Green’s functions for chordal SLE curves

Rezaei, Mohammad A.; Zhan, Dapeng

doi:10.1007/s00440-017-0802-0

Cited by 8 publications

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References 20 publications

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“…Let E(w, δ) be the event that η intersects all of these m balls. Using the n-point Green's function for chordal SLE (see [47,Proposition 2.3]), we get that the probability of E(w, δ) is at most an absolute constant times δ m(2−d)(α−ζ) . We then choose m big enough so that m(2 − d)(α − ζ) + 2γ > 2.…”

Section: 21mentioning

confidence: 99%

Uniqueness of the Welding Problem for Sle and Liouville Quantum Gravity

McEnteggart¹,

Miller²,

Qian³

2019

J. Inst. Math. Jussieu

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We give a simple set of geometric conditions on curves η, η in H from 0 to ∞ so that if ϕ : H → H is a homeomorphism which is conformal off η with ϕ(η) = η then ϕ is a conformal automorphism of H. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if η is a non-space-filling SLEκ curve in H from 0 to ∞, and ϕ is a homeomorphism which is conformal on H \ η, and ϕ(η), η are equal in distribution, then ϕ is a conformal automorphism of H. Applying this result for κ = 4 establishes that the welding operation for critical (γ = 2) Liouville quantum gravity (LQG) is well-defined. Applying it for κ ∈ (4, 8) gives a new proof that the welding of two independent κ/4-stable looptrees of quantum disks to produce an SLEκ on top of an independent 4/ √ κ-LQG surface is well-defined.

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Section: 21mentioning

confidence: 99%

Uniqueness of the Welding Problem for Sle and Liouville Quantum Gravity

McEnteggart¹,

Miller²,

Qian³

2019

J. Inst. Math. Jussieu

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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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SLE loop measures

Zhan

2020

Probab. Theory Relat. Fields

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Green’s functions for chordal SLE curves

Cited by 8 publications

References 20 publications

Uniqueness of the Welding Problem for Sle and Liouville Quantum Gravity

Uniqueness of the Welding Problem for Sle and Liouville Quantum Gravity

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

SLE loop measures

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