There is an essentially unique way to associate to any Riemann surface a measure on its simple loops, such that the collection of measures satisfy a strong conformal invariance property. Wendelin Werner constructed these random simple loops as boundaries of Brownian loops, and so they correspond in the zoo of statistical mechanics models to central charge $0$, or Schramm-Loewner Evolution (SLE) parameter $κ=8/3$. The goal of this paper is to construct a family of measures on simple loops on Riemann surfaces that satisfies a conformal covariance property, and that would correspond to SLE parameter $κ=2$ (central charge $-2$). On planar annuli, this loop measure was already built by Adrien Kassel and Rick Kenyon. We will give an alternative construction of this loop measure on planar annuli, investigate its conformal covariance, and finally extend this measure to general Riemann surfaces. This gives an example of a Malliavin-Kontsevich-Suhov loop measure in non-zero central charge.
We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_κ$ loop measures for $κ\in(0,8)$. First, we construct rooted SLE$_κ$ loop measures in the Riemann sphere $\widehat{\mathbb C}$, which satisfy Möbius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its $(1+\frac κ8)$-dimensional Minkowski content. Second, by integrating rooted SLE$_κ$ loop measures, we construct the unrooted SLE$_κ$ loop measure in $\widehat{\mathbb C}$, which satisfies Möbius invariance and reversibility. Third, w…
We define the Schramm-Loewner evolution (SLE) in multiply connected domains for kappa \leq 4 using the Brownian loop measure. We show that in the case of the annulus, this is the same measure obtained recently by Dapeng Zhan. We use the loop formulation to give a different derivation of the partial differential equation for the partition function for the annulus.
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 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…
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.
We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In [LPW21], the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE$_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE$_2$. The conclusion is a generalization of [HLW20,Theorem 1.6] where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE$_2(-1,-1; -1, -1)$. However, the limiting…
In recent work we have shown that loop-erased random walk (LERW) connecting two boundary points of a domain converges to the chordal Schramm-Loewner evolution (SLE(2)) in the sense of curves parametrized by Minkowski content. In this note we explain how to derive the analogous result for LERW from a boundary point to an interior point, converging towards radial SLE(2).
We review some recently completed research that establishes the scaling limit of Fomin's identity for loop-erased random walk on Z^2 in terms of the chordal Schramm-Loewner evolution (SLE) with parameter 2. In the case of two paths, we provide a simplified proof of the identity for loop-erased random walk and simple random walk, and prove directly that the corresponding identity holds for chordal SLE(2) and Brownian motion. We also include a brief introduction to SLE and discussion of the relationship between SLE(2) and loop-erased random walk.
We derive a rate of convergence of the Loewner driving function for planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). The proof uses a new estimate of the difference between the discrete and continuous Green's functions that is an improvement over existing results for the class of domains we consider. Using the rate for the driving process convergence along with additional information about SLE(2), we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.
We show how to connect together the loops of a simple Conformal Loop Ensemble (CLE) in order to construct samples of chordal SLE(κ) processes and their SLE(κ,ρ) variants, and we discuss some consequences of this construction.
We prove that the SLE$_κ$ loop measure arises naturally from the conformal welding of two $γ$-Liouville quantum gravity (LQG) disks for $γ^2 = κ\in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE$_{8/3}$ loop. Our result is also a key input to recent work of the first and third coauthors on …
In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain simple Poissonian percolation clusters: Recall that the Brownian loop-soup (introduced in the paper arxiv:math.PR/0304419 with Greg Lawler) with intensity c defines a Poissonian collection of (simple if one focuses only on the outer boundary) loops in a domain. This random family of (possibly intersecting) loops is conformally invariant (and there are almost surely infinitely many small loops in any sample). We show that there exists a critical value a in (0,1] such that if one colors all the …
In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the …
We give a direct construction of the conformally invariant measure on self-avoiding loops in Riemann surfaces (Werner measure) from chordal $\text{SLE}_{8/3}$. We give a new proof of uniqueness of the measure and use Schramm's formula to construct a measure on boundary bubbles encircling an interior point. After establishing covariance properties for this bubble measure, we apply these properties to obtain a measure on loops by integrating measures on boundary bubbles. We calculate the distribution of the conformal radius of boundary bubbles encircling an interior point and deduce from it expl…
In the second article of this series, we establish the convergence of the loop ensemble of interfaces in the random cluster Ising model to a conformal loop ensemble (CLE) --- thus completely describing the scaling limit of the model in terms of the random geometry of interfaces. The central tool of the present article is the convergence of an exploration tree of the discrete loop ensemble to a branching SLE$(16/3,-2/3)$. Such branching version of the Schramm's SLE not only enjoys the locality property, but also arises logically from the Ising model observables.
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.
We construct a natural measure mu supported on the intersection of a chordal SLE(kappa) curve gamma with the real line R, in the range 4 < kappa < 8. The measure is a function of the SLE path in question. Assuming that boundary measures transform in a ``d-dimensional'' way (where d is the Hausdorff dimension of gamma intersected with R), we show that the measure we construct is (up to multiplicative constant) the unique measure-valued function of the SLE path that satisfies the Domain Markov property.
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 …
We provide another construction of the natural parametrization of SLE$_κ$ for $κ< 4$. We construct it as the expectation of the quantum time, which is a random measure carried by SLE in an ambient Gaussian free field. This quantum time was built as the push forward on the SLE curve of the Liouville boundary measure, which is a natural field-dependent measure supported on the boundary of the domain. We moreover show that the quantum time can be reconstructed as a chaos on any measure on the trace of SLE with the right Markovian covariance property. This provides another proof that natural param…
We define multiple-paths Schramm-Loewner evolution ($SLE_κ$) in multiply connected domains when $κ\leq 4$ and prove that in annuli, the partition function is smooth. Moreover, we give up-to-constant estimates for the partition function of bi-chordal annulus $SLE_κ$ measure and we establish a connection between this measure and two-sided $SLE_κ$ in the unit disk.
We show that, for $κ\le 4$, the integral of the two-sided radial SLE$_κ$ measures over all interior points is chordal SLE$_κ$ biased by the path's natural length, which is its $(1+κ/8)$-dimensional Minkowski content.
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…
We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. This coupling is different than the ones previously considered in this context. Our recent work (arXiv:1603.05203) on the convergence of LERW parametrized by length to SLE(2) parameterized by Minkowski content uses specific features of the coupling constructed here.
In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal points and the CLE carpet/gasket. As an application, we also show the equivalence between our natural measures defined in this paper (i.e. CLE pivotal and gasket measures) and their discrete analogs of counting measures in critical continuum planar Bernoulli percolation in [Garban-Pete-Schramm, J. Amer. Math. Soc.,2013]. Although the existence and uniqueness for the natural measure for CLE carpet/gasket…
We define multiple chordal SLEs in a simply connected domain by considering a natural configurational measure on paths. We show how to construct these measures so that they are conformally covariant and satisfy certain boundary perturbation and Markov properties, as well as a cascade relation. As an example of our construction, we derive the scaling limit of Fomin's identity in the case of two paths directly; that is, we prove that the probability that an SLE(2) and a Brownian excursion do not intersect can be given in terms of the determinant of the excursion hitting matrix. Finally, we defin…
Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix κ\in (4,8) and let η be an SLE_κprocess from x to y in D. We prove that the law of the time-reversal of ηis, up to reparameterization, an SLE_κprocess from y to x in D. More generally, we prove that SLE_κ(ρ_1;ρ_2) processes are reversible if and only if both ρ_i are at least κ/2-4, which is the critical threshold at or below which such curves are boundary filling. Our result supplies the missing ingredient needed to show that for all κ\in (4,8) the so-called conformal loop ensembles CLE_κ are canonicall…
The CFT of SLE loop measures and the Kontsevich--Suhov conjecture
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；CNRS & CEREMADE, Université Paris-Dauphine, PSL University, France（法国国家科学研究中心与CEREMADE，巴黎第九大学，PSL大学）
For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural conformal restriction property that is conjecturally satisfied by the scaling limits of interfaces in models from statistical physics. This property is basically the combination of conformal invariance and the locality of the interaction in the model. Unlike the Markov property that Schramm used to characterize SLE curves (which involves conditioning on partially generated interfaces up to arbitrary stopping times), this property only involves conditioning on entire loops and thus appears a…
We consider a transient chordal SLE$_κ(ρ_1,\dots,ρ_m)$ curve $η$ in $\mathbb{H}$ from $w$ to $\infty$ with force points $ v_1> \cdots >v_m$ in $(-\infty,w^-]$, which intersects and is not boundary-filling on $(-\infty,v_m)$. The main result is that there is an atomless locally finite Borel measure $μ_η$ on $η\cap (-\infty,v_m]$ such that for any $v<v_m$, the $d$-dimensional Minkowski content of $η\cap [v,v_m]$ exists and equals $μ_η[v,v_m]$, where $d=\frac{(\sum ρ_j+4)(κ-4-2\sum ρ_j)}{2κ}$ is the Hausdorff dimension of $η\cap [v,v_m]$. In the case that all $ρ_j=0$, this measure agrees with the…
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 prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D…
Loop-erased random walk, abbreviated LERW, is one of the most well-studied critical lattice models. It is the self-avoiding random walk one gets after erasing the loops from a simple random walk in order or alternatively by considering the branches in a uniformly chosen spanning tree. This paper proves that planar LERW parametrized by renormalized length converges in the lattice size scaling limit to SLE(2) parametrized by 5/4-dimensional Minkowski content. In doing this we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theore…
The two-dimensional Brownian loop-soup is a Poissonian random collection of loops in a planar domain with an intensity parameter c. When c is not greater than 1, we show that the outer boundaries of the loop clusters are disjoint simple loops (when c>1, there is almost surely only one cluster) that satisfy certain conformal restriction axioms. We prove various results about loop-soups, cluster sizes, and the c=1 phase transition. Combining this with the results of another paper of ours on the Markovian characterization of simple conformal loop ensembles (CLE), this proves that these outer boun…
We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved. In the main body of the paper these results are proved while assuming, as argued by Schramm and Smirnov, that the percolation exploration path converges in distribution to the trace of chordal SLE(6). Then, i…
We give a simple set of geometric conditions on curves $η$, $\tildeη$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $φ\colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $η$ with $φ(η) = \tildeη$ then $φ$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $η$ is a non-space-filling SLE$_κ$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $φ$ is a h…
We construct and study the conformal loop ensembles CLE(kappa), defined for all kappa between 8/3 and 8, using branching variants of SLE(kappa) called exploration trees. The conformal loop ensembles are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties. We conjecture that they are the scaling limits of various random loop models from statistical physics, including the O(n) loop models.
We study a one-dimensional SDE that we obtain by performing a random time change of the backward Loewner dynamics in $\mathbb{H}$. The stationary measure for this SDE has a closed-form expression. We show the convergence towards its stationary measure for this SDE, in the sense of random ergodic averages. The precise formula of the density of the stationary law gives a phase transition at the value $κ=8$ from integrability to non-integrability, that happens at the same value of $κ$ as the change in behavior of the $SLE_κ$ trace from non-space filling to space-filling curve. Using convergence i…
We derive the large deviation principle for radial Schramm-Loewner evolution ($\operatorname{SLE}$) on the unit disk with parameter $κ\rightarrow \infty$. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures $\{ϕ_t^2 (ζ)\, dζ\}_{t \in [0,1]}$ on the unit circle and equals $\int_0^1 \int_{S^1} |ϕ_t'|^2/2\,dζ\,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.
We consider some probabilistic and analytic realizations of Virasoro highest-weight representations. Specifically, we consider measures on paths connecting points marked on the boundary of a (bordered) Riemann surface. These Schramm-Loewner Evolution (SLE)- type measures are constructed by the method of localization in path space. Their partition function (total mass) is the highest-weight vector of a Virasoro representation, and the action is given by Virasoro uniformization. We review the formalism of Virasoro uniformization, which allows to define a canonical action of Virasoro generators o…
Loop-erased random walk and it's scaling limit, Schramm--Loewner evolution, have found numerous applications in mathematics and physics. We present a 2 dimensional analogue of LERW, the loop erased random surface. We do this by defining a 2 dimensional spanning tree and declaring that LERS should have the same relation to these 2 trees as LERW has to ordinary spanning trees. Furthermore we present numerical evidence that the growth rate for LERS on a $δ$ fine grid as $δ\to 0$ is $2.5269 \pm 0.0017$ and we hypothesize that it has an exact value of 48/19. This suggests the possibility of a fract…
