We characterize and describe all random subsets $K$ of a given simply connected planar domain (the upper half-plane $\H$, say) which satisfy the ``conformal restriction'' property, i.e., $K$ connects two fixed boundary points (0 and $\infty$, say) and the law of $K$ conditioned to remain in a simply connected open subset $D$ of $\H$ is identical to that of $Φ(K)$, where $Φ$ is a conformal map from $\H$ onto $D$ with $Φ(0)=0$ and $Φ(\infty)=\infty$. The construction of this family relies on the stochastic Loewner evolution (SLE) processes with parameter $κ\le 8/3$ and on their distortion under conformal maps. We show in particular that SLE(8/3) is the only random simple curve satisfying conformal restriction and relate it to the outer boundaries of planar Brownian motion and SLE(6).
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…
The Schramm-Loewner evolution (SLE) can be simulated by dividing the time interval into N subintervals and approximating the random conformal map of the SLE by the composition of N random, but relatively simple, conformal maps. In the usual implementation the time required to compute a single point on the SLE curve is O(N). We give an algorithm for which the time to compute a single point is O(N^p) with p<1. Simulations with kappa=8/3 and kappa=6 both give a value of p of approximately 0.4.
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.
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…
We describe all random sets that satisfy the radial conformal restriction property, therefore providing the analogue in the radial case of results of Lawler, Schramm and Werner in the chordal case.
We discuss the partition function point of view for chordal Schramm-Loewner evolutions and their relationship with correlation functions in conformal field theory. Both are closely related to crossing probabilities and interfaces in critical models in two-dimensional statistical mechanics. We gather and supplement previous results with different perspectives, point out remaining difficulties, and suggest directions for future studies.
We review some of the results related to conformal restriction: the chordal case and the radial case. We describe Brownian intersection exponents, conformal restriction property and SLE, and study their properties.
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path…
From conformal field theory on the Riemann sphere, we implement its boundary version in a simply-connected domain using the Schottky double construction. We consider the statistical fields generated by background charge modification of the Gaussian free field with Dirichlet boundary condition under the OPE multiplications. We prove that the correlation functions of such fields with symmetric background charges form a collection of martingale-observables for (forward) chordal/radial SLE with force points and spins. We also present the connection between conformal field theory with Neumann bound…
We show how to relate Schramm-Loewner Evolutions (SLE) to highest-weight representations of infinite dimensional Lie Algebras using the conformal restriction properties studied by Lawler, Schramm and Werner in the paper arXiv:math.PR/0209343. This confirms the prediction from theoretical physics and conformal field theory that two-dimensional critical systems are related to such degenerate representations.
This paper is based on mini-courses given in July 2003. Its goal is to give a self-contained sketchy and heuristic survey of the recent results concerning conformal restriction, that were initiated in our joint work with Greg Lawler and Oded Schramm, and further investigated in the last year in joint work with Roland Friedrich, Greg Lawler, and by Julien Dubedat. These notes can be viewed as complementary to my Saint-Flour notes.
In this note, we show how to relate the Schramm-Loewner Evolution processes (SLE) to highest-weight representations of the Virasoro Algebra. The conformal restriction properties of SLE that have been recently studied in the paper arXiv:math.PR/0209343 by G. Lawler, O. Schramm and the second author play an instrumental role. In this setup, various considerations from conformal field theory can be interpreted and reformulated via SLE. This enables to make a concrete link between the two-dimensional discrete critical systems from statistical physics and conformal field theory.
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…
The study of conformal restriction properties in two-dimensions has been initiated by Lawler, Schramm and Werner who focused on the natural and important chordal case: They characterized and constructed all random subsets of a given simply connected domain that join two marked boundary points and that satisfy the additional restriction property. The radial case (sets joining an inside point to a boundary point) has then been investigated by Wu. In the present paper, we study the third natural instance of such restriction properties, namely the "trichordal case", where one looks at random sets …
Two-pointed quantum disks with a weight parameter $W > 0$ are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with an independent chordal $\mathrm{SLE}_κ(ρ_-;ρ_+)$ curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees…
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 show that for the conformal restriction measure with exponent $b$ in the unit disk on hulls $γ$ connecting $e^{ix}$ to 1 the probability of the event that $γ$ avoids the disk of radius $q$ centered at zero decays like $\exp(-bπx/(1-q))$ if either $b\in[5/8,1]\cup[5/4,\infty)$ and $x\in(0,π]$, or if $b\in(1,5/4)$, $x\in(0,π)$, and $bx\leπ$.
We resolve a conjecture of Sheffield that $\SLE(4)$, a conformally invariant random curve, is the universal limit of the chordal zero-height contours of random surfaces with isotropic, uniformly convex potentials. Specifically, we study the \emph{Ginzburg-Landau $\nabla ϕ$ interface model} or \emph{anharmonic crystal} on $D_n = D \cap \tfrac{1}{n} \Z^2$ for $D \subseteq \C$ a bounded, simply connected Jordan domain with smooth boundary. This is the massless field with Hamiltonian $\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$ with $\CV$ symmetric and uniformly convex and $h(x) = ϕ(x)$ for $x \in …
We show that when observing the range of a chordal SLE$_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…
Uniqueness of generalized conformal restriction measures and Malliavin-Kontsevich-Suhov measures for $c \in (0,1]$
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Institute for Theoretical Sciences, Westlake University（西湖大学理论科学研究所）
We develop a version of dipolar conformal field theory based on the central charge modification of the Gaussian free field with the Dirichlet boundary condition and prove that correlators of certain family of fields in this theory are martingale-observables for dipolar SLE. We prove the restriction property of dipolar SLE(8/3) and Friedrich-Werner's formula in the dipolar case.
The aim of these notes is threefold. First, we discuss geometrical aspects of conformal covariance in stochastic Schramm-Loewner evolutions (SLEs). This leads us to introduce new ``dipolar'' SLEs, besides the known chordal, radial or annular SLEs. Second, we review the main features of our approach connecting SLEs to conformal field theories (CFTs). It is based on using boundary CFTs to probe the SLE hulls. Finally, we study zig-zag probabilities and their relation with CFT correlation functions. We suggest a putative link between the braiding of SLE samples and that of CFT correlation functio…
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 present a relation between conformal field theories (CFT) and radial stochastic Schramm-Loewner evolutions (SLE) similar to that we previously developed for the chordal SLEs. We construct an important local martingale using degenerate representations of the Virasoro algebra. We sketch how to compute derivative exponants and the restriction martingales in this framework. In its CFT formulation, the SLE dual Fokker-Planck operator acts as the two-particle Calogero hamiltonian on boundary primary fields and as the dilatation operator on bulk primary fields localized at the fixed point of the S…
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…
Schramm-Loewner Evolutions ($\SLE$) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of $\SLE$ and the free field with appropriate boundary conditions; this involves $ζ$-regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of $\SLE$ with the free field, show…
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…
The backward chordal Schramm-Loewner Evolution naturally defines a conformal welding homeomorphism of the real line. We show that this homeomorphism is invariant under the automorphism $x\mapsto -1/x$, and conclude that the associated solution to the welding problem (which is a natural renormalized limit of the finite time Loewner traces) is reversible. The proofs rely on an analysis of the action of analytic circle diffeomorphisms on the space of hulls, and on the coupling techniques of the second author.
Miller and Sheffield introduced the notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $\mathbb{C}$ and Gaussian free fields (GFFs) on them under the conformal equivalence. They considered the situation in which the conformal maps are given by a chordal Schramm--Loewner evolution (SLE). In the present paper, we construct GFF-valued processes on $\mathbb{H}$ (the upper half-plane) and $\mathbb{O}$ (the first orthant of $\mathbb{C}$) by coupling a GFF with a multiple SLE evolving in time on each domain. We prove that a GFF on $\mathbb{H}$ and…
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…
