SLE boundary visits
We study the probabilities with which chordal Schramm-Loewner Evolutions (SLE) visit small neighborhoods of boundary points. We find formulas for general chordal SLE boundary visiting probability amplitudes, also known as SLE boundary zig-zags or order refined SLE multi-point Green’s functions on the boundary. Remarkably, an exact answer can be found to this important SLE question for an arbitrarily large number of marked points. The main technique employed is a spin chain - Coulomb gas correspondence between tensor product representations of a quantum group and functions given by Dotsenko-Fateev type integrals. We show how to express these integral formulas in terms of regularized real integrals, and we discuss their numerical evaluation.
The results are universal in the sense that apart from an overall multiplicative constant the same formula gives the amplitude for many different formulations of the SLE boundary visit problem. The formula also applies to renormalized boundary visit probabilities for interfaces in critical lattice models of statistical mechanics: we compare the results with numerical simulations of percolation, loop-erased random walk, and Fortuin-Kasteleyn random cluster models at and , and find good agreement.
- 1 Introduction
- 2 The problem: partial differential equations and asymptotics
3 Quantum group and integral formulas
- 3.1 Coulomb gas integrals
- 3.2 Quantum group
- 3.3 Spin chain - Coulomb gas correspondence
- 3.4 Linear problem in quantum group representations
- 3.5 Solutions in terms of quantum group representations
- 4 Regularized real integrals and evaluation of the formulas
5 Notions of SLE boundary visits and applications
- 5.1 Definition of chordal SLE in half-plane
- 5.2 Conformal covariance of boundary visit amplitudes
- 5.3 Different definitions of SLE boundary visits
- 5.4 Applications of the results and universal and non-universal aspects
- 6 Comparisons with lattice model simulations
- 7 Conclusions and outlook
- A SLE derivations of the exponent and a PDE
- B Conformal field theory considerations
- C Some explicit quantum group formulas
- D Numerical evaluation of the integrals
1.1 SLE curves
Schramm-Loewner evolutions (SLE) are conformally invariant random
fractal curves in the plane, whose most important characteristics
are determined by one parameter . They were introduced
by Oded Schramm [Sch00] as the only plausible
candidates for the scaling limits of random interfaces in statistical
mechanics models that are expected to display conformal invariance,
with different models corresponding to different values of the parameter
The fundamental example of SLEs is the chordal [LSW01, RS05]. For a given simply connected domain with two marked boundary points , the chordal in from to is an oriented but unparametrized random curve in the closure of starting from and ending at . Its two characterizing properties are conformal invariance and domain Markov property:
Conformal invariance states that the image of a chordal SLE under a conformal map is a chordal SLE in the image domain.
Domain Markov property states that given an initial segment of a chordal SLE, the conditional law of the continuation is a chordal SLE in the remaining subdomain.
Some features of SLEs vary continuously in , notably the Hausdorff dimension of the fractal curve is given by for [Bef08]. On the other hand, some qualitative properties of SLEs show abrupt phase transitions with respect to the parameter . For the present purposes, it is important to distinguish the following three phases [RS05]:
The chordal is a simple curve, i.e., the curve does not have double points, see Figure 1.1 (left). The curve does not touch the boundary of the domain except at the starting point and the end point . The curve avoids any given point of the domain with probability one.
The chordal is a non self-traversing curve with double points, see Figure 1.1 (right). The intersection of the curve with the boundary of the domain is a random Cantor set. The curve still avoids any given point of the domain or of its boundary with probability one.
The chordal is a space-filling curve; any point of the domain is on the curve.
The behavior in the case is somewhat pathological.
No interfaces in statistical mechanics models are expected to correspond
1.2 Chordal SLE boundary visits
The main goal in this article is to find formulas for the probabilities with which the chordal SLE visits small neighborhoods of given boundary points. Partial answers to similar questions have been obtained in [BB03a, SZ10, AS08, AS09, Law14].
It is easiest to illustrate the question in the upper half-plane
with the chordal curve starting from the origin and ending at infinity. We will briefly recall the precise definition of chordal in from to in Section 5.1, and we refer the reader to [RS05] for more thorough background.
Denote the half-disk of radius centered at a boundary point by
Given points and radii , the probability that the curve visits all of , , tends to zero as a power law as the radius is taken small. More precisely, the scaling exponent of the power law is
(see Appendix A), and we are
interested in the limit
of probabilities of events illustrated schematically in Figure 1.2.
In the spirit of [Law10, LS11, AKL12, LW13, LZ13, LR15],
it is appropriate to call the limit (1.2)
an SLE boundary Green’s function. We emphasize that one could in principle choose
to define a boundary visit of SLE differently, for example for one could
ask the curve to touch a boundary segment of length , or one could choose
the neighborhood shape to be something other than a half-disk.
Yet, independently of the precise formulation, the limit remains universal apart from
a multiplicative constant which depends on the details of the chosen formulation.
Recalling that is an oriented curve, we may even specify the order of the boundary visits, i.e., require that the curve first reaches the chosen small neighborhood of , then the neighborhood of and so on until reaching the neighborhood of . The order refinement of the SLE boundary Green’s function is the limit
where any increasing parametrization of the curve is chosen, and we denote by
the stopping time at which the curve first reaches the -neighborhood
of . Obviously one can recover the complete correlation function
from the ordered ones
by summing over all possible orders of visits
In the general form with the order of visits specified, the question of finding the asymptotic amplitudes of the visiting probabilities of chordal was posed in [BB03a], where these quantities were called “(boundary) zig-zag probabilities”.
Depending on the details of the precise formulation of the boundary visit question, one would obtain a different non-universal multiplicative constant in the SLE boundary Green’s function (1.2) and its order refinement (1.3). We therefore prefer to use a generic notation for a quantity of this type, for which we are free to choose a more convenient multiplicative normalization. We also prefer to make explicit the dependence of the question on the starting point of the chordal curve, but the end point of the curve will always be kept at infinity. In the rest of this article,
Similarly, we denote by
a complete (boundary) correlation function, so that in particular
with the same proportionality constant.
Explicit formulas for the above types of quantities are known in the following two special cases:
The one-point function () behaves simply as a power law, as follows immediately from the invariance under dilatations () of the chordal in
In this article we present a method for finding the solutions in the general case. We write down a system of partial differential equations (PDEs) motivated by conformal field theory (CFT) for the quantities of interest, and . Our solutions for them are written in terms of Coulomb gas integrals (Dotsenko-Fateev integrals [DF84]) and are found by quantum group calculations. This technique is developed in the present article and in [KP14], we call it the spin chain - Coulomb gas correspondence. Our primary goal here is to find the explicit formulas and show their wide applicability: the functions and answer various formulations of boundary visit questions for SLEs as well as for interfaces in lattice models. We also compare the results to numerical simulations of various lattice models, and outline a strategy of proof that our formulas give the (order refined) SLE boundary Green’s functions.
We emphasize that it is very rarely possible to find the exact solution
for an SLE problem involving a large number of marked points — the
few existing solutions to such problems rely on finding tricks that
appear particular to each problem
[Hag09, HD08, SZK09, SK11, SKFZ11, BI12, AKL12, FKZ12, FK15].
1.3 Organization of the article
The rest of the article is organized as follows.
In Section 2 we formulate the PDE problem which we solve in the subsequent sections to find the zig-zag amplitudes and the complete correlation functions :
The functions and are conformally covariant.
The functions and satisfy a second order PDE and third order PDEs.
The boundary conditions depend on the order of visits: they are written in terms of asymptotic behaviors of and their inhomogeneous terms involve the in a recursive manner.
In Section 3 we discuss the spin chain - Coulomb gas correspondence, by which the PDE problem is translated to a linear problem in representations of a quantum group:
We associate functions defined by Coulomb gas integrals to vectors in a finite-dimensional tensor product representation of the quantum group .
The functions associated to highest weight vectors are solutions to the partial differential equations of Section 2, and for particular highest weights they also have the correct conformal covariance.
Projections to subrepresentations in consecutive tensorands determine the asymptotic behaviors of the functions.
There are unique highest weight vectors of the correct highest weights whose subrepresentation projections correspond to the boundary conditions imposed on the zig-zag amplitudes .
In Section 4 the integrals obtained in the spin chain - Coulomb gas correspondence are rewritten as regularized real integrals. The transformation to real integrals concretely exhibits the needed closed homology properties of our solutions.
In Section 5 we discuss basic properties, applications, interpretations, and universality of the SLE boundary visit question and outline a strategy of proof.
In Section 6 we compare our formula numerically to simulations of lattice models of statistical mechanics. We study random interfaces in percolation, random cluster model, and loop-erased random walk. We perform computer simulations of them and collect frequencies of multi-point boundary visits of the interfaces, and compare renormalized frequencies to the zig-zag amplitudes .
We conclude the article by discussion and outlook in Section 7.
The article is complemented with several appendices. Appendix A provides two derivations of the value of the scaling exponent (1.1), and a derivation of the second order PDE. Appendix B contains relevant background on conformal field theory. Our normalization conventions for some quantum group representations and some explicit four-point solutions are contained in Appendix C. Numerical evaluation of the integrals of Sections 3 and 4 is treated in Appendix D.
2 The problem: partial differential equations and asymptotics
We find the boundary visit amplitudes and by solving a PDE problem. The system of partial differential equations is given below in Section 2.1. This part is the same for and for , and moreover the system is the same for all boundary zig-zag amplitudes corresponding to different orders of visits to the same set of points. The results will be different, however, as each of the functions satisfies different boundary conditions, detailed in Section 2.2.
2.1 Differential equations for boundary visit amplitudes
The linear homogeneous system of PDEs below contains essentially three different types of partial differential equations — all of them can be argued to hold by conformal field theory (see Appendix B.2), but from the point of view of SLE analysis, the argument leading to each of them is different. For the system reads:
and we have used the parameters and .
The first order PDEs (2.1)
and (2.2) express the translation
invariance and homogeneity of the amplitudes. More general conformal
covariance of the answer will be discussed in Section 5.2
and again from a conformal field theory point of view in Appendix
B.1. The second order PDE (2.3)
can be interpreted either in terms of the SLE process as the statement
of a local martingale property of the answer, see Appendix A.3,
or in terms of conformal field theory as a conformal Ward identity
associated to a second order degeneracy of the boundary field located
at , as will be discussed in Appendix B.2.
The third order PDEs (2.4)
are similarly the conformal Ward identities associated to third order
degeneracies of the boundary fields located at , ,
see Appendix B.2. Unlike for the first and
second order equations we do not know how to explain the third order
equations by SLE analysis directly. As a partial justification, however,
we note that Equations (2.4)
coincide with the third order partial differential equations [Dub15b]
derived by Dubédat
for limiting cases of multiple SLE partition functions, which morally
describe the same configurations of curves as our boundary visiting SLEs.
Ultimately, the validity of all of the above equations
for the SLE boundary visit amplitudes would need to be established
by first finding the explicit answer, which is the main task in the
present article, and then proving that it gives the SLE boundary Green’s
function following the strategy that will be outlined in Section 5.4.3.
2.2 Asymptotics for boundary visit amplitudes
The system of differential equations of Section 2.1 has a large space of solutions. To pin down the correct solution we need boundary conditions, which will be specified in the form of asymptotic behavior of the boundary zig-zag amplitudes. Considerations of the possible asymptotics allowed by conformal field theory can be found in Appendix B.3. The particular requirements that finally specify the solutions are given below.
Consider the question of visiting the neighborhoods of in this order. Some notation and terminology is needed to conveniently describe the specific asymptotics of in this case. We say that points such that are on the left and points such that are on the right. We say that the points are in an outwards increasing order if for any on the left we have that implies and for any on the right we have that implies , in other words that among points on the same side, the point further away from starting point is visited later.
The boundary visit amplitude vanishes unless the points are in an
outwards increasing order — a visit to a small neighborhood of a
point further away on the same side almost disconnects the future
passage of the curve to the point that would need to be visited later.
It is convenient to use a separate ordering for the points on the left and right. Denote therefore the points on the left in a decreasing order (in the order of visits) and the points on the right in an increasing order (in the order of visits). The following notation makes the arguments of the zig-zag amplitude appear in the same order as they are on the real axis,
where is a sequence of “”-symbols specifying the sequence of sides of the visits in the sense that (resp. ) if is on the left (resp. on the right). If we fix the number of points on the left and the number of points on the right, , then the number of different outwards increasing orders is , corresponding to the choices of with “”-symbols and “”-symbols. The complete correlation function is the sum of these zig-zag amplitudes. In the particular case when all the points are on the same side, the complete correlation function coincides with the zig-zag amplitude.
The specific asymptotics depend on the order of visits, and to describe them we need a few separate cases. We call the consecutive points and on the same side () successively visited points on the same side if for some we have and .
We claim that for any outwards increasing order the boundary
zig-zag amplitude satisfies the asymptotics conditions
Asymptotics for successively visited points: If and are successively visited points on the same side, then
Asymptotics for non-successively visited points: If and are non-successively visited consecutive points on the same side, then
Asymptotics for the first points on the left and right: For the first point to be visited we have
For the first point on the opposite side, i.e., for , we have
The constants in (2.5) and (2.7)
are different, but for different pairs of successively visited consecutive points,
the constant in (2.5) should be the same.
We conjecture that the solution space to the partial differential equations (2.1), (2.2), (2.3), (2.4) is finite-dimensional and that its dimension is exactly the multiplicity of a certain irreducible direct summand in a tensor product, see Section 3.5.5. Under this assumption, it could be shown with the techniques introduced in Section 3, that recursively in the asymptotics conditions (2.5), (2.6), (2.7), (2.8) specify uniquely, up to a multiplicative constant, solutions for all outwards increasing orders of visits .
Our choice of normalization of and will be determined recursively by fixing the constant appearing in Equation (2.7), see Section 3.4. Once this natural choice is made, the different -point functions obtain correct relative normalizations, with the universal ratios referred to in Section 5.4. In particular, the constant appearing in Equation (2.5) gets automatically fixed as well.
3 Quantum group and integral formulas
3.1 Coulomb gas integrals
The main tool that allows us to solve the PDE problem of Section 2 and therefore to find the explicit formula for the SLE boundary visit amplitudes is the spin chain - Coulomb gas correspondence. In this article, for the sake of concreteness, we describe only the case relevant to the problem of boundary visit amplitudes — a more general treatment can be found in [KP14].
Standard Coulomb gas integrals and their properties
The Coulomb gas formalism of conformal field theory, or Dotsenko-Fateev integrals [DF84], is a way of producing solutions to systems of differential equations of the type of Section 2.1 by integrating an auxiliary function, which in our case takes the form
Consider the function
where is a closed -surface avoiding the points . The integral of course only depends on the homotopy type of the surface . The function is defined such that while the contour of the -variables may depend on the positions of , the choice is locally constant. One then observes:
translation invariance: satisfies Equation (2.1).
scale covariance: is homogeneous of degree , and in particular if it satisfies Equation (2.2).
second order differential equation: satisfies Equation (2.3).
third order differential equations: satisfies Equations (2.4).
The translation invariance follows immediately from the translation invariance of the integrand by considering a shift of the variables small enough so that the integration contour can be kept constant, and then the same shift of the integration contour, which now does not change the homotopy type. The scaling covariance is shown similarly, starting with scaling close enough to identity. The relevant scaling covariance of the integrand reads
and an extra factor comes from the change of variables in the integration — the formal proofs can be found in [KP14, Lemma 3.3 and Theorem 4.17].
The second and third order differential equations rely more crucially on the fact that the integration surface is closed. One again starts from a property satisfied by the integrand alone. Starting from the second order equation, let
be the differential operator we want to show annihilates . It is a matter of straightforward verification to see that the integrand satisfies
and to notice that this can also be read as
Thus when acting on by the differential operator , we may take the operator inside the integral, and rewrite the integrand as a sum of total derivatives. The integral of these vanish because the contour was assumed to be closed. Hence one gets the second order differential equation for . The third order differential equations are shown to hold similarly — the formal proof of a more general statement can be found in [KP14, Proposition 4.12 and Theorem 4.17].
Spin chain - Coulomb gas basis functions
Our solution will eventually be of the form (3.2), with . As in [KP14, KP15], we need to unveil an underlying quantum group structure, which will be useful for calculations, and in particular crucial for dealing with the asymptotics. For this purpose, we introduce the functions
indexed by and , which are defined by the integrals
The integration surface is shown in Figure 3.1. The dimension of the integration surface, i.e., the number of integration variables , is . In the functions appearing in our final answer this will always be . The contour of each integration variable is a loop based at an anchor point to the left of all of the variables, and the loop encircles one of the points in the positive direction. The loops of the first variables encircle the point , the next variables encircle the point and so on. The loops encircling the same point are nested. The loops encircling different points avoid each other so that the contours to a point further on the right go below.
We make the following remarks about the role and properties of the above functions:
Individually the surfaces are not closed, but our solution will be a linear combination which is closed in the appropriate homology [FW91].
In the spin chain - Coulomb gas correspondence defined in Section 3.3.1, we will make basis vectors in a quantum group representation correspond to the functions . In Sections 3.3.2 and 3.3.3 we explain how straightforward quantum group calculations will allow us to decide about the asymptotics of the functions as well as the closedness of the surfaces in an appropriate homology — see also [FW91, KP14].
3.2 Quantum group
We need to recall some facts and fix some notation for the quantum group . It should be thought of as a deformation of (the universal enveloping algebra of) the Lie algebra , with a deformation parameter — with a suitable normalization when one recovers from the definitions we give below.
We let , and assume that is generic
in the sense that .
Since we assume , all -integers with are non-zero.
Definition of the quantum group
The quantum group is the algebra over with generators and relations
Moreover, is equipped with the unique Hopf algebra structure such that the coproducts of the generators are
The coproduct determines the action of the quantum group in tensor product of two representations and , for example . The tensor product of representations is then associative but not commutative: multiple tensor products are well defined, for example , but the order of the tensorands is important.
Representations of the quantum group
The quantum group is semisimple (for not a root of unity) in the sense that any finite dimensional representation is the direct sum of its irreducible subrepresentations. In fact, the representation theory essentially just deforms that of . We recall the following standard facts, the proofs of which can be found in, e.g., [KP14, Lemmas 2.3 and 2.4].
For any , there exists a -dimensional irreducible representation with a basis such that the action of the generators on the basis vectors is given by
|(with interpretation )|
|(with interpretation )|
This representation is the appropriate deformation of the -dimensional irreducible representation of (“the spin- representation”). The tensor products of decompose according to the formula
Our calculations will require some specific cases of such (quantum) Clebsch-Gordan decompositions to be made explicit. Formulas for those cases are given in Appendix C.1.
The one-dimensional irreducible is the trivial representation, it acts as a neutral element of the tensor products: for any representation we have the isomorphisms . This allows us to omit in tensor products, when needed.
3.3 Spin chain - Coulomb gas correspondence
Definition of the correspondence
With the above preparations we can now define the correspondence. The spin chain - Coulomb gas correspondence linearly associates to vectors
in a tensor product of representations of a function, so that for the natural tensor product basis vectors the associated functions are those defined in Section 3.1.2:
Note that in our convention, the order of the variables of the function is the reverse of the order of the corresponding factors in the tensor product.
Asymptotics via the correspondence
A key property of the spin chain - Coulomb gas correspondence is that the asymptotics of the functions can be straightforwardly read from the projections to subrepresentations of the corresponding vectors in . For these projections, we use below the notation and normalization conventions of Appendix C.1.
Let and let be the function associated to by the correspondence of Section 3.3.1. The correspondence of asymptotics and subrepresentations is stated precisely in the following:
Consider two consecutive points on the right or left (superscript “” or “”, respectively). For , denote accordingly by the projection to -dimensional subrepresentation of acting on the :th and :st components on the appropriate side.
Suppose that is in the singlet of the components corresponding to , that is . Then as , we have
where the variables have been removed from the right hand side, the function is the function of two variables less associated to the vector interpreted as a vector in either or , and the constant is the generalized beta-function
Suppose that is in the triplet of the components corresponding to , that is . Then as , we have
where on the right hand side the two variables have been removed and replaced by one , the function is the function of one variable less associated to the vector interpreted as a vector in either or and the constant is the beta-function
Suppose that is in the quintuplet of the components corresponding to , that is . Then as , we have
where on the right hand side the two variables have been removed and replaced by one . We will not need any properties of the function , but we nevertheless remark that with a generalization of the present method it becomes in principle explicit (see [KP14, Proposition 4.4] for details).
Consider the point and the first point on the right or left (superscript “” or “”, respectively). For , denote accordingly by the projection to -dimensional subrepresentation of or acting on the middle factor and the on the appropriate side of it.
Suppose that is in the doublet of the components corresponding to , that is . Then as , we have