Random conformal weldings

Astala, Kari; Kupiainen, Antti; Saksman, Eero; Jones, Peter W.

doi:10.1007/s11511-012-0069-3

Cited by 77 publications

(186 citation statements)

References 34 publications

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“…It is natural to conjecture that zipping up gives a welding that respects this measure, and also that SLE 4 is removable. We note that a closely related result has been established by Tecu in [Tec12], which builds on the work of [AJKS11].…”

supporting

confidence: 56%

“…Astala, Jones, Kupiainen, and Saksman showed that the construction Jones proposed produces a well-defined curve [AJKS11,AJKS10], but Binder and Smirnov recently announced a proof (involving multifractal exponents) that this curve is not a form of SLE, and hence the original Jones conjecture is false [Smi] (see Section 1.5). Our construction shows that a simple variant of the Jones conjecture is in fact true.…”

Section: Overviewmentioning

confidence: 99%

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Conformal weldings of random surfaces: SLE and the quantum gravity zipper

2016

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We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path-decorated) random planar maps have (SLEdecorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.

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supporting

confidence: 56%

Section: Overviewmentioning

confidence: 99%

Conformal weldings of random surfaces: SLE and the quantum gravity zipper

2016

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“…In the stochastic setting of the Schramm-Loewner evolution SLE κ where the driving term is λ t = √ κB t and B t is a standard one-dimensional Brownian motion, the conformal welding approach leads to interesting and difficult problems, see [36] and [1] for related deep results. In [31], the almost sure continuity of the SLE κ hulls was proved by different means:…”

Section: The Existence Of Loewner Curves When ||λ||mentioning

confidence: 99%

Regularity of Loewner Curves

Lind¹,

Tran²

2016

Indiana Univ. Math. J.

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The regularity of Loewner curves Huy V. Tran Chair of the Supervisory Committee:Professor Steffen Rohde MathematicsThe Loewner differential equation, a classical tool that has attracted recent attention due to Schramm-Loewner evolution (SLE), provides a unique way of encoding a simple 2-dimensional curve into a continuous 1-dimensional driving function. In this thesis we study the curve in three cases according to the regularity of driving function: weakly Hölder-1/2, Hölder-1/2 with norm less than 4 and C α with α ∈ (1/2, ∞). In the first case, given the existence of the curve we show that the standard algorithm simulating the curve converges in a strong sense. One direct application is for simulating SLE. In the second case, wegive another proof of Marshall, Rohde [26] and Lind [22] in which the curve exists and is a quasi-arc. A sufficient condition for the rectifiability of the curve is also given. In the final case, we show that the Loewner curve is in C α+1/2 . The thesis is a combination of three projects [39], [32] and [21] which are joint work with

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“…They also come into play to construct processes close to curves SLE κ by the conformal welding method [65].…”

Section: E(e Iξρ(b) ) = Exp ψ(ξ)λ(B)mentioning

confidence: 99%

Mandelbrot's cascades: a legendary destiny

Barral¹,

Peyrière²

2015

Benoit Mandelbrot

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The original density is 1 for t ∈ (0, 1), b is an integer base (b ≥ 2), and p ∈ (0, 1) is a parameter. The first construction stage divides the unit interval into b subintervals and multiplies the density in each subinterval by either 1 or −1 with the respective frequencies of 1 2 + p 2 and 1 2 − p 2 . It is shown that the resulting density can be renormalized so that, as n → ∞ (n being the number of iterations) the signed measure converges in some sense to a nondegenerate limit. If H = 1 + log b p > 1/2, hence p > b −1/2 , renormalization creates a martingale, the convergence is strong, and the limit shares the Hölder and Hausdorff properties of the fractional Brownian motion of exponent H. If H ≤ 1/2, hence p ≤ b −1/2 , this martingale does not converge. However, a different normalization can be applied, for H ≤ 1 2 to the martingale itself and for H > 1 2 to the discrepancy between the limit and a finite approximation. In all cases the resulting process is found to converge weakly to the Wiener Brownian motion, independently of H and of b. Thus, to the usual additive paths toward Wiener measure, this procedure adds an infinity of multiplicative paths.

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Random conformal weldings

Cited by 77 publications

References 34 publications

Conformal weldings of random surfaces: SLE and the quantum gravity zipper

Conformal weldings of random surfaces: SLE and the quantum gravity zipper

Regularity of Loewner Curves

Mandelbrot's cascades: a legendary destiny

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