On the quantitative boundary behavior of SLE

Kang, Namgyu

doi:10.1090/fic/050/08

Cited by 3 publications

(6 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They both have been previously published in [13] and [21] correspondingly. Both results can be easily derived from the properties of the spectrum (see [17]) and Theorem 2.3.…”

Section: Results Conjectures and Organization Of The Papermentioning

confidence: 99%

Harmonic Measure and SLE

Beliaev

Smirnov

2009

Commun. Math. Phys.

180

View full text Add to dashboard Cite

In this paper we study the multifractal structure of Schramm's SLE curves. We derive the values of the (average) spectrum of harmonic measure and prove Duplantier's prediction for the multifractal spectrum of SLE curves. The spectrum can also be used to derive estimates of the dimension, Hölder exponent and other geometrical quantities. The SLE curves provide perhaps the only example of sets where the spectrum is non-trivial yet exactly computable

show abstract

“…They both have been previously published in [13] and [21] correspondingly. Both results can be easily derived from the properties of the spectrum (see [17]) and Theorem 2.3.…”

Section: Results Conjectures and Organization Of The Papermentioning

confidence: 99%

Harmonic Measure and SLE

Beliaev

Smirnov

2009

Commun. Math. Phys.

180

View full text Add to dashboard Cite

show abstract

“…A calculation using Itô's formula gives d S t (z) r/2 = S t (z) r/2 (2ar + r 2 2 + r 2 )X t (z) 2 − r 2 Y t (z) 2 |Z t (z)| 4 dt − rX t (z) |Z t (z)| 2 dB t .…”

Section: Studying the Flowmentioning

confidence: 99%

“…The product rule combined with (3), (4), and (5) shows that M t = M t (z) is a nonnegative local martingale satisfying (9). (Note that |h ′ t (z)| λ , Y t (z) ζ are differentiable quantities so there are no covariation terms.)…”

Section: Studying the Flowmentioning

confidence: 99%

“…It is known (see, e.g., [10,4]) that estimates forf ′ t are often more easily derived by considering the reverse (time) Loewner flow. This is how the one-point estimate is derived.…”

Section: Reverse Timementioning

confidence: 99%

“…Using only the Loewner equation (1) and the time-reversibility of Brownian motion (see Section 10.3), one can show that the distribution of h t (z) − U t is the same as the distribution off t (z). In many ways the reverse Loewner flow (2) is easier to analyze than the forward flow (1), see, e.g., [10,4,9]. In this paper we will study the moments of |h ′ t | and show how they can be studied using relatively standard methods of stochastic calculus.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Schramm-Loewner evolution

Lawler¹

2008

Mathematical Surveys and Monographs

View full text Add to dashboard Cite

Abstract. The Schramm-Loewner evolution (SLE) is a one-parameter family of conformally invariant processes that are candidates for scaling limits for two-dimensional lattice models in statistical physics. Analysis of SLE curves requires estimating moments of derivatives of random conformal maps. We show how to use the Girsanov theorem to study the moments for the reverse Loewner flow. As an application, we give a new proof of Beffara's theorem about the dimension of SLE curves.

show abstract

Quantum Loewner evolution

Miller¹,

Sheffield⋆²

2016

Duke Math. J.

124

View full text Add to dashboard Cite

What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the η-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter γ ∈ [0, 2].In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ 2 , η). QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion ν t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of ν t using an SPDE. For each γ ∈ (0, 2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of ν t .We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation.We propose QLE(2, 1) as a scaling limit for DLA on a random spanningtree-decorated planar map, and QLE(8/3, 0) as a scaling limit for the Eden model on a random triangulation. We propose using QLE(8/3, 0) to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE(8/3, 0), up to a fixed time, as a metric ball in a random metric space.

show abstract

On the quantitative boundary behavior of SLE

Cited by 3 publications

References 0 publications

Harmonic Measure and SLE

Harmonic Measure and SLE

Schramm-Loewner evolution

Quantum Loewner evolution

Contact Info

Product

Resources

About