Construction of the Schramm-Loewner Evolution - A Review

I first learned about the theory of Schramm-Loewner Evolutions (SLE) after learning about the conjectured conformal invariance of two-dimensional critical percolation through the article Percolation: Slipping through the Cracks. This phenomenon on the triangular lattice has been proven by Stanislav Smirnov, which won him the Fields Medal. Now, more than a year later, I am finally learning the rudiments of $\mathsf{SLE}$ theory.

Classical Loewner theory was developed purely for geometric function theoretic purposes: to resolve the Bieberbach Conjecture (now de Branges’ Theorem). The chordal Loewner theory on the upper half-plane was a very general one. Essentially any type of model which scales to a curve of local growth (in particular, this requires it to be continuous and non-self-crossing) can be described by Loewner theory. This observation is slightly disappointing, as SLE turned out to be a less special object than I had imagined. The conformal invariance part? This direct from definition! These were my initial thoughts. But as I thought a bit more, I realised that this is not entirely correct: the $\mathsf{SLE}$ is quite special, but perhaps still not as special as I thought.

First, the non-self-crossing property is quite strong, which is why the simple random walk does not scale to a $\mathsf{SLE}$ but models of non-self-intersecting paths potentially do. For example, the loop-erased random walk ($\mathsf{SLE}\_2$) and the self avoiding walk ($\mathsf{SLE}\_\frac{8}{3}$) are two such models.

Another special feature of $\mathsf{SLE}$ stems from using the Brownian motion as its driving function. From my understanding of Loewner theory (which is miniscule), driving functions are not really supposed to look like a particle oscillating on the real line. Perhaps geometric function theorists think this is an abomination. I have no idea. But although some of these models are clearly related to the simple random walk, as far as I know, there was no obvious way to precisely state their relationships to the Brownian motion. Well, the $\mathsf{SLE}$ did provide that. The phase transitions caused by adjusting the diffusivity $\kappa$ are also indeed a sight to behold.

It is also nice that all the $\kappa$‘s are so neat, although I don’t have any understanding of this since this is related to the specific proofs of the models scaling to the $\mathsf{SLE}$.

Still, the $\mathsf{SLE}$ is not an all powerful object. The theory’s largest limitation is in its adoption of the Loewner theory to describe curves, which is restricted to two dimensions. But then I am not sure how possible a generalise to higher dimensions, especially in view of the implications to conformal field theory (which I know nothing about, but isn’t two dimensions the most well-studied?).

An additional issue lies in the extreme difficulty to prove convergence of these models to the $\mathsf{SLE}$. For example, Smirnov’s proof relies on special properties of the triangular lattice (I didn’t look at the proof myself, but I heard this personally from a very well respected researcher), and universality suggests that conformal invariance does still hold for other lattices, but I don’t see any progress on that front.

Anyway full disclosure I haven’t actually read Schramm’s original paper titled ‘Scaling limits of loop-erased random walks and uniform spanning trees’. So this whole post is hot garbage. Whatever.

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Comment 2025-06-05:

On scaling limits in higher dimensions

A large part of the recent development on the loop-erased random walk (LERW) have been focused on its scaling limit in three dimensions. As of now, the biggest problem is in describing this scaling limit directly in the continuum, analogous to the SLE in two dimensions. As far as I know, there is not much progress ¹.

Footnotes

1. Remark 1.5.2, Convergence of three-dimensional loop-erased random walk in the natural parametrization, Li & Shiraishi ↩