scaling limits, we provide two a priori results to guarantee the precompactness of the dis…

The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$

We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE$_κ$ as $κ$ tends to 0 and the known reversibility of the SLE$_κ$ curves for small $κ$, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from B to A in D.

A new approach to SLE phase transition

It is well know that $SLE_κ$ curves exhibit a phase transition at $κ=4$. For $κ\le 4$ they are simple curves with probability one, for $κ>4$ they are not. The standard proof is based on the analysis of the Bessel SDE of dimension $d=1+4/κ$. We propose a different approach which is based on the analysis of the Bessel SDE with $d=1-4/κ$. This not only gives a new perspective, but also allows to describe the formation of the SLE `bubbles' for $κ>4$.

奇异路径空间及其应用

Singular paths spaces and applications

本文研究零点处带量化奇异性的路径与粗糙路径，建立奇异Hölder空间及相应积分理论，并用于解释SLE边界效应、统一粗糙波动率奇异核及归约重整化问题。

The Length of an SLE - Monte Carlo Studies

The scaling limits of a variety of critical two-dimensional lattice models are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the parameter kappa. These lattice models have a natural parametrization of their random curves given by the length of the curve. This parametrization (with suitable scaling) should provide a natural parametrization for the curves in the scaling limit. We conjecture that this parametrization is also given by a type of fractal variation along the curve, and present Monte Carlo simulations to support this conjecture. Then we show by simulations that …

绘制随机Loewner曲线的分裂算法与范数收敛性

Splitting algorithm and normed convergence for drawing the random Loewner curves

本文提出一种带严格范数收敛分析的分裂算法来模拟随机Loewner曲线，并将其扩展至分数阶SLE和噪声增强SLE，为分形维数等统计现象提供理论预测。

用于绘制随机分形Loewner曲线的分裂算法与范数收敛性

Splitting algorithm and normed convergence for drawing the random fractal Loewner curves

该研究提出用Ninomiya-Victoir分裂算法逼近$SLE_κ$迹线，证明其依概率依上确界范数一致收敛，还研究了$L^p$范数收敛性，并验证了基于布朗驱动力线性插值方案的一致收敛性。

Non-simple SLE curves are not determined by their range

We show that when observing the range of a chordal SLE$_κ$ curve for $κ\in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related results about conformal loop ensembles (CLE): (i) The loops in a CLE$_κ$ for $κ\in (4,8)$ are not determined by the CLE$_κ$ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE$_κ$ for $κ\in (8/3, 4)$ (we defined these percolation interfaces in previous work, and showed there that they are SLE$_{16/κ}$ curves) are not determined by the CLE$_κ$ carpet th…

带螺旋的多径向SLE：重采样性质与边界扰动

Multiradial SLE with spiral: resampling property and boundary perturbation

该研究推广Healey与Lawler的构造方法，构造带螺旋的多径向SLE曲线，证明其暂态性、重采样性质及边界扰动性质，且其配分函数含电和磁荷。

Three phases of multiple SLE driven by non-colliding Dyson's Brownian motions

The present paper is concerned with properties of multiple Schramm--Loewner evolutions (SLEs) labelled by a parameter $κ\in (0,8]$. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson's Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the…

Dimension of the SLE light cone, the SLE fan, and SLE$_κ(ρ)$ for $κ\in (0,4)$ and $ρ\in [\tfracκ{2}-4,-2)$

Suppose that $h$ is a Gaussian free field (GFF) on a planar domain. Fix $κ\in (0,4)$. The SLE$_κ$ light cone ${\mathbf L}(θ)$ of $h$ with opening angle $θ\in [0,π]$ is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field $e^{i h/χ}$, $χ= \tfrac{2}{\sqrtκ} - \tfrac{\sqrtκ}{2}$, with angles in $[-\tfracθ{2},\tfracθ{2}]$. We derive the Hausdorff dimension of ${\mathbf L}(θ)$. If $θ=0$ then ${\mathbf L}(θ)$ is an ordinary SLE$_κ$ curve (with $κ< 4$); if $θ= π$ then ${\mathbf L}(θ)$ is the range of an SLE$_{κ'}$ curve ($κ' = 16/κ> 4$). In …

A continuous proof of the existence of the SLE$_8$ curve

Suppose that $η$ is a whole-plane space-filling SLE$_κ$ for $κ\in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $η(0) = 0$. For each $T > 0$ and $κ\in (4,8)$ we let $μ_{κ,T}$ denote the law of $η|_{[0,T]}$. We show for each $ν, T > 0$ that the family of laws $μ_{κ,T}$ for $κ\in [4+ν,8)$ is compact in the weak topology associated with the space of probability measures on continuous curves $[0,T] \to {\mathbf C}$ equipped with the uniform distance. As a direct byproduct of this tightness result (taking a limit as $κ\uparrow 8$), we obtain a new proof o…

A multifractal boundary spectrum for SLE$_κ(ρ)$

We study SLE$_κ(ρ)$ curves, with $κ$ and $ρ$ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed "angle" and determine the almost sure Hausdorff dimension of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps $g_t$, by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.

Conformal removability of SLE$_4$

We consider the Schramm-Loewner evolution (SLE$_κ$) with $κ=4$, the critical value of $κ> 0$ at or below which SLE$_κ$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable, answering a question posed by Sheffield. Such curves arise as the conformal welding of a pair of independent critical ($γ=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\m…

Green's function for cut points of chordal SLE attached with boundary arcs

Let $κ\in(4,8)$. Let $γ$ be an SLE$_κ$ curve in a Jordan domain $D$ connecting $a_1\ne a_2\in\partial D$. We attach $γ$ with two open boundary arcs $A_1,A_2$ of $D$, which share end points $b_1\ne b_2\in\partial D\setminus\{a_1,a_2\}$, and consider for each $z_0\in D$ the limit $$ \lim_{r \downarrow 0}r^{1-\frac 38κ} \mathbb{P}[γ\cup A_1\cup A_2 \mbox{ has a cut point in }\{|z-z_0|<r\}].$$ We prove that the limit converges, derive a rate of convergence, and obtain the exact formula of the limit up to a multiplicative constant depending only on $κ$.

离散曲线的幂律收敛速率：框架与应用

Power rate of convergence of discrete curves: framework and applications

本文提出一个通用框架，通过鞅可观测量和交叉事件条件推导离散曲线收敛到 SLE 曲线的幂律速率，并应用于六边形晶格临界渗流，证明其以幂律收敛到 SLE$_6$。

Transforming fixed-length self-avoiding walks into radial SLE_8/3

We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE with kappa=8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and then apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo s…