Basic properties of SLE

By Steffen Rohde, Oded Schramm. 2001

658 citations recorded by the source corpus.

SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f314d80feb827ea9050ca2d5da16dc097422d91

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

In Mathematical Physics

Citations in the corpus

Duality of chordal SLE (2007, background): “, the convergence of self-avoiding walk to SLE(8/3) [8], and double domino tilling to SLE(4) [11].”
Harmonic explorer and its convergence to SLE4 (2003, background): “) It has been further conjectured [13] that if colors are instead chosen from a critical FK cluster model (where one weights configurations according to the total number of clusters and the lengths of their interfaces), then γ will converge to some SLEκ with 4 < κ < 8, where κ depends on the weight ”
Reversibility of chordal SLE (2008, background): “From [9], the chordal Loewner trace β(t), 0 ≤ t <∞, driven by ξ exists almost surely.”
Almost sure multifractal spectrum for the tip of an SLE curve (2009, methodology): “The starting point of our analysis is estimation of moments of the derivative of ft using the reverse-time Loewner flow; this was started by Rohde and Schramm in [18] and extended in many places, e.”
Multiple Schramm–Loewner Evolutions and Statistical Mechanics Martingales (2005, background): “Let us briefly recall what is meant by the chordal SLE — detailed studies can be found in [2] or [3].”
Towards conformal invariance of 2D lattice models (2007, background): “visits every point in C+) – see [22], [30] for these and other properties.”
The scaling limits of planar LERW in finitely connected domains. (2006, background): “For the properties of SLE, see [5, 15] and [19].”
The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$ (2016, background): “ the image of the SLE in the upper half-plane via some conformal map from (H,0,∞) onto (D,a,b) to deﬁne SLE fromatobinD. WearegoingtousethefollowingknownfeaturesofSchrammLoewnerEvolutions: Theorem F ([25]). Forκ∈[0,4],SLE κisalmostsurelyasimplecurve(γ t) t≥0 starting at0. Almostsurely|γ t|→∞ast→∞. R”
THE SELF-AVOIDING WALK (2006, methodology): “Now, we will introduce Loewner’s equation based on Lawler [15] or Rohde and Schramm [22].”
The Random-Cluster Model (2002, background): “General properties of SLEκ, viewed as a function of κ have been studied in [124], and a beautiful theory has emerged.”
The dimension of the SLE curves (2002, methodology): “Rohde and Schramm ([14]) used this strategy to show that almost surely the Hausdor dimension of the SLE trace is not larger than 1 + =8 when 8, and they conjectured that this bound was sharp.”
Exploration trees and conformal loop ensembles (2006, background): “The fact that SLE(κ) is continuous appears in [11], but the proof has never been extended to SLE(κ; ρ) processes.”
Two-Dimensional Critical Percolation: The Full Scaling Limit (2006, background): “If z is a deterministic point in D, then with probability one, z is not touched by γ [30] and so it belongs to a unique domain in D \ γD,a,b[0, 1] that we denote Da,b(z).”
2D growth processes: SLE and Loewner chains (2006, methodology): “or the following Sections. They – the properties and the comments – should be understood with the standard proviso “almost surely” or “with probability 1”. We start with a surprisingly diﬃcult result [117,89,90,91,114,19]. • Whatever the value of κ, the pre-image of the driving parameter γt ≡ lim ε→”
Contour lines of the two-dimensional discrete Gaussian free field (2006, background): “Analogous results about winding exponents and fractal dimension have now been proved rigorously for SLE [Sch00, RS05 , Bef02].”
Conformal weldings of random surfaces: SLE and the quantum gravity zipper (2010, methodology): “ ([0;t]). The curve is almost surely a simple curve when 2[0;4], a self-intersecting but non-space-lling curve when 2(4;8), and a space-lling curve (ultimately hitting every point in H) when 8 [RS05]. The maps f t(z) := g t(z) W t satisfy df t(z) = 2 f t(z) dt p dB t; and f t((t)) = 0. Throughout th”