Random planar curves and Schramm-Loewner evolutions

By Wendelin Werner. 2003

279 citations recorded by the source corpus.

We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1b6d17325a72901f4270a3e1bbbc9d6fd99e8ab8

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

The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$ (2016, background): “For further SLE background, readers can also refer to [10], [28].”
The scaling limits of planar LERW in finitely connected domains. (2006, unknown): “The Annals of Probability 2008, Vol. 36, No. 2, 467–529 DOI: 10.1214/07-AOP342 © Institute of Mathematical Statistics, 2008”
Towards conformal invariance of 2D lattice models (2007, background): “We do not discuss the general mathematical theory of SLE curves or their connections to physics, for which interested reader can consult the expository works [5], [10], [15], [41] and the book [22].”
Multiple Schramm–Loewner Evolutions and Statistical Mechanics Martingales (2005, unknown): “3.1 The basic equations . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Arch probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 8”
Boundary Conformal Field Theory (2004, background): “How are these boundary states to be characterized? Using the transformation law ( 9 ) the conformal boundary condition applied to the circle implies that Ln = Lin.”
Commutation relations for Schramm‐Loewner evolutions (2004, background): “Though, it is not quite clear how to define this scaling limit and what conformal invariance exactly means.”
Two-Dimensional Critical Percolation: The Full Scaling Limit (2006, methodology): “The work of Lawler, Schramm, Sheffield and Werner has provided tools to define such loop ensembles in the continuum and to study some of their properties.”
Almost sure multifractal spectrum for the tip of an SLE curve (2009, background): “We refer the reader to [7, 8, 22] for surveys and further references.”
Critical percolation exploration path and SLE6: a proof of convergence (2006, methodology): “Here we describe the chordal version of SLEκ ; for more, see [27] as well as the fine reviews by Lawler [18], Kager and Nienhuis [15], and Werner [37], and Lawler’s book [19].”