Crossing Probabilities of Multiple Ising Interfaces
We prove that in the scaling limit, the crossing probabilities of multiple interfaces in the critical planar Ising model with alternating boundary conditions are conformally invariant expressions given by the pure partition functions of multiple with .
Keywords: Ising model, Schramm-Loewner evolution (SLE), crossing probability
MSC: 82B20, 60J67, 60K35
- 1 Introduction
- 2 Partition Functions of Multiple SLEs
- 3 Analysis of Multiple SLE Partition Functions for the Ising Model
- 4 Loewner Chains Associated to Partition Functions When
- 5 Crossing Probabilities in Critical Ising Model
- A Appendix: Crossing Probabilities in Critical Percolation
- B Appendix: Connection Probabilities for Level Lines of GFF
- C Appendix: Connection Probabilities for Loop-Erased Random Walks
The planar Ising model is arguably one of the most studied lattice models in statistical physics. It was introduced in the 1920s by W. Lenz as a model for magnetic materials (and could hence be also termed the “Lenz-Ising model”). In the 1920s, this model was further studied by Lenz’s student, E. Ising, who proved that there is no phase transition in dimension one, leading him to conjecture that this is the case also in higher dimensions. However, as R. Peierls showed in 1936, in two dimensions an order-disorder phase transition in fact occurs at some critical temperature, identified by L. Onsager in 1944. During the next decades, renormalization group arguments and the introduction of conformal field theory suggested that, due to its continuous (second-order) phase transition, at criticality the Ising model should enjoy conformal invariance in the scaling limit [BPZ84, Car96]. Ever since, there has been active research to understand the 2D Ising model at criticality, with recent success towards proving the conformal invariance via methods of discrete complex analysis [Smi06, Smi10, CS12, CI13, HS13, CDCH14].
In this article, we consider the critical Ising model on a -scaled square lattice approximation of a simply connected subdomain of the plane, with small. We impose alternating boundary conditions, that is, we divide the boundary into segments, of which have spin and spin , and the different type segments alternate — see Section 5 for details. With such boundary conditions, macroscopic interfaces connect pairwise the marked boundary points, as illustrated in Figure 1.1. Supplementing the results of [CDCH14], it was proved in [Izy17, BPW18] that these interfaces converge in the scaling limit to a multiple Schramm-Loewner evolution, -, with .
The interfaces can form a Catalan number of possible planar connectivities, as also illustrated in Figure 1.1. We label them by (planar) link patterns , where , and we denote by the set of the link patterns of links . We also denote by the planar connectivity formed by the discrete interfaces. We are interested in the crossing probabilities with , as functions of the marked boundary points.
In this article, we prove that for any number of marked boundary points and any link pattern , the crossing probability converges in the scaling limit to a conformally invariant function expressed in terms of the pure partition functions of multiple . Furthermore, we find that the scaling limit of the interfaces is in fact the global multiple which is the sum of the extremal multiple probability measures associated to the various possible connectivity patterns of the interfaces.
To state our main result, we fix a simply connected domain and distinct boundary points in counterclockwise order, lying on locally connected boundary segments. We call a (topological) polygon, and when , we also call a Dobrushin domain. Suppose now that is bounded. A sequence of discrete polygons, defined in Section 5, is said to converge to as in the Carathéodory sense if there exist conformal maps (resp. ) from the unit disc to (resp. from to ) such that on any compact subset of , and for all , we have .
We define the pure partition functions of multiple SLEs in Section 2.2.
theoremIsingTHM Let discrete polygons on converge to as in the Carathéodory sense. Consider the critical Ising model in with alternating boundary conditions. Denote by the connectivity pattern formed by the discrete interfaces. Then we have
and are the pure partition functions of multiple .
The result in Theorem 1 is very natural from the point of view of conformal field theory and statistical physics, knowing that variants of s describe scaling limits of critical planar interfaces. Indeed, this result has been conjectured for example in [BBK05, FK15b, FSKZ17, PW17] and is widely believed to be true, however lacking rigorous proof. The purpose of the present article is to fill this gap.
The proof of Theorem 1 has three main inputs: 1: convergence of Ising interfaces in a local sense, including convergence of their Loewner driving functions, from K. Izyurov’s work [Izy17] that generalizes the approach of [CDCH14], 2: continuity of the scaling limit curves up to and including the swallowing time of the marked points (Proposition 5.1 of the present article); and 3: continuity of the Loewner chain associated to the multiple partition function up to and including the same swallowing time (Theorem 4.1 of the present article). We prove Theorem 1 in Section 5.3.
We discuss this formula in Section 3.2. The pure partition functions are uniquely characterized as solutions of certain boundary value problems for partial differential equations, given by Equations (2.1), (2.2), and (2.4) in Section 2.2.
Analogues of Theorem 1 also hold for other critical planar statistical mechanics models whose boundary conditions are symmetric under cyclic permutations of the marked boundary points (rotationally symmetric). In the appendices to the present article, we discuss the following examples:
critical percolation, whose exploration process converges in the scaling limit to with ;
Gaussian free field, whose level lines are curves with ;
chordal loop-erased random walks, which converge in the scaling limit to with .
Explicit formulas for connection probabilities for loop-erased random walks, the double-dimer model, and the Gaussian free field were found in [KW11a], and further related to s in [KKP17, PW17]. The other lattice models seem to be less exactly solvable: explicit formulas for crossing probabilities for the Ising model are not known in general, (special cases appear in [Izy17]), and for percolation, only the case of two curves is known explicitly (given by Cardy’s formula [Car92, Smi01]).
In the random cluster representation of the Ising model (i.e., FK-Ising model), each connection probability describes a natural percolation event, but the boundary conditions are not rotationally symmetric. K. Izyurov showed in [Izy15] that at criticality, probabilities of certain unions of such events have conformally invariant scaling limits, expressed by quadratic irrational functions. In general, however, the problem of finding connection probabilities for the FK-Ising case remains open. With judiciously found total partition function for this model, an analogue of Theorem 1 should also hold.
Outline. The article is organized as follows. Section 2 is devoted to preliminaries: we briefly discuss Schramm-Loewner evolutions (), their basic properties, and define the multiple (pure) partition functions. In the next Section 3, we focus on the case of and prove crucial results concerning the multiple partition function . Section 4 consists of the analysis of the Loewner chain associated to this partition function, leading to Theorem 4.1. In Section 5, we first briefly discuss the Ising model and some existing results on the convergence of Ising interfaces, and then prove the main result of this article, Theorem 1. Finally, Appendices A, B, and C respectively discuss results similar to Theorem 1 for other critical planar models: critical percolation, Gaussian free field, and loop-erased random walks.
Acknowledgments. We thank K. Izyurov for pointing out the very useful identity (3.11). We have also enjoyed discussions with V. Beffara and K. Kytölä on this topic. Parts of this work was completed during E.P.’s visit at the Institut Mittag-Leffler, and during the authors’ visit to the MFO, which we cordially thank for hospitality. This paper was finished during and after the workshop “Random Conformal Geometry and Related Fields” at Seoul, and we kindly thank the organizers for the inspiring meeting.
2 Partition Functions of Multiple SLEs
Fix a simply connected domain . We define a Dobrushin domain to be a triple with two distinct points on locally connected boundary segments. In general, given distinct boundary points in counterclockwise order (on locally connected boundary segments), we call a (topological) polygon. We say that is a sub-polygon of if is simply connected and and agree in neighborhoods of , and in the case of , we also call it a Dobrushin subdomain. Finally, when the boundary of is sufficiently regular (e.g., for some ) in neighborhoods of all of the marked points , we say that is nice.
2.1 Schramm-Loewner Evolutions
For , the (chordal) Schramm-Loewner Evolution, , can be thought of as a family of probability measures on curves, indexed by Dobrushin domains . Each measure is supported on continuous unparameterized curves in from to . This family is uniquely determined by the following two properties [Sch00]:
Conformal invariance: Given any conformal map such that and , we have if .
Domain Markov property: if is a stopping time, then, given an initial segment of the curve , the conditional law of the remaining piece is the law of the from the tip to in the unbounded component of .
Explicitly, the curves can be generated using random Loewner evolutions. Consider a family of maps obtained by solving the Loewner equation: for each ,
where is a real-valued continuous function, which we call the driving function. Let
be the swallowing time of . Denote . Then, is the unique conformal map from onto with the following normalization: . We say that is normalized at . The collection of hulls associated with such maps is called a Loewner chain.
Now, the in is the random Loewner chain driven by where is the standard Brownian motion. By the results of S. Rohde and O. Schramm [RS05], the family of hulls is almost surely generated by a continuous transient curve , in the sense that is the unbounded connected component of for each , and . This random curve is the trace in from to , also referred to as the in . The are simple curves when , but have self-touchings when . For , the curves are space-filling. We only consider the range in this article.
2.2 Partition Functions of Multiple SLEs
Next, we discuss the characterization of the multiple SLE pure partition functions, appearing in the formula (1.1) for the crossing probabilities. We frequently use the following parameters (mostly focusing on the case of , with ):
Multiple partition functions are important objects in the theory of SLEs. They are positive smooth functions of real variables , satisfying the following two properties:
Partial differential equations of second order:
Möbius covariance: For all Möbius maps of such that , we have
By the results of J. Dubédat [Dub07], there is a one-to-one correspondence between the set of multiple partition functions (up to multiplicative constant) and the space of so-called “local” multiple processes . These processes are described by random Loewner evolutions as follows. For each , the curve is a Loewner chain associated to starting from in the sense that its Loewner driving function satisfies the SDEs
This process is well-defined up to the first time when either or is swallowed. Note that is the evolution of the marked point . It coincides with for smaller than the swallowing time of . From properties (2.1) and (2.2), we know that the law of the Loewner chain associated to is that of the in from to weighted by the following local martingale:
The PDE system (2.1) and Möbius covariance (2.2) guarantee conformal invariance as well as a “stochastic reparameterization invariance” (i.e., “commutation”) property, which roughly amounts to saying that, up to a certain stopping time, one may grow the curves in any order using the Loewner evolutions (2.3).
The pure partition functions are special partition functions with the following asymptotics property (that serves as a boundary condition for the PDE system (2.1)):
Asymptotics: Denoting for by the link pattern in , we have and for all , for all , and for all and , we have
where denotes the link pattern obtained from by removing the link and relabeling the remaining indices by the first positive integers.
The multiple pure partition functions have been extensively studied [BBK05, Dub06, Dub07, FK15b, KP16, PW17, Wu17]. The existence of the collection for was proved in [PW17, Theorem 1.1] and [Wu17, Theorem 1.6], and under a technical power law bound [FK15a], the functions are unique. The functions are called “pure” partition functions because, for any , the collection forms a basis for a -dimensional space of multiple partition functions, and the multiple probability measures associated to these basis functions are the extremal points of a convex set of such measures [KP16, PW17].
More generally, the multiple partition functions are defined for any nice polygon via their conformal image: if is any conformal map such that , we set
We note that when , we have only one multiple partition function, namely
where is the boundary Poisson kernel, that is, the unique function determined by the properties
for any conformal map . The boundary Poisson kernel also has the following useful monotonicity property: for any Dobrushin subdomain of , we have
To end this section, we collect some useful properties of the multiple partition functions. First, we set and define, for all , and , and for all , the following functions:
More generally, for a nice polygon , we set
where is again a conformal map from onto such that .
In applications, the following strong bound for the pure partition functions is very important: for , and for any nice polygon , we have
This bound was proved in [PW17, Theorem 1.1] and [Wu17, Theorem 1.6], and it was used in [PW17] to prove that the pure partition functions with give formulas for connection probabilities of the level lines of the Gaussian free field with alternating boundary data (see also Appendix B of the present article). Properties of the bound functions were crucial in that proof, and they will also play an essential role in the present article, where we consider the case .
3 Analysis of Multiple SLE Partition Functions for the Ising Model
In this section, we consider the symmetric partition function (2.9) for , denoted by , which appears in the denominator of the formula (1.1) for the Ising crossing probabilities. We prove key results needed in the proof of the main Theorem 1, concerning the following properties of :
For all and , we have
For any , we have
These bounds are not sharp in general, but they are nevertheless sufficient to our purposes.
Another important result in this section is Proposition 3.11, concerning the boundary behavior of the ratios of partition functions when the variables evolve under a Loewner evolution. All of these results are crucial for proving Theorem 1, but also interesting by their own right. Furthermore, some of these results also hold for other values of , thereby providing with tools to investigate crossing probabilities for other statistical mechanics models — see Appendices A, B, and C.
In Section 3.1, we collect general identities concerning the bound functions and . Then, we focus on the case . We prove Propositions 3.1 and 3.2 respectively in Sections 3.2 and 3.3. Finally, we state and prove Proposition 3.11 in Section 3.4. Its proof relies on the above Propositions 3.1 and 3.2.
3.1 Properties of the Bound Functions
To begin, we consider the bound functions and defined in Section 2.3. In particular, they satisfy properties similar to those appearing in Propositions 3.1 and 3.2 — see Lemmas 3.3 and 3.4. These results can in fact be applied to analyze multiple partition functions for any , and already in the article [PW17], related results were used to prove that connection probabilities of level lines of the Gaussian free field are given by multiple pure partition functions with (see also Appendix B).
[PW17, Lemma A.2] For all and , we have
Fix . For any , we have
Using the above observation, we first prove the upper bound in (3.2):
|[by the ind. hypothesis]|
Then, we prove the lower bound in (3.2): first, we have
Second, we note that , which implies . This gives the desired lower bound and completes the proof. ∎
Fix and let be the symmetric partition function (2.9). Then we have
Plugging in , Corollary 3.6 gives immediately the upper bound in Proposition 3.2. However, the ratio can be arbitrarily small, so the lower bound in Proposition 3.2 cannot be derived easily from the lower bound in Lemma 3.4. To obtain the lower bound in Proposition 3.2, we first prove an identity for in Lemma 3.8.
We record a trivial but helpful inequality here: for , we have
Fix and denote by for and for . Then, we have
In particular, we have
3.2 Asymptotics of the Ising Partition Function
The symmetric partition function has an explicit Pfaffian formula, already well-known in the physics literature, and appearing, e.g., in [KP16, Izy17, PW17] in the context of s. To state it, we use the following notation: we let denote the set of all pair partitions of the set , that is, partitions of this set into disjoint two-element subsets , with the convention that