Conformal Loop Ensembles Research Articles

The fuzzy Potts model in the plane: scaling limits and arm exponents

AbstractWe consider a critical Fortuin–Kasteleyn (FK) percolation with cluster weight $$q \in [1,4)$$ q ∈ [ 1 , 4 ) in the plane, and color its clusters in red (respectively blue) with probability $$r \in (0,1)$$ r ∈ ( 0 , 1 ) (respectively $$1-r$$ 1 - r ), independently of each other. We study the resulting fuzzy Potts model, which corresponds to the critical Ising model in the special case $$q=2$$ q = 2 and $$r=1/2$$ r = 1 / 2 . We show that under the assumption that the critical FK percolation converges to a conformally invariant scaling limit (which is known to hold for the FK-Ising model,i.e. $$q=2$$ q = 2 ), the obtained coloring converges to variants of Conformal Loop Ensembles constructed, described and studied by Miller, Sheffield and Werner. Based on discrete considerations, we also show that the arm exponents for this coloring in the discrete model are identical to the ones of the continuum model. Using the values of these arm exponents in the continuum, we determine the arm exponents for the fuzzy Potts model.

Liouville quantum gravity weighted by conformal loop ensemble nesting statistics

Liouville quantum gravity weighted by conformal loop ensemble nesting statistics

Conformal uniformization of planar packings by disk packings

A Sierpiński packing in the 2-sphere is a countable collection of disjoint, non-separating continua with diameters shrinking to zero. We show that any Sierpiński packing by continua whose diameters are square-summable can be uniformized by a disk packing with a packing-conformal map, a notion that generalizes conformality in open sets. Being special cases of Sierpiński packings, Sierpiński carpets and some domains can be uniformized by disk packings as well. As a corollary of the main result, the conformal loop ensemble (CLE) carpets can be uniformized conformally by disk packings, answering a question of Rohde–Werness.

The SLE loop via conformal welding of quantum disks

We prove that the SLEκ loop measure arises naturally from the conformal welding of two γ-Liouville quantum gravity (LQG) disks for γ2=κ∈(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 SLE8∕3 loop. Our result is also a key input to recent work of the first and third coauthors on the integrability of the conformal loop ensemble.

Simple conformal loop ensembles on Liouville quantum gravity

We show that, when one draws a simple conformal loop ensemble (CLEκ for κ∈(8/3,4)) on an independent κ-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corresponding to the interior of the CLE loops) that are cut out form a Poisson point process of quantum disks. This construction allows us to make direct links between CLE on LQG, (4/κ)-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be −cos(4π/κ) which can be interpreted as a “density” of CLE loops in the CLE on LQG setting. Positive jumps correspond to the discovery of a CLE loop (where the LQG length of the loop is given by the jump size) and negative jumps correspond to the moments where the discovery process splits the remaining to be discovered domain into two pieces. Some consequences of this result are the following: (i) It provides a construction of a CLE on LQG as a patchwork/welding of quantum disks. (ii) It allows us to construct the “natural quantum measure” that lives in a CLE carpet. (iii) It enables us to derive some new properties and formulas for SLE processes and CLE themselves (without LQG) such as the exact distribution of the trunk of the general SLEκ(κ−6) processes. The present work deals directly with structures in the continuum and makes no reference to discrete models, but our calculations match those for scaling limits of O(N) models on planar maps with large faces and CLE on LQG. Indeed, our Lévy-tree descriptions are exactly the ones that appear in the study of the large-scale limit of peeling of discrete decorated planar maps, such as in recent work of Bertoin, Budd, Curien and Kortchemski. The case of nonsimple CLEs on LQG is studied in another paper.

On the scaling behaviour and conformal properties of triangular Ising model with three-spin interactions at the critical point

The scaling behaviour and conformal properties of a triangular Ising model with three-spin interactions at the critical point are studied using extensive Monte Carlo simulations. We show that in the continuum limit the external perimeter of spin clusters are conformally invariant and statistically equivalent to Schramm–Loewner evolution (SLE κ ) curves with diffusivity κ = 4, indicating that the loop ensembles associated with the critical Ising model with three-spin interactions, belonging to the universality class of domain walls in the four-states Potts model. The left-passage probabilities, distribution of the winding angle, and the driving function distribution for the interfaces are consistent with each other. We also study many geometrical properties of the critical interfaces to characterize the fractal structure and the other scaling properties of conformal loop ensembles. To this end, the fractal dimension of contours, the scaling properties of length and areas distributions and Zipf’s laws, are demonstrated numerically. Our results are compatible with the hyper-scaling relation for spin models.

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble hbox {CLE}_{kappa '} for kappa ' in (4, 8) that is drawn on an independent gamma -LQG surface for gamma ^2=16/kappa '. The results are similar in flavor to the ones from our companion paper dealing with hbox {CLE}_{kappa } for kappa in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the hbox {CLE}_{kappa '} in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a hbox {CLE}_{kappa '} independently into two colors with respective probabilities p and 1-p. This description was complete up to one missing parameter rho . The results of the present paper about CLE on LQG allow us to determine its value in terms of p and kappa '. It shows in particular that hbox {CLE}_{kappa '} and hbox {CLE}_{16/kappa '} are related via a continuum analog of the Edwards-Sokal coupling between hbox {FK}_q percolation and the q-state Potts model (which makes sense even for non-integer q between 1 and 4) if and only if q=4cos ^2(4pi / kappa '). This provides further evidence for the long-standing belief that hbox {CLE}_{kappa '} and hbox {CLE}_{16/kappa '} represent the scaling limits of hbox {FK}_q percolation and the q-Potts model when q and kappa ' are related in this way. Another consequence of the formula for rho (p,kappa ') is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

Duality of random planar maps via percolation

We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\alpha\in(1,2]$. We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes $\alpha \in (3/2,2]$, and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime $\alpha \in (1,3/2)$, with parameter \[\alpha':= \frac{2\alpha+3}{4\alpha-2}.\] This is the counterpart in random planar maps of the duality property $\kappa \leftrightarrow 16/\kappa$ of Schramm--Loewner Evolutions and Conformal Loop Ensembles, recently established by Miller, Sheffield and Werner. As a byproduct, we identify the scaling limit of the boundary of the percolation cluster conditioned to have a large perimeter. The cases of subcritical and supercritical percolation are also discussed. In particular, we establish the sharpness of the phase transition through the tail distribution of the size of the percolation cluster.

Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)

Building upon recent results of Dubédat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations \Omega^\delta to a simply connected domain \Omega\subset\mathbb C we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on \Omega^\delta as \delta\to 0 . More precisely, let \lambda_1,\dots,\lambda_n\in\Omega and L be a macroscopic lamination on \Omega\setminus\{\lambda_1,\dots,\lambda_n\} , i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities P_L^\delta that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on \Omega^\delta converge to a conformally invariant limit P_L as \delta \to 0 , for each L . Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety Hom (\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C) and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do not use any RSW-type arguments for double-dimers. The limits P_L of the probabilities P_L^\delta are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock–Goncharov lamination basis on the representation variety. The fact that P_L coincides with the probability of obtaining L from a sample of the nested CLE(4) in \Omega requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.

Conformal Invariance of CLEΚ on the Riemann Sphere for Κ ∈(4,8)

Abstract The conformal loop ensemble (${\textrm{CLE}}$) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in $\mathbbm{C}$ and is indexed by a parameter $\kappa \in (8/3,8)$. We consider ${\textrm{CLE}}_\kappa $ on the whole-plane in the regime in which the loops are self-intersecting ($\kappa \in (4,8)$) and show that it is invariant under the inversion map $z \mapsto 1/z$. This shows that whole-plane ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ defines a conformally invariant measure on loops on the Riemann sphere. The analogous statement in the regime in which the loops are simple ($\kappa \in (8/3,4]$) was proven by Kemppainen and Werner and together with the present work covers the entire range $\kappa \in (8/3,8)$ for which ${\textrm{CLE}}_\kappa $ is defined. As an intermediate step in the proof, we show that ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ on an annulus, with any specified number of inner-boundary-surrounding loops, is well defined and conformally invariant.

Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model

AbstractWe consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two‐point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near‐critical scaling limit Euclidean magnetization field. For the lattice model with a = 1, the mass (inverse correlation length) is of order h8/15 as h ↓ 0; for the Euclidean field, it equals exactly Ch8/15 for some C. Although there has been much progress in the study of critical scaling limits, results on near‐critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near‐critical scaling limits. Thus we provide the first substantial application of measure ensembles. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Non-simple SLE curves are not determined by their range

We show that when observing the range of a chordal SLE _\kappa curve for \kappa \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 _\kappa for \kappa \in (4,8) are not determined by the CLE _\kappa gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE _\kappa for \kappa \in (8/3,4) (we defined these percolation interfaces in an earlier paper, and showed there that they are SLE _{16/\kappa} curves) are not determined by the CLE _\kappa carpet that they are defined in.

Conditioning a Brownian loop-soup cluster on a portion of its boundary

Dans le present article, nous etudions certaines proprietes des amas de lacets browniens, dans une soupe de lacets browniens d’intensite $c$ dans un domaine du plan, pour toute intensite $c\le 1$. Notre principal resultat dit que si l’on decouvre de maniere Markovienne une portion $\partial $ du bord exterieur d’un tel amas, alors dans le domaine restant, la loi conditionnelle de l’union de tous les lacets dans $L$ qui touchent $\partial $ satisfait la propriete de restriction conforme tandis que les autres lacets dans $L$ forment une soupe de lacets independante. Ceci implique en particulier l’existence d’une transition de phase a $c=14/15$ pour la connectivite de l’ensemble des lacets qui touchent $\partial $. Nos resultats constituent une extension de certains resultats de notre papier (J. Eur. Math. Soc. (2019) to appear) dans les deux directions suivantes: Dans (J. Eur. Math. Soc. (2019) to appear), un cluster de lacets est conditionne par son bord exterieur entier tandis que nous decouvrons ici seulement une partie de ce bord. En outre, dans (J. Eur. Math. Soc. (2019) to appear), nous expliquons que la description que nous donnons de l’ensemble des lacets qui touchent ce bord via un processus ponctuel de Poisson d’excursions est specifique au cas de la soupe de lacets critique ($c=1$), nous montrons ici que dans les cas sous-critiques $c<1$, une propriete plus faible de restriction conforme reste neanmoins vraie.

Convergence of the two-dimensional random walk loop-soup clusters to CLE

We consider the random walk loop-soup of subcritical intensity parameter on the discrete half-plane \mathtt{H}:=\mathbb{Z}\times\mathbb{N} . We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is a CLE _{\kappa} conformal loop ensemble.

Double dimers, conformal loop ensembles and isomonodromic deformations

The double-dimer model consists in superimposing two independent, identically distributed perfect matchings on a planar graph, which produces an ensemble of non-intersecting loops. In [20], Kenyon established conformal invariance in the small mesh limit by considering topological observables of the model parameterized by SL _2 (\mathbb C) representations of the fundamental group of the punctured domain. The scaling limit is conjectured to be CLE _4 , the Conformal Loop Ensemble at \kappa = 4 [36]. In support of this conjecture, we prove that a large subclass of these topological correlators converge to their putative CLE _4 limit. Both the small mesh limit of the double-dimer correlators and the corresponding CLE _4 correlators are identified in terms of the -functions introduced by Jimbo, Miwa and Ueno [14] in the context of isomonodromic deformations.

Connection Probabilities for Conformal Loop Ensembles

The goal of the present paper is to explain, based on properties of the conformal loop ensembles {{rm CLE}_kappa} (both with simple and non-simple loops, i.e., for the whole range {kappa in (8/3, 8)}), how to derive the connection probabilities in domains with four marked boundary points for a conditioned version of {{rm CLE}_kappa} which can be interpreted as a {{rm CLE}_kappa} with wired/free/wired/free boundary conditions on four boundary arcs (the wired parts being viewed as portions of to-be-completed loops). In particular, in the case of a square, we prove that the probability that the two wired sides of the square hook up so that they create one single loop is equal to {1/(1 - 2 {rm cos} (4 pi / kappa ))} . Comparing this with the corresponding connection probabilities for discrete O(N) models. For instance, indicates that if a dilute O(N) model (respectively a critical FK(q)-percolation model on the square lattice) has a non-trivial conformally invariant scaling limit, then necessarily this scaling limit is {{rm CLE}_kappa} where {kappa} is the value in (8/3, 4] such that {-2 {rm cos} (4 pi / kappa )} is equal to N (resp. the value in [4,8) such that {-2 {rm cos} (4pi / kappa)} is equal to {sqrt q}). On the one hand, Our arguments and computations build on Dubédat’s SLE commutation relations (as developed and used by Dubédat, Zhan or Bauer-Bernard-Kytölä) and on the other hand, on the construction and properties of the conformal loop ensembles and their relation to Brownian loop-soups, restriction measures, and the Gaussian free field (as recently derived in works with Sheffield and with Qian).

Percolation on Triangulations, and a Bijective Path to Liouville Quantum Gravity

We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by certain 2D lattice paths. Our bijection parallels in the discrete setting the \emph{mating-of-trees} framework of LQG and Schramm-Loewner evolutions (SLE) introduced by Duplantier, Miller, and Sheffield. Combining these two correspondences allows us to relate uniform site-percolated triangulations to $\sqrt{8/3}$-LQG and SLE$_6$. In particular, we establish the convergence of several functionals of the percolation model to continuous random objects defined in terms of $\sqrt{8/3}$-LQG and SLE$_6$. For instance, we show that the exploration tree of the percolation converges to a branching SLE$_6$, and that the collection of percolation cycles converges to the conformal loop ensemble CLE$_6$. We also prove convergence of counting measure on the pivotal points of the percolation. Our results play an essential role in several other works, including a program for showing convergence of the conformal structure of uniform triangulations and works which study the behavior of random walk on the uniform infinite planar triangulation.

Coupling the Gaussian Free Fields with Free and with Zero Boundary Conditions via Common Level Lines

We point out a new simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary-touching zero-level lines (these signs are alternating for the zero-boundary GFF) in order to obtain a free boundary GFF. Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are also discussed. Such considerations show for instance that in a domain with an axis of symmetry, if one looks at the overlay of a single usual Conformal Loop Ensemble CLE3 with its own symmetric image, one obtains the CLE4-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain.

Coupling of c = −2 and c and c = 0 conformal field theories: the geometrical point of view

The coupling of the c = −2, and c = 0 conformal field theories are numerically considered in this paper. As the prototypes of the couplings, and , we consider the Bak–Tang–Weisenfeld (BTW) model on the 2D square critical site-percolation and the BTW model on Ising-correlated percolation lattices respectively. Some geometrical techniques are used to characterize the presumable conformal symmetry of the resultant systems. Based on the numerical analysis of the diffusivity parameter (κ) in the Schramm–Loewner evolution (SLE) theory we propose that the algebra of the central charges of the coupled models is closed. This result is based on the analysis of the conformal loop ensemble (CLE) analysis. The diffusivity parameter in each case is obtained by calculating the fractal dimension of loops (and the corresponding exponent of mean-square root distance), the direct SLE mapping method, the left passage probability and the winding angle analysis. More precisely we numerically show that the coupling results to 2D self-avoiding walk (SAW) fixed point corresponding to c = 0 conformal field theory, whereas the coupling results to the 2D critical Ising fixed point corresponding to the conformal field theory.

ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble$\text{CLE}_{4}$and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the$\text{CLE}_{4}$carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of$\text{CLE}_{4}$) are in fact measurable functions of the GFF.