Conformal Loop Ensembles Research Articles

CLE PERCOLATIONS

Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set — a random and conformally invariant analog of the Sierpinski carpet or gasket.In the present paper, we derive a direct relationship between the CLEs with simple loops ($\text{CLE}_{\unicode[STIX]{x1D705}}$ for $\unicode[STIX]{x1D705}\in (8/3,4)$, whose loops are Schramm’s $\text{SLE}_{\unicode[STIX]{x1D705}}$-type curves) and the corresponding CLEs with nonsimple loops ($\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ with $\unicode[STIX]{x1D705}^{\prime }:=16/\unicode[STIX]{x1D705}\in (4,6)$, whose loops are $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-type curves). This correspondence is the continuum analog of the Edwards–Sokal coupling between the $q$-state Potts model and the associated FK random cluster model, and its generalization to noninteger $q$.Like its discrete analog, our continuum correspondence has two directions. First, we show that for each $\unicode[STIX]{x1D705}\in (8/3,4)$, one can construct a variant of $\text{CLE}_{\unicode[STIX]{x1D705}}$ as follows: start with an instance of $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$, then use a biased coin to independently color each $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loop in one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loops as interfaces of a continuum analog of critical Bernoulli percolation within $\text{CLE}_{\unicode[STIX]{x1D705}}$ carpets — this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by $\text{SLE}_{6}$ and $\text{CLE}_{6}$.These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized $\text{SLE}_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$ curves for $\unicode[STIX]{x1D70C}<-2$, such as their decomposition into collections of $\text{SLE}_{\unicode[STIX]{x1D705}}$-type ‘loops’ hanging off of $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-type ‘trunks’, and vice versa (exchanging $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D705}^{\prime }$). We also define a continuous family of natural $\text{CLE}$ variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize $\text{CLE}$s, and that should be scaling limits of critical models with special boundary conditions. We extend the $\text{CLE}_{\unicode[STIX]{x1D705}}$/$\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence to a $\text{BCLE}_{\unicode[STIX]{x1D705}}$/$\text{BCLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence that makes sense for the wider range $\unicode[STIX]{x1D705}\in (2,4]$ and $\unicode[STIX]{x1D705}^{\prime }\in [4,8)$.

Quantum gravity and inventory accumulation

We begin by studying inventory accumulation at a LIFO (last-in-first-out) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on $\mathbb{Z}^{2}$. In more interesting versions, a $p$ fraction of customers orders the “freshest available” product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on $p$. We then turn our attention to the critical Fortuin–Kastelyn random planar map model, which gives, for each $q>0$, a probability measure on random (discretized) two-dimensional surfaces decorated by loops, related to the $q$-state Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loop-decorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformal loop ensemble, with parameters depending on $q$. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at $p=1/2$, $q=4$.

Imaginary geometry III: reversibility of \SLE_\kappa for \kappa \in (4,8)

Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix \kappa \in (4,8) and let be an SLE_\kappa process from x to y in D. We prove that the law of the time-reversal of \eta is, up to reparameterization, an SLE_\kappa process from y to x in D. More generally, we prove that SLE_\kappa(\rho_1;\rho_2) processes are reversible if and only if both \rho_i are at least \kappa/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 \kappa \in (4,8) the so-called conformal loop ensembles CLE_\kappa are canonically defined, with almost surely continuous loops. It also provides an interesting way to couple two Gaussian free fields (with different boundary conditions) so that their difference is piecewise constant and the boundaries between the constant regions are SLE_\kappa curves.

Level lines of Gaussian Free Field I: Zero-boundary GFF

We study level lines of Gaussian Free Field h emanating from boundary points. The article has two parts. In the first part, we show that the level lines are random continuous curves which are variants of SLE4 path. We show that the level lines with different heights satisfy the same monotonicity behavior as the level lines of smooth functions. We prove that the time-reversal of the level line coincides with the level line of −h. This implies that the time-reversal of SLE4(ρ¯) process is still an SLE4(ρ¯) process. We prove that the level lines satisfy “target-independent” property. In the second part, we discuss the relation between Gaussian Free Field and Conformal Loop Ensemble (CLE). A CLE is a collection of disjoint SLE-loops. Since the level lines of GFF are SLE4 paths, the collection of level loops of GFF corresponds to CLE4. We study the coupling between GFF and CLE4 with time parameter which sheds lights on the conformally invariant metric on CLE4.

Extreme nesting in the conformal loop ensemble

The conformal loop ensemble $\operatorname {CLE}_{\kappa}$ with parameter $8/3<\kappa<8$ is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain. Given $\kappa$ and $\nu$, we compute the almost-sure Hausdorff dimension of the set of points $z$ for which the number of CLE loops surrounding the disk of radius $\varepsilon$ centered at $z$ has asymptotic growth $\nu\log (1/\varepsilon )$ as $\varepsilon \to0$. By extending these results to a setting in which the loops are given i.i.d. weights, we give a CLE-based treatment of the extremes of the Gaussian free field.

Random walk loop soups and conformal loop ensembles

AbstractThe random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm–Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, “large” lattice loops from the random walk loop soup converge to “large” loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of “large” lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of “large” lattice loops converges to CLE.

The nested simple conformal loop ensembles in the Riemann sphere

Simple conformal loop ensembles (CLE) are random collections of simple non-intersecting loops that are of particular interest in the study of conformally invariant systems. Among other things related to these CLEs, we prove the invariance in distribution of their nested full-plane versions under the inversion $$z \mapsto 1/z$$ .

The conformal loop ensemble nesting field

The conformal loop ensemble CLE$_\kappa$ with parameter $8/3 < \kappa < 8$ is the canonical conformally invariant measure on countably infinite collections of non-crossing loops in a simply connected domain. We show that the number of loops surrounding an $\varepsilon$-ball (a random function of $z$ and $\varepsilon$) minus its expectation converges almost surely as $\varepsilon\to 0$ to a random conformally invariant limit in the space of distributions, which we call the nesting field. We generalize this result by assigning i.i.d. weights to the loops, and we treat an alternate notion of convergence to the nesting field in the case where the weight distribution has mean zero. We also establish estimates for moments of the number of CLE loops surrounding two given points.

The Hausdorff dimension of the CLE gasket

The conformal loop ensemble $\mathrm{CLE}_{\kappa}$ is the canonical conformally invariant probability measure on noncrossing loops in a proper simply connected domain in the complex plane. The parameter $\kappa$ varies between $8/3$ and $8$; $\mathrm{CLE}_{8/3}$ is empty while $\mathrm{CLE}_{8}$ is a single space-filling loop. In this work, we study the geometry of the $\mathrm{CLE}$ gasket, the set of points not surrounded by any loop of the $\mathrm{CLE}$. We show that the almost sure Hausdorff dimension of the gasket is bounded from below by $2-(8-\kappa)(3\kappa-8)/(32\kappa)$ when $4<\kappa<8$. Together with the work of Schramm–Sheffield–Wilson [Comm. Math. Phys. 288 (2009) 43–53] giving the upper bound for all $\kappa$ and the work of Nacu–Werner [J. Lond. Math. Soc. (2) 83 (2011) 789–809] giving the matching lower bound for $\kappa\le4$, this completes the determination of the $\mathrm{CLE}_{\kappa}$ gasket dimension for all values of $\kappa$ for which it is defined. The dimension agrees with the prediction of Duplantier–Saleur [Phys. Rev. Lett. 63 (1989) 2536–2537] for the FK gasket.

Hypotrochoids in conformal restriction systems and Virasoro descendants

A conformal restriction system is a commutative, associative, unital algebra equipped with a representation of the groupoid of univalent conformal maps on connected open sets of the Riemann sphere, along with a family of linear functionals on subalgebras, satisfying a set of properties including conformal invariance and a type of restriction. This embodies some expected properties of expectation values in conformal loop ensembles CLEκ (at least for 8/3 < κ ≤ 4). In the context of conformal restriction systems, we study certain algebra elements associated with hypotrochoid simple curves (a family of curves including the ellipse). These have the CLE interpretation of being ‘renormalized random variables’ that are nonzero only if there is at least one loop of hypotrochoid shape. Each curve has a center w, a scale ϵ and a rotation angle θ, and we analyze the renormalized random variable as a function of u = ϵeiθ and w. We find that it has an expansion in positive powers of u and , and that the coefficients of pure u () powers are holomorphic in w (). We identify these coefficients (the ‘hypotrochoid fields’) with certain Virasoro descendants of the identity field in conformal field theory, thereby showing that they form part of a vertex operator algebraic structure. This largely generalizes works by the author (in CLE), and the author with his collaborators Riva and Cardy (in SLE8/3 and other restriction measures), where the case of the ellipse, at the order u2, led to the stress–energy tensor of CFT. The derivation uses in an essential way the Virasoro vertex operator algebra structure of conformal derivatives established recently by the author. The results suggest in particular the exact evaluation of CLE expectations of products of hypotrochoid fields as well as nontrivial relations amongst them through the vertex operator algebra, and further shed light onto the relationship between CLE and CFT.

Ising interfaces and free boundary conditions

The authors generalize the result of D. Chelkak et al. [C. R., Math., Acad. Sci. Paris 352, No. 2, 157{161 (2014; Zbl 06265643)] to the case when free boundary conditions enter the picture. The proof is related to the rigorous computation of a (dual) boundary conformal eld theory correlation function, which is obtained by using both resent results concerning the boundary correlation functions of the model and existing SLE results for dual models. The result is applied for the conformal invariance of crossing probabilities. Another application is the proof that the collection of the Ising model interfaces converges to the conformal loop ensemble.

On Conformally Invariant CLE Explorations

We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles with simple loops introduced in earlier papers by Sheffield, and by Sheffield and Werner. One outcome is a conformally invariant way to measure a distance of a CLE(4) loop to the boundary "within" the CLE(4), when one identifies all points of each loop.

From CLE($\kappa$) to SLE($\kappa,\rho$)'s

We show how to connect together the loops of a simple Conformal Loop Ensemble (CLE) in order to construct samples of chordal SLE$_{\kappa}$ processes and their SLE$_{\kappa}(\rho)$ variants, and we discuss some consequences of this construction.

Conformal Loop Ensembles and the Stress–Energy Tensor

We give a construction of the stress-energy tensor of conformal field theory (CFT) as a local "object" in conformal loop ensembles CLE_\kappa, for all values of \kappa in the dilute regime 8/3 < \kappa <= 4 (corresponding to the central charges 0 < c <= 1, and including all CFT minimal models). We provide a quick introduction to CLE, a mathematical theory for random loops in simply connected domains with properties of conformal invariance, developed by Sheffield and Werner (2006). We consider its extension to more general regions of definition, and make various hypotheses that are needed for our construction and expected to hold for CLE in the dilute regime. Using this, we identify the stress-energy tensor in the context of CLE. This is done by deriving its associated conformal Ward identities for single insertions in CLE probability functions, along with the appropriate boundary conditions on simply connected domains; its properties under conformal maps, involving the Schwarzian derivative; and its one-point average in terms of the "relative partition function." Part of the construction is in the same spirit as, but widely generalizes, that found in the context of SLE_{8/3} by the author, Riva and Cardy (2006), which only dealt with the case of zero central charge in simply connected hyperbolic regions. We do not use the explicit construction of the CLE probability measure, but only its defining and expected general properties.

Conformal loop ensembles: the Markovian characterization and the loop-soup construction

For random collections of self-avoiding loops in two-dimensional domains, we dene a simple and natural conformal restriction property that is conjecturally satised 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 at rst glance to be weaker. Our rst main result is that there exists exactly a one-dimensional family of random loop collections with this property | one for each 2 (8=3; 4] | and that the loops are forms of SLE . The proof proceeds in two steps. First, uniqueness is established by showing that every such loop ensemble can be generated by an \exploration process based on SLE. Second, existence is obtained using the two-dimensional Brownian loopsoup, which is a Poissonian random collection of loops in a planar domain. When the intensity parameterc of the loop-soup is less 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 the conformal restriction axioms. We prove various results about loop-soups, cluster sizes, and the c = 1 phase transition. Taken together, our results imply that the following families are equiv

Calculus on manifolds of conformal maps and CFT

In conformal field theory (CFT) on simply connected domains of the Riemann sphere, the natural conformal symmetries under self-maps are extended, in a certain way, to local symmetries under general conformal maps, and this is at the basis of the powerful techniques of CFT. Conformal maps of simply connected domains naturally have the structure of an infinite-dimensional groupoid, which generalizes the finite-dimensional group of self-maps. We put a topological structure on the space of conformal maps on simply connected domains, which makes it into a topological groupoid. Further, we (almost) extend this to a local manifold structure based on the infinite-dimensional Fréchet topological vector space of holomorphic functions on a given domain A. From this, we develop the notion of conformal A-differentiability at the identity. Our main conclusion is that quadratic differentials characterizing cotangent elements on the local manifold enjoy properties similar to those of the holomorphic stress–energy tensor of CFT; these properties underpin the local symmetries of CFT. Applying the general formalism to CFT correlation functions, we show that the stress–energy tensor is exactly such a quadratic differential. This is at the basis of constructing the stress–energy tensor in conformal loop ensembles. It also clarifies the relation between Cardy's boundary conditions for CFT on simply connected domains, and the expression of the stress–energy tensor in terms of metric variations.

Geometric Exponents of Dilute Loop Models

The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is $\beta=-2 \cos(\pi/\bar{\kappa})$ where the parameter $\bar{\kappa}$ is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges $c(\bar{\kappa})=13-6(\bar{\kappa}+\bar{\kappa}^{-1})$ and, for the minimal models of interest here, $\bar{\kappa}=p/p'$ where p and p′ are two coprime integers. The geometric exponents of the hull and external perimeter are studied for the pairs (p,p′)=(1,1),(2,3),(3,4),(4,5),(5,6),(5,7), and that of the red bonds for (p,p′)=(3,4). Monte Carlo upgrades are proposed for these models as well as several techniques to improve their speeds. The measured fractal dimensions are obtained by extrapolation on the lattice size H,V→∞. The extrapolating curves have large slopes; despite these, the measured dimensions coincide with theoretical predictions up to three or four digits. In some cases, the theoretical values lie slightly outside the confidence intervals; explanations of these small discrepancies are proposed.

Random soups, carpets and fractal dimensions

We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are deterministic and equal to their expectation dimension. We also estimate their low-intensity limiting behavior. This applies in particular to the "conformal loop ensembles" defined via Poissonian clouds of Brownian loops for which the expectation dimension has been computed by Schramm, Sheffield and Wilson.

Poisson point processes, excursions and stable processes in two-dimensional structures

Itô’s contributions lie at the root of stochastic calculus and of the theory of excursions. These ideas are also very useful in the study of conformally invariant two-dimensional structures, via conformal loop ensembles, excursions of Schramm–Loewner evolutions and Poisson point processes of Brownian loops.

Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus

A Fortuin-Kasteleyn cluster on a torus is said to be of type {a,b},a,b in Z , if it is possible to draw a curve belonging to the cluster that winds a times around the first cycle of the torus as it winds -b times around the second. Even though the Q -Potts models make sense only for Q integers, they can be included into a family of models parametrized by beta = square root of Q for which the Fortuin-Kasteleyn clusters can be defined for any real beta(0,2] . For this family, we study the probability pi({a,b}) of a given type of clusters as a function of the torus modular parameter tau=tau(r)+itau(i). We compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin. For example, the behavior of pi({1,0}) is studied for tau(i) --> infinity . Exponents describing these behaviors are defined and related to weights h(r,s) of the extended Kac table for r and s integers, but also half-integers. Numerical simulations are also presented. Possible relationship with recent works and conformal loop ensembles is discussed.