# A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability.

A quick google search gave a lot of references on SLE that are exactly the opposite of what I'm looking for, in the sense that they assume strong background in probability and no knowledge of complex analysis.

EDIT (In response to Timothy Chow's comment) :

I guess what I'm looking for is a reference that

(a) does not assume that the reader is familiar with the probability theory needed for SLE (except for the basics), so it should contain the required material on stochastic calculus, brownian motions, etc.

(b) describes in details the classical (non-stochastic) case

(c) contains an introduction to the stochastic case, which should focus more on the complex-analytic aspects than the probabilistic ones.

Thank you, Malik

• for one thing you can have Lowener evolution without being stochastic. also the theory of univalent functions is quite important - issues I couldn't appreciate at the time. and the koebe $\frac{1}{4}$ theorem on conformal mappings. the SLE theory * should * have the same flavor of the theory of conformal mappings. indeed Oded Schramm was a student of Bill Thurston. – john mangual Oct 18 '16 at 10:47
• – john mangual Oct 18 '16 at 10:55

As far as I'm aware, the most current in-depth book reference on SLE is Greg Lawler's book Conformally Invariant Processes in the Plane, a PDF of which is available here. From experience, SLE theory is extremely hard to grasp without a global understanding of what's going on, both on the complex analysis side and the probability side, in addition to its applications.

There's also a set of lecture notes from the PIMS summer school, more in the context of self-avoiding-walks.

• Yes, it also seems that the most recent in-depth book is Lawler's, although it seems to be more from the probabilistic point of view. I'll look into it, thanks – Malik Younsi Oct 3 '14 at 21:07
• @MalikYounsi: I think you'd be doing yourself a disservice if you're just interested in the complex analysis aspects of SLE. It's like eating a brownie without ice cream! For what it's worth, Lawler's book does review all of the probability that goes into SLE and talks about the complex analysis involved in disjoint sections (then mixing them together later). Are you interested in SLE primarily for developments in Loewner equation theory or stuff like Koebe's theorem/ Bieberbach Conjecture? – Alex R. Oct 3 '14 at 21:16
• ah, good point..! I guess Lawler's book looks pretty good for what I'm looking for. Thanks for the suggestion! – Malik Younsi Oct 3 '14 at 23:38
• Just for clarification, what is desired, (a) a text that assumes a stronger background in complex analysis than in probability, or (b) a text that focuses more on the complex-analytic aspects than the probabilistic aspects? The two are quite different. – Timothy Chow Oct 4 '14 at 2:26
• your link is not all of Lawler's book. in fact -- it's not the book at all. it is some really old lecture notes dating to 2002 – john mangual Oct 18 '16 at 11:32

Nathanael Berestycki and James Norris have written up notes on SLE, based on a Part III (ie. masters level) lecture course at Cambridge. http://www.statslab.cam.ac.uk/~james/Lectures/

I think they might partially match at least parts b) and c) of your requirements. Best, Alan.

• Looks interesting! It does indeed seem to agree with b) and c). I guess some combination of these notes and Lawler's book would be good. Thanks! – Malik Younsi Oct 5 '14 at 15:38

I recall seeing this post half a year ago while I was searching for knowledge myself regarding the very same topic. I thought I would throw my own master's dissertation into the picture. It can be found by clicking here.

The aim while writing was to make a rigorous, self-contained and readable introduction to the SLE, for anybody who has obtained knowledge in advanced complex analysis and stochastic calculus. It is my guess that you will never find a rigorous text that does not assume familiarity with at least stochastic calculation on application-level.

It treats compact hulls, and their relation to the Loewner equation. The overall method is taken directly from the brilliant notes written by Nathanael Berestycki and James Norris which may be found here. Then I introduce randomness through the Brownian motion, and present the proof of existence of the SLE trace, which was introduced in the article by Schramm & Rohde Annals Math. 161 (2005) 879-920. This may be the only part of my dissertation where the level of detail between my references and my own work is considerably different. Thus this may be the part that can save you some time.

I also include proofs of a "weak" phases characterization, which can be found in the Cambridge notes, among others. Finally, I discuss the simulation method presented by Tom Kennedy in his article J.Statist.Phys.128:1125-1137,2007

NB: I should of course stress that this is a master's dissertation and not an article from a journal, which means that it has not been through any approval-procedure apart from grading.

You'll correct me if I'm mistaken, but it seems to me that Vincent Beffara's lecture notes might be just what you're searching for.

The goal of these lectures is to provide a self-contained introduction to SLE and related objects, but some motivation is needed before introducing SLE as such; so it seems natural to start with a quick review of a few two-dimensional discrete models. The focus of the first part will be, for each model, to arrive at the question of scaling limits as quickly as possible, and to justify conformal invariance where it is known to hold in the limit.

In the second part we discuss Schramm’s insight that, under mild (and reasonable) assumptions in addition to conformal invariance, the limit has to be distributed as one of a one-parameter family of measures on curves, which he named Stochastic Loewner Evolutions. They are now universally known as Schramm-Loewner Evolutions. We give a precise definition of these random curves and present a few of their fundamental properties.

• These notes look interesting, but as the author writes in the foreword, they are more an introduction with essentially no detailed proofs, which isn't really what I'm looking for. Thank you for the reference though! – Malik Younsi Oct 3 '14 at 21:05