Boundary Correlations in Planar LERW and UST

Karrila, Alex; Kytölä, Kalle; Peltola, Eveliina

doi:10.1007/s00220-019-03615-0

Cited by 25 publications

(29 citation statements)

References 63 publications

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“…We emphasize that in CFT, the fields themselves might not be analytically well-defined objects, but nevertheless, their correlation functions are well-defined functions of several complex variables. Moreover, some correlation functions have been rigorously related to lattice model correlations (see, e.g., [HS13, CHI15, CHI19+] for the Ising model) and SLE curves (see, e.g., [KP16,KKP17,PW18], and Section 3).…”

Section: Conformal Field Theory (Cft)mentioning

confidence: 99%

“…To begin, we discuss the close connection of the pure partition functions Z α to crossing probabilities in critical models. We give the statement for the critical Ising model -see [KKP17,PW18,PW19] for other known results. We also state convergence results for critical Ising interfaces, proved in [CDCH + 14, Izy15, BPW18].…”

Section: Relation To Schramm-loewner Evolutions and Critical Modelsmentioning

confidence: 99%

“…Indeed, different connectivity patterns of multiple SLEs can be encoded in so-called pure partition functions, which form a distinguished basis in the space of SLE partition functions [KP16,PW19]. These basis functions, in turn, are also naturally related to probabilities of non-local crossing events, e.g., for the critical Ising model [KKP17,PW18].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, these functions admit a beautiful hierarchy of fusion rules, which can be thought of as a rigorous operator product expansion, one of the cornerstones of conformal field theory. In fact, in some cases the fusion can also be related to actual observables in critical models [GC05,KKP17] Organization of this article. We discuss the SLE, its relation to critical lattice models, and basics of CFT in Section 2.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Toward a conformal field theory for Schramm-Loewner evolutions

Peltola

2019

Journal of Mathematical Physics

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We discuss the partition function point of view for chordal Schramm-Loewner evolutions and their relationship with correlation functions in conformal field theory. Both are closely related to crossing probabilities and interfaces in critical models in two-dimensional statistical mechanics. We gather and supplement previous results with different perspectives, point out remaining difficulties, and suggest directions for future studies.

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Section: Conformal Field Theory (Cft)mentioning

confidence: 99%

Section: Relation To Schramm-loewner Evolutions and Critical Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Toward a conformal field theory for Schramm-Loewner evolutions

Peltola

2019

Journal of Mathematical Physics

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“…A natural question regarding multiple curves between fixed boundary points consists in the computation of crossing or connection probabilities. The former have been discussed for percolation [11] and the critical Ising model [1], while the latter have been computed for the loop-erased random walk and the uniform spanning tree [26,28], the double dimer model [28] and the discrete Gaussian free field [42].…”

Section: Introductionmentioning

confidence: 99%

Schramm’s formula for multiple loop-erased random walks

Poncelet¹

2018

J. Stat. Mech.

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We revisit the computation of the discrete version of Schramm's formula for the loop-erased random walk derived by Kenyon. The explicit formula in terms of the Green function relies on the use of a complex connection on a graph, for which a line bundle Laplacian is defined. We give explicit results in the scaling limit for the upper half-plane, the cylinder and the Möbius strip. Schramm's formula is then extended to multiple loop-erased random walks.

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Two-Curve Green’s Function for 2-SLE: The Interior Case

Zhan

2020

Commun. Math. Phys.

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We prove that for a 2-SLE κ pair (η 1 , η 2 ) in a simply connected domain D, whose boundary is C 1 near z 0 ∈ ∂D, there is some α > 0 such that lim r→0 + r −α P[dist(z 0 , η j ) < r, j = 1, 2] converges to a positive number, called the boundary two-curve Green's function. The exponent α equals 2( 12 κ − 1) if z 0 is not one of the endpoints of η 1 and η 2 ; and otherwise equals 12 κ − 1. We also prove the existence of the boundary (one-curve) Green's function for a single-boundary-force-point SLE κ (ρ) curve, for κ and ρ in some range. In addition, we find the convergence rate and the exact formula of the above Green's functions up to multiplicative constants. To derive these results, we construct a family of two-dimensional diffusion processes, and use orthogonal polynomials to obtain their transition density.converges to a positive number, where the exponent α equals α 0 := (12−κ)(κ+4) 8κ . The limit G(z 0 ) is called the (interior) two-curve Green's function for (η 1 , η 2 ). The paper [22] also derived the convergence rate and the exact formula of G(z 0 ) up to an unknown constant.In this paper we study the limit in the case that z 0 ∈ ∂D, assuming that ∂D is C 1 near z 0 , for some suitable exponent α. Below is our first main theorem.Theorem 1.1. Let κ ∈ (0, 8). Let (η 1 , η 2 ) be a 2-SLE κ in a simply connected domain D. Let z 0 ∈ ∂D. Suppose ∂D is C 1 near z 0 . We have the following results. PreliminaryWe first fix some notation. Let H = {z ∈ C : Im z > 0}. For z 0 ∈ C and S ⊂ C, let rad z 0 (S) = sup{|z − z 0 | : z ∈ S ∪ {z 0 }}. If a function f is absolutely continuous on I, and f ′ = g a.e. on I, then we write f ′ ae = g on I. This means that f (x 2 ) − f (x 1 ) = Let K be a non-empty H-hull. Let K doub = K ∪ {z : z ∈ K}, where K is the closure of K, and z is the complex conjugate of z. By Schwarz reflection principle, there is a compact set S K ⊂ R such that g K extends to a conformal map from C \ K doub onto C \ S K . LetExample. Let x 0 ∈ R, r > 0. Then H := {z ∈ H : |z − x 0 | ≤ r} is an H-hull with g H (z) = z + r 2 z−x 0 , hcap(H) = r 2 , a H = x 0 − r, b H = x 0 + r, H doub = {z ∈ C : |z − x 0 | ≤ r}, c H = x 0 − 2r, d H = x 0 + 2r. The next proposition combines Lemmas 5.2 and 5.3 of [28].

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Boundary Correlations in Planar LERW and UST

Cited by 25 publications

References 63 publications

Toward a conformal field theory for Schramm-Loewner evolutions

Toward a conformal field theory for Schramm-Loewner evolutions

Schramm’s formula for multiple loop-erased random walks

Two-Curve Green’s Function for 2-SLE: The Interior Case

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