On the Uniqueness of Global Multiple SLEs

On the Uniqueness of Global Multiple SLEs

Abstract

This article focuses on the characterization of global multiple Schramm-Loewner evolutions (SLE). The chordal SLE process describes the scaling limit of a single interface in various critical lattice models with Dobrushin boundary conditions, and similarly, global multiple SLEs describe scaling limits of collections of interfaces in critical lattice models with alternating boundary conditions. In this article, we give a minimal amount of characterizing properties for the global multiple SLEs: we prove that there exists a unique probability measure on collections of pairwise disjoint continuous simple curves with a certain conditional law property. As a consequence, we obtain the convergence of multiple interfaces in the critical Ising and FK-Ising models.



1 Introduction

At the turn of the millennium, O. Schramm introduced random fractal curves in the plane which he called “stochastic Loewner evolutions” ([Sch00, RS05], and which have since then been known as Schramm-Loewner evolutions. He proved that these probability measures on curves are the unique ones that enjoy the following two properties: their law is conformally invariant, and, viewed as growth processes (via Loewner’s theory), they have the domain Markov property — a memorylessness property of the growing curve. These properties are natural from the physics point of view, and in many cases, it has been verified that interfaces in critical planar lattice models of statistical physics converge in the scaling limit to type curves, see [Smi01, LSW04, CN07, SS09, CS12, CDCH14] for examples.

In the chordal case, there is a one-parameter family of such curves, parameterized by a real number that is believed to be related to universality classes of the critical models, and the central charges of the corresponding conformal field theories. In this article, we consider several interacting curves, multiple s. We prove that when , there exists a unique multiple measure on families of curves with a given connectivity pattern, as detailed in Theorem 1.2. Such measures have been considered in many works [BBK05, Dub07, Gra07, KL07, Law09], but we have not found a conceptual approach in the literature, in terms of a minimal number of characterizing properties in the spirit of Schramm’s classification.

Results of convergence of a single discrete interface to an curve in the scaling limit are all rather involved. On the other hand, after our characterization of the multiple s, it is relatively straightforward to extend these convergence results to prove that multiple interfaces also converge to the multiple . Indeed, the relative compactness of the interfaces in a suitable topology can be verified with little effort using results in [CDCH16, DCST17, KS17], and the main problem is then to identify the limit (i.e., to prove that the subsequential limits are given by a unique collection of random curves).

As an application, we prove that the chordal interfaces in the critical Ising model with alternating boundary conditions converge to the multiple with parameter , in the sense detailed in Sections 1.2 and 4.1. In contrast to the previous work [Izy17] of K. Izyurov, we work on the global collection of curves and condition on the event that the interfaces form a given connectivity pattern — see also Figure 1.1 for an illustration. We also identify the marginal law of one curve in the scaling limit as a weighted chordal . As an input in our proof, together with results from [CDCH16, DCST17, KS17] for the relative compactness, we also use the convergence of a single critical Ising interface to the chordal from [CS12, CDCH14].

The explicit construction of the global multiple s given in [KL07, Law09, PW17] and in Section 3 of the present article fails for . Nevertheless, we discuss in Section 4 how, using information from discrete models, one could extend the classification of the multiple s of our Theorem 1.2 to the range . More precisely, we prove that the convergence of a single interface in the critical random-cluster model combined with relative compactness implies the existence and uniqueness of a multiple , where is related to the cluster weight by Equation (4.7). In the special case of the FK-Ising model (), using the results of [CS12, CDCH14, CDCH16, DCST17, KS17], we obtain the convergence of any number of chordal interfaces to the unique multiple . However, for general , this result remains conditional on the convergence of a single interface (we note that the case corresponds to critical percolation, where the convergence is also known [Smi01, CN07]).

1.1 Global Multiple SLEs

Throughout, we let denote a simply connected domain with distinct points appearing in counterclockwise order along the boundary. We call the -tuple a (topological) polygon. We consider curves in each of which connects two points among . These curves can have various planar (i.e., non-crossing) connectivities. We describe the connectivities by planar pair partitions , where . We call such (planar) link patterns and we denote the set of them by . Given a link pattern and , we denote by the link pattern in obtained by removing from and then relabeling the remaining indices so that they are the first integers.

We let denote the set of continuous simple unparameterized curves in connecting and such that they only touch the boundary in . When , the chordal curve belongs to this space almost surely. Also, when , we let denote the set of families of pairwise disjoint curves, where for all .

Definition 1.1.

Let . For and for any link pattern , we call a probability measure on the families a global - associated to if, for each , the conditional law of the curve given is the chordal connecting and in the component of the domain containing the endpoints and of on its boundary.

Theorem 1.2.

Let and let be a polygon with . For any , there exists a unique global - associated to .

The existence part of Theorem 1.2 is already known — see [KL07, Law09, PW17]. We briefly review the construction in Section 3.1. The uniqueness part of Theorem 1.2 for was proved in [MS16b, Theorem 4.1], where the authors used a coupling of the and the Gaussian free field. Unfortunately, this proof does not apply in general for commuting s, which is the case of the present article. We first give a different proof for the uniqueness when by a Markov chain argument (in Section 3.2), and then generalize it for all (in Section 3.3).

We note that it follows immediately from Definition 1.1 that global multiple s have the following cascade property. Suppose has the law of a global multiple - associated to the link pattern . Assume also that for some . Then, the conditional law of given is a global - associated to .

1.2 Multiple Interfaces in Critical Planar Ising Model

\floatbox

[\capbeside\thisfloatsetupcapbesideposition=right,center,capbesidewidth=0.5]figure[\FBwidth]

Figure 1.1: Simulation of the critical Ising model with alternating boundary conditions. There are eight marked points on the boundary of the polygon and therefore, four interfaces connecting the marked points pairwise. We only illustrate one possible connectivity of the curves (the reader may verify that there are different topological possibilities).

Next, we consider critical Ising interfaces in the scaling limit. Assuming that is bounded, we let discrete domains on the square lattice approximate as (we will provide the details of the approximation scheme in Section 4), and we consider the critical Ising model (which we also define in Section 4) on with the following alternating boundary conditions (see Figure 1.1):

(1.1)

with the convention that and . Then, interfaces connect the boundary points , forming a planar connectivity encoded in a link pattern .

To understand the scaling limit of the interfaces, we must specify the topology in which the convergence of the curves as occurs. Thus, we let denote the set of planar oriented curves, that is, continuous mappings from to modulo reparameterization. More precisely, we equip with the metric

(1.2)

where the infimum is taken over all increasing homeomorphisms . Then, the metric space is complete and separable.

{restatable}

propositionisingmultiplecvg Let . Then, as , conditionally on the event , the law of the collection of critical Ising interfaces converges weakly to the global - associated to . In particular, as , the law of a single curve in this collection connecting two points and converges weakly to a conformal image of the Loewner chain with driving function given by Equation (3.16) in the end of Section 3, with .

We prove Proposition 1.2 in Section 4.1, where we also define the Ising model and discuss some of its main features. The key ingredients in the proof are [CS12, CDCH14, KS17] and Theorem 1.2. In addition, we also need sufficient control on six-arm events and an appropriate Russo-Seymour-Welsh bound proved in [CDCH16] in order to rule out certain undesired behavior of the interfaces.

1.3 Multiple Interfaces in Critical Planar FK-Ising Model

Finally, we consider critical FK-Ising interfaces in the scaling limit. More generally, in Section 4 we study the random-cluster model, whose interfaces conjecturally converge to curves with . We define these models in Section 4.2. We consider the critical FK-Ising model in with the following alternating boundary conditions (see Figure 4.2):

(1.3)

with the convention that and . As in the case of the Ising model, interfaces connect the boundary points , forming a planar connectivity encoded in a link pattern . However, this time the scaling limits are not simple curves, and we need to extend the definition of a global multiple to include the range . For this, we let denote the closure of the space in the metric topology of . Note that the curves in may have multiple points but no self-crossings. In particular, the chordal curve belongs to this space almost surely for all .

Now, for and , we denote by the collection of curves such that, for each , we have and has a positive distance from the arcs and . Note that is not complete. In Definition 1.1, the global - was defined for  — we can now extend this definition to all by replacing with in Definition 1.1. We remark that this definition would still formally make sense in the range , but since for such values of , the process is described by a Peano curve, uniqueness of multiple clearly fails, as one can specify the common boundaries of the different curves in an arbitrary way while preserving the conditional distributions of individual curves.

Proposition 1.3.

Theorem 1.2 also holds for , and for any , as , conditionally on the event , the law of the collection of critical FK-Ising interfaces converges weakly to the global - associated to .

We prove Proposition 1.3 in Section 4.3. To show that the scaling limit is a global multiple , we again use results from the literature [CS12, CDCH14, KS17] combined with a Russo-Seymour-Welsh bound proved in [DCST17] and six-arm estimates. To prove the uniqueness of the limit, we use a Markov chain argument similar to the proof of Theorem 1.2.

Remark 1.4.

Combining the same argument that we use in Section 4.3 with the results of [Smi01, CN07] one can check that there also exists a unique global multiple for with any given connectivity pattern, and Proposition 1.3 holds for the critical site percolation on the triangular lattice with .


Acknowledgments. We thank C. Garban, K. Izyurov, A. Kemppainen, and F. Viklund for interesting and useful discussions. VB is supported by the ANR project ANR-16-CE40-0016. During this work, EP and HW were supported by the ERC AG COMPASP, the NCCR SwissMAP, and the Swiss NSF, and HW was also later supported by the startup funding no. 042-53331001017 of Tsinghua University. The first version of this paper was completed while EP and HW visited MFO as “research in pair”.

2 Preliminaries

In this section, we give some preliminary results, for use in the subsequent sections. In Section 2.1, we discuss Brownian excursions and the Brownian loop measure. These concepts are needed frequently in this article. In Sections 2.2 and 2.3, we define the chordal and study its relationships in different domains via so-called boundary perturbation properties. Then, in Section 2.4, we give a crucial coupling result for s in different domains. This coupling is needed in the proof of Theorem 1.2 in Section 3.

2.1 Brownian Excursions and Brownian Loop Measure

We call a polygon with two marked points a Dobrushin domain. Also, if is a simply connected subdomain that agrees with in neighborhoods of and , we say that is a Dobrushin subdomain of . For a Dobrushin domain , the Brownian excursion measure is a conformally invariant measure on Brownian excursions in with its two endpoints in the arc — see [LW04, Section 3] for definitions. It is a -finite infinite measure, with the following restriction property: for any Dobrushin subdomain , we have

(2.1)

For , we call a Poisson point process with intensity a Brownian excursion soup.

Suppose that and lie on analytic boundary segments of . Then, the boundary Poisson kernel is a conformally invariant function which in the upper-half plane with is given by

(2.2)

(we do not include here), and in it is defined via conformal covariance: we set

(2.3)

for any conformal map .

Lemma 2.1.

Let be a Dobrushin domain with on analytic boundary segments. Let be two Dobrushin subdomains that agree with in a neighborhood of the arc . Then we have

(2.4)
(2.5)
Proof.

The inequality (2.4) follows from (2.3). To prove (2.5), let be a Brownian excursion soup with intensity . Then we have

(2.6)

Now, denote by the collection of excursions in that are contained in . By (2.1), we know that is a Brownian excursion soup with intensity . Equation (2.5) now follows from

The Brownian loop measure is a conformally invariant measure on unrooted Brownian loops in — see, e.g., [LW04, Sections 3 and 4] for the definition. It is a -finite infinite measure, which has the following restriction property: for any subdomain , we have

For , we call a Poisson point process with intensity a Brownian loop soup. This notion will be needed in Section 2.4.

Given two disjoint subsets , we denote by the Brownian loop measure of loops in that intersect both and . In other words,

If are at positive distance from each other, both of them are closed, and at least one of them is compact, then we have . Furthermore, the measure is conformally invariant: for any conformal transformation , we have .

Also, for disjoint subsets of , we denote by the Brownian loop measure of loops in that intersect all of . Again, provided that are closed and at least one of them is compact, is finite. This quantity will be needed in Section 3.

2.2 Loewner Chains and SLE

An -hull is a compact subset of such that is simply connected. By Riemann’s mapping theorem, for any hull , there exists a unique conformal map from onto such that . Such a map is called the conformal map from onto normalized at . By standard estimates of conformal maps, the derivative of this map satisfies

(2.7)

In fact, this derivative can be viewed as the probability that the Brownian excursion in from to avoids the hull — see [Vir03, LSW03].

Consider a family of conformal maps which solve the Loewner equation: for each ,

where is a real-valued continuous function, which we call the driving function.

Denote , where is the swallowing time of the point . Then, is the unique conformal map from onto normalized at . The collection of -hulls associated with such maps is called a Loewner chain. We say that is generated by the continuous curve if for any , the unbounded component of coincides with .

In this article, we are concerned with particular hulls generated by curves. For , the random Loewner chain driven by , where is a standard Brownian motion, is called the (chordal) Schramm-Loewner Evolution in from to . S. Rohde and O. Schramm proved in [RS05] that this Loewner chain is almost surely generated by a continuous transient curve , with . This random curve exhibits the following phase transitions in the parameter : when , it is a simple curve; whereas when , it has self-touchings, being space-filling if . The law of the curve is a probability measure on the space , and we denote it by .

By conformal invariance, we can define the probability measure in any simply connected domain with two marked boundary points (around which is locally connected) via pushforward of a conformal map: if , then we have , where is any conformal map such that and . In fact, by the results of O. Schramm [Sch00], the are the only probability measures on curves satisfying conformal invariance and the following domain Markov property: given an initial segment of the curve up to a stopping time , the conditional law of the remaining piece is the law of the in the complement of the hull of the initial segment, from the tip to .

2.3 Boundary Perturbation of SLE

The chordal curve has a natural boundary perturbation property, where its law in a Dobrushin subdomain of is given by weighting by a factor involving the Brownian loop measure and the boundary Poisson kernel. More precisely, when , the is a simple curve only touching the boundary at its endpoints, and its law in the subdomain is absolutely continuous with respect to its law in , as we state in the next Lemma 2.2. However, for , we cannot have such absolute continuity since the has a positive chance to hit the boundary of . Nevertheless, in Lemma 2.3 we show that if we restrict the two processes in a smaller domain, then we retain the absolute continuity for .

Throughout this article, we use the following real parameters:

(2.8)
Lemma 2.2.

Let . Let be a Dobrushin domain and a Dobrushin subdomain. Then, the in connecting and is absolutely continuous with respect to the in connecting and , with Radon-Nikodym derivative given by

Proof.

See [LSW03, Section 5] and [KL07, Proposition 3.1]. ∎

The next lemma is a consequence of results in [LSW03, LW04]. We briefly summarize the proof.

Lemma 2.3.

Let . Let be a Dobrushin domain. Let be Dobrushin subdomains such that and agree in a neighborhood of the arc and . Then, we have

Proof.

By conformal invariance, we may assume that . Let , let be its driving function, and the corresponding conformal maps. Let be the conformal map from onto normalized at . On the event , define to be the first time that disconnects from .

Denote by the hull of . For , let be the conformal map from onto normalized at , and let be the conformal map from onto normalized at . Then we have . Now, define for ,

where is the Schwarzian derivative1. It was proved in [LSW03, Proposition 5.3] that is a local martingale. Furthermore, using Itô’s formula, one can show that the law of weighted by is up to time . Also, it follows from [Law05, Proposition 5.22] (see also [LW04, Section 7]) that

Now, on the event , there exists such that for , we have . When , we have and , and thus, on the event , we have . When , we have and and in this case, . In either case, is uniformly bounded on the event , and as , we have almost surely and thus,

The assertion follows taking into account that and recalling (2.3). ∎

2.4 A Crucial Coupling Result for SLEs

We finish this preliminary section with a result from [WW13], which says that we can construct s using the Brownian loop soup and the Brownian excursion soup. This gives us a coupling of s in two Dobrushin domains , which will be crucial in our proof of Theorem 1.2.

Let be a Dobrushin domain. Let be a Brownian loop soup with intensity , and a Brownian excursion soup with intensity , with and defined in (2.8) and . We note that then we have and .

We say that two loops and in are in the same cluster if there exists a finite chain of loops in such that , , and for . We denote by the family of all closures of the loop-clusters and by the family of all outer boundaries of the outermost elements of . Then, forms a collection of disjoint simple loops called the for , see [SW12].

Finally, define to be the right boundary of the union of all excursions and to be the boundary of the union of and all loops in that it intersects, as illustrated in Figure 2.1.

Figure 2.1: In the left panel, is the right boundary of all Brownian excursions in . In the middle panel, is the family of all outer boundaries of the outermost elements of the clusters of Brownian loops in . In the right panel, is the right boundary of the union of and all loops in that intersect . By [WW13, Theorem 1.1], we find that .
Lemma 2.4.

[WW13, Theorem 1.1]. Let . Let be a Dobrushin domain and define , , , , and as above. Then, has the law of the in connecting and .

From Lemma 2.4, we see that we can couple in different domains in the following way. Let be a Dobrushin domain and a Dobrushin subdomain that agrees with in a neighborhood of the arc . Take , , , , and as in the above lemma. Let and respectively be the collections of excursions in and loops in that are contained in . Define to be the right boundary of the union of all excursions , define to be the collection of all outer boundaries of the outermost clusters of , and to be the right boundary of the union of and all loops in that it intersects.

Corollary 2.5.

Let be a Dobrushin domain and a Dobrushin subdomain that agrees with in a neighborhood of the arc . There exists a coupling of and such that, almost surely, stays to the left of and

Proof.

Lemma 2.4 and the above paragraph give the sought coupling. ∎

In fact, the coupling of Corollary 2.5 is the coupling which maximizes the probability .

3 Global Multiple SLEs

Global -s associated to all link patterns and all were constructed in the works [KL07, PW17]. This immediately gives the existence part of Theorem 1.2. In Section 3.1, we briefly recall the main idea of this construction. Then we prove the uniqueness part of Theorem 1.2 in Sections 3.2 and 3.3.

3.1 Construction of Global Multiple SLEs for

Let be a polygon. For a link pattern , we let denote the product measure of independent chordal curves,

and denote the expectation with respect to . A global - associated to can be constructed as the probability measure which is absolutely continuous with respect to , with explicit Radon-Nikodym derivative given in Equation (3.2) below. This formula involves a combinatorial expression of Brownian loop measures, obtained by an inclusion-exclusion procedure that depends on . More precisely, for a configuration , we define

(3.1)

where the sum is over all the connected components (c.c.) of the complement of the curves, and

is a combinatorial expression associated to the c.c. , where

denotes the set of indices for which the curve is a part of the boundary of .

Now, we define the probability measure via

(3.2)

By [PW17, Proposition 3.3], this measure satisfies the defining property of a global multiple . The expectation of defines a conformally invariant and bounded function of the marked boundary points:

(3.3)

If is a polygon and a simply connected subdomain that agrees with in neighborhoods of , we say that is a sub-polygon of . When the marked points lie on analytic boundary segments of , for all integers and link patterns , we may define

(3.4)

where is the boundary Poisson kernel introduced in Section 2.1. Since , we see that

(3.5)

The functions are called pure partition functions for multiple s. Explicit formulas for them have been obtained when  [KKP17, Theorem 4.1] and  [PW17, Theorem 1.5].

The multiple probability measure has a useful boundary perturbation property. It serves as an analogue of Lemma 2.2 in our proof of Theorem 1.2.

Proposition 3.1.

[PW17, Proposition 3.4] Let . Let be a polygon and a sub-polygon. Then, is absolutely continuous with respect to , with Radon-Nikodym derivative given by

Moreover, when and lie on analytic boundary segments of , we have

(3.6)

3.2 Uniqueness for a Pair of Commuting SLEs

Next, we prove that the global - measures are unique. This result was first proved by J. Miller and S. Sheffield [MS16b, Theorem 4.1] by using a coupling of the s with the Gaussian free field (). We present another proof not using this coupling. Our proof also generalizes to the case of commuting curves, whereas couplings with the seem not to be useful in that case.

In this section, we focus on polygons with . We call such a polygon a quad. We also say that is a sub-quad of if is a sub-polygon of .

Because the two connectivities of the curves are obtained from each other by a cyclic change of labeling of the marked boundary points, we may without loss of generality consider global -s associated to . Hence, throughout this section, we consider pairs of simple curves such that , and , and . We denote the space of such pairs by Now, a probability measure supported on these pairs is a global - if the conditional law of given is that of the connecting and in the connected component of containing and on its boundary, and vice versa with and interchanged.

Proposition 3.2.

For any , there exists a unique global - on .

We prove Proposition 3.2 in the end of this section, after some technical lemmas. The idea is to show that the global - is the unique stationary measure of a Markov chain which at each discrete time resamples one of the two curves according to its conditional law given the other one. We have already seen a construction of such a measure in the previous section, so we only need to prove that there exists at most one stationary measure. To this end, we use couplings of Markov chains — see e.g. [MT09] for a general background.

The next lemma is crucial in our proof. In this lemma, we prove that the chordal in always has a uniformly positive probability of staying in a subdomain of in the following sense.

Lemma 3.3.

Let and let be a Dobrushin domain. Let be Dobrushin subdomains such that , , and agree in a neighborhood of the arc . Suppose . Then, there exists a constant independent of such that .

Proof.

We prove the lemma separately for and .

When