SLE Boundary Visits

Jokela, Niko; Järvinen, Matti; Kytölä, Kalle

doi:10.1007/s00023-015-0452-7

Cited by 9 publications

(24 citation statements)

References 57 publications

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“…Such asymptotic behavior is believed to specify the solution to the PDEs up to a multiplicative constant. Admitting this, our formulas coincide with the SLE boundary zig-zag amplitude prediction of [KP16,JJK16].…”

Section: Introductionsupporting

confidence: 73%

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Boundary Correlations in Planar LERW and UST

Karrila

Kytölä

Peltola

2019

Commun. Math. Phys.

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We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. We show that both probabilities, when suitably renormalized, converge in the scaling limit to conformally covariant functions which satisfy partial differential equations of second and third order, as predicted by conformal field theory. The scaling limit connectivity probabilities also provide formulas for the pure partition functions of multiple SLEκ at κ = 2. e inê

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Section: Introductionsupporting

confidence: 73%

“…Asymptotics of the scaling limits of the boundary visit probabilities. To finish this section, we prove asymptotics properties for the boundary visit probabilities, predicted in [JJK16].…”

Section: 4mentioning

confidence: 99%

Boundary Correlations in Planar LERW and UST

Karrila

Kytölä

Peltola

2019

Commun. Math. Phys.

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“…The (ASY) part is again suitable for this purpose: the decomposition M 3 ⊗ M 3 ∼ = M 1 ⊕ M 3 ⊕ M 5 applies to the case |y ± i+1 − y ± i | → 0, and the decompositions M 3 ⊗ M 2 ∼ = M 2 ⊕ M 4 and M 2 ⊗ M 3 ∼ = M 2 ⊕ M 4 apply respectively to the cases |y + 1 − x| → 0 and |x − y − 1 | → 0. SLE boundary visits are treated in more detail in [JJK16], and the requirements for the vectors v ω are shown to have a unique solution in general in [KP16, Section 5]. Again, the main results of the present article are thus instrumental for finding the explicit formulas for these order-refined multi-point boundary visit probabilities of the chordal SLE.…”

Section: (Pde)mentioning

confidence: 93%

“…. , y + R ) can then be defined, see [JJK16] for details. The task is to find an explicit expression for it.…”

Section: (Pde)mentioning

confidence: 99%

“…In this article, we establish properties which are directly relevant for solving the PDE problems arising in the theory of Schramm-Loewner evolutions (SLE). In two companion articles, the results are applied to produce explicit answers to the questions of boundary visit probabilities of chordal SLEs [JJK16] and the pure geometries of multiple SLEs [KP16]. With appropriate modifications, the method can also be applied to bulk correlation functions -in [FP18b] it is applied to the construction of single-valued bulk correlation functions of conformal field theory.…”

Section: Introductionmentioning

confidence: 99%

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Conformally covariant boundary correlation functions with a quantum group

Kytölä¹,

Peltola

2019

J. Eur. Math. Soc.

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Particular boundary correlation functions of conformal field theory are needed to answer some questions related to random conformally invariant curves known as Schramm-Loewner evolutions (SLE). In this article, we introduce a correspondence and establish its fundamental properties, which are used in the companion articles [JJK16, KP16] for explicitly solving two such problems. The correspondence associates Coulomb gas type integrals to vectors in a tensor product representation of a quantum group, a q-deformation of the Lie algebra sl 2 . We show that desired properties of the functions are guaranteed by natural representation theoretical properties of the vectors. denote the vector obtained by identifying v as a vector in an (n − 1)-fold tensor product representation. Then, as |x j+1 − x j | → 0, the function F[v] has the asymptotic behaviorwhere the constant B = B dj ,dj+1 d and the exponent ∆ = ∆ dj ,dj+1 d are explicit. The analogous statement holds also for F (x0) [v].In practice, this theorem is applied as follows. In specific problems, we are looking for particular solutions to systems of PDEs of conformal field theory. Typically, the sought solution has specific Möbius covariance properties and specific boundary conditions as the distance of some of its arguments tend to zero. By our correspondence, the task of finding a function with these properties is translated to the problem of finding a corresponding vector in the tensor product representation. The different parts (PDE), (COV), and (ASY) of the theorem state that a careful choice of the vector would ensure the desired properties of the function -even the most delicate boundary conditions for the function can be guaranteed by (ASY) if the vector has appropriate projections to certain subrepresentations. All of the requirements are explicit linear conditions on the vector living in a finite dimensional vector space, and we moreover have a variety of representation theoretical tools at our disposal to solve for such a vector. By outlining a few case studies in Sections 1.4 and 1.5, we exemplify how the correspondence thus allows us to translate the original, possibly rather complicated problem to an explicitly solvable one, and to eventually express the function of interest explicitly as a linear combination of integral form functions.Before example applications, we make a few further observations about the interpretation of the constructed correspondence, and comparisons to related research.

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A Solution Space for a System of Null-State Partial Differential Equations: Part 1

Flores

Kleban

2014

Commun. Math. Phys.

104

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This article is the first of four that completely and rigorously characterize a solution space S N for a homogeneous system of 2N + 3 linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE κ ). In CFT, these are null-state equations and conformal Ward identities. They govern partition functions for the continuum limit of a statistical cluster or loop-gas model, such as percolation, or more generally the Potts models and O(n) models, at the statistical mechanical critical point. (SLE κ partition functions also satisfy these equations.) For such a lattice model in a polygon P with its 2N sides exhibiting a free/fixed side-alternating boundary condition ϑ, this partition function is proportional to the CFT correlation functionwhere the w i are the vertices of P and where ψ c 1 is a one-leg corner operator. (Partition functions for "crossing events" in which clusters join the fixed sides of P in some specified connectivity are linear combinations of such correlation functions.) When conformally mapped onto the upper half-plane, methods of CFT show that this correlation function satisfies the system of PDEs that we consider.In this first article, we use methods of analysis to prove that the dimension of this solution space is no more than C N , the N th Catalan number. While our motivations are based in CFT, our proofs are completely rigorous. This proof is contained entirely within this article, except for the proof of Lemma 14, which constitutes the second article Kleban, in Commun Math Phys, arXiv:1404.0035, 2014). In the third article Kleban, in Commun Math Phys, arXiv:1303.7182, 2013), we use the results of this article to prove that the solution space of this system of PDEs has dimension C N and is spanned by solutions constructed with the CFT Coulomb gas (contour integral) formalism. In the fourth article (Flores and Kleban, in Commun Math Phys, arXiv:1405.2747, we prove further CFT-related properties about these 390 S. M. Flores, P. Kleban solutions, some useful for calculating cluster-crossing probabilities of critical lattice models in polygons.

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SLE Boundary Visits

Cited by 9 publications

References 57 publications

Boundary Correlations in Planar LERW and UST

Boundary Correlations in Planar LERW and UST

Conformally covariant boundary correlation functions with a quantum group

A Solution Space for a System of Null-State Partial Differential Equations: Part 1

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