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 that they are defined in.
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.
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.
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…
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 …
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…
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.
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…
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.
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…
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…
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 $0$.
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.
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.
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.
SLE$_κ(ρ)$ is a variant of SLE$_κ$ where $ρ$ characterizes the repulsion (if $ρ>0$) or attraction $(ρ<0)$ from the boundary. This paper examines the probabilities of SLE$_κ(ρ)$ to get close to the boundary. We show how close the chordal SLE$_κ(ρ)$ curves get to the boundary asymptotically, and provide an estimate for the probability that the SLE$_κ(ρ)$ curve hits graph of functions. These generalize the similar result derived by Schramm and Zhou for standard SLE$_κ$ curves.
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…
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 $\operatorname {Min}(2,1+κ/8)$.
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 …
In this paper, we will show that the higher moments of the natural parametrization of SLE curves in any bounded domain in the upper half plane is finite. We prove this by estimating the probability that an SLE curve gets near n given points.
We first prove that, for $κ\in(0,4)$, a whole-plane SLE$(κ;κ+2)$ trace stopped at a fixed capacity time satisfies reversibility. We then use this reversibility result to prove that, for $κ\in(0,4)$, a chordal SLE$_κ$ curve stopped at a fixed capacity time can be mapped conformally to the initial segment of a whole-plane SLE$(κ;κ+2)$ trace. A similar but weaker result holds for radial SLE$_κ$. These results are then used to study the ergodic behavior of an SLE curve near its tip point at a fixed capacity time. The proofs rely on the symmetry of backward SLE laminations and conformal removabilit…
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.
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.
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…
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.
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.
Following the strategy proposed by Makarov and Smirnov in arXiv:0909.5377, we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up of arXiv:0909.5377 appeared since then, we believe that such a treatment might be of interest for the community. We do not require any regularity of the limiting planar domain $Ω$ near its degenerate prime ends $a$ and $b$ except that $(Ω^δ,a^δ,b^δ)$ are assumed to be `close discrete approximations' to $(Ω,a,b)$ near $a$ and $b$ in the sense of a recent work arXiv:1810.05608.
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…
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…
We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under welding via SLE martingales. As an application, we construct the natural quantum length and boundary intersection measures on the SLE curve itself.
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…
In this paper we mingle the Gaussian free field, the Schramm-Loewner evolution and the KPZ relation in a natural way, shedding new light on all of them. Our principal result shows that the level lines and the SLE$_κ$ flow lines of the Gaussian free field do not satisfy the usual KPZ relation. In order to prove this, we have to make a technical detour: by a careful study of a certain diffusion process, we provide exact estimates of the exponential moments of winding of chordal SLE curves conditioned to pass nearby a fixed point. This extends previous results on winding of SLE curves by Schramm.
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.
We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.
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 $κ>4$.
Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called Stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which …
In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally-invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishin…
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…
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…
We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner's equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to f…
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-…
We derive a surprising correspondence between SLE$_κ(ρ)$ processes and light cones of the Gaussian free field (GFF). Recall that (one-sided, chordal, origin-seeded) SLE$_κ(ρ)$ processes are in some sense the simplest and most natural variants of the Schramm-Loewner evolution. They were originally defined only for $ρ> -2$, but one can use Lévy compensation to extend the definition to any $ρ> -2-\tfracκ{2}$ and to obtain qualitatively different curves. The triangle $T = \{(κ, ρ): (-2-\tfracκ{2})\vee (\tfracκ{2}-4) < ρ< -2 \}$ is the primary focus of this paper. When $(κ, ρ) \in T$, the SLE$_κ(ρ)…
Conformally-invariant curves that appear at critical points in two-dimensional statistical mechanics systems, and their fractal geometry have received a lot of attention in recent years. On the one hand, Schramm has invented a new rigorous as well as practical calculational approach to critical curves, based on a beautiful unification of conformal maps and stochastic processes, and by now known as Schramm-Loewner evolution (SLE). On the other hand, Duplantier has applied boundary quantum gravity methods to calculate exact multifractal exponents associated with critical curves. In the first par…
We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.
We prove upper bounds for the probability that a radial SLE$_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.
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.
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.
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…
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…
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.
