Schramm–Loewner evolution
conformally invariant stochastic process
In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter κ and a domain U in the complex plane, it gives a family of random curves in U, with κ controlling how much the curve turns. There are two main variants of SLE, chordal SLE which gives a family of random curves from two fixed boundary points, and radial SLE, which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property.
It was discovered by Oded Schramm (2000) as a conjectured scaling limit of the planar uniform spanning tree (UST) and the planar loop-erased random walk (LERW) probabilistic processes, and developed by him together with Greg Lawler and Wendelin Werner in a series of joint papers.
Besides UST and LERW, the Schramm–Loewner evolution is conjectured or proven to describe the scaling limit of various stochastic processes in the plane, such as critical percolation, the critical Ising model, the double-dimer model, self-avoiding walks, and other critical statistical mechanics models that exhibit conformal invariance. The SLE curves are the scaling limits of interfaces and other non-self-intersecting random curves in these models. The main idea is that the conformal invariance and a certain Markov property inherent in such stochastic processes together make it possible to encode these planar curves into a one-dimensional Brownian motion running on the boundary of the domain (the driving function in Loewner's differential equation). This way, many important questions about the planar models can be translated into exercises in Itô calculus. Indeed, several mathematically non-rigorous predictions made by physicists using conformal field theory have been proven using this strategy.
Begin with the source’s own compact description: “Schramm–Loewner evolution” is conformally invariant stochastic process. The dossier treats that line as a proposition to test through Schramm, Loewner and evolution, not as a finished interpretation.
Why this record matters
The phrase “conformally invariant stochastic process” supplies a clear boundary for inquiry. It also exposes the unanswered questions: who defined that boundary, when it became stable and which sources sit outside it.
Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated Aug 11, 2026. The linked authority identifier is Q7432797. None of the 0 selected statements returned an explicit reference. The first chronological checks are 2000.
A concise general-reference account can conceal disagreements about scope, terminology or the weight assigned to individual sources. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Schramm–Loewner evolution”, its source revision and the description used here.
- Expand the search: follow Schramm–Loewner evolution primary sources, Schramm–Loewner evolution archive and Schramm research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Schramm–Loewner evolution”?
- Which cited source is closest to the event, object or claim?
- What terminology or title could unlock a more precise catalogue search?
Search terms from this dossier
This entry incorporates text from “Schramm–Loewner evolution” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.