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2D growth processes: SLE and Loewner chains

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Michel Bauer, Denis Bernard

Mathematical Physics · Submitted 20 Feb 2006 · 142 references

Pith's verdict: A solid, honest review of SLE and its CFT connections — no new results, but a genuinely useful bridge between the math and physics literatures.

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arXiv IDmath-ph/0602049 (version 1)
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CitationDenis Bernard, Michel Bauer. "2D growth processes: SLE and Loewner chains." arXiv:math-ph/0602049 (2006). https://pith.science/paper/ISC4NCAF
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The paper

The paper claims

A pedagogical review of SLE and Loewner chains, covering their derivation from conformal invariance, connections to statistical mechanics and CFT, and tools for computing with SLE.

The reading

This paper is a review, not a new research result. It explains how certain random curves that appear in two-dimensional physics, such as the boundaries between regions in a critical material, can be described by a mathematical process called Schramm-Loewner evolution (SLE). The authors start with simple examples like percolation and random walks, then introduce Loewner chains, which encode the growth of a shape through complex analysis. They then present Schramm's argument that conformally invariant random curves must be driven by a single Brownian motion, leading to the one-parameter family SLE(kappa). A large part of the review connects SLE to conformal field theory (CFT), showing how SLE probabilities can be expressed as CFT correlation functions, and how the central charge of the CFT is determined by kappa. They also discuss special values of kappa that correspond to known models: kappa=2 for loop-erased random walks, kappa=6 for percolation, and kappa=8/3 for self-avoiding walks. The review includes appendices on probability theory and CFT to make it self-contained. It does not present new theorems or data; instead, it synthesizes existing results, many from the authors' own prior papers, into a coherent pedagogical narrative.
Extended reading notes

Core claim

Chordal SLE, defined by the Loewner equation driven by Brownian motion, is the unique conformally invariant random curve satisfying the domain Markov property, and the SLE/CFT correspondence identifies the boundary changing operator with scaling dimension h1;2=(6-kappa)/(2*kappa) and central charge c=(6-kappa)*(3*kappa-8)/(2*kappa). (Sections 4.2 and 5.3, eq. 25).

Load-bearing premise

The SLE/CFT correspondence (Section 5) assumes that the ratio of lattice partition functions conditioned on interface shapes converges, in the continuum limit, to ratios of CFT correlation functions in the cut domain (eq. 24). The authors explicitly note they 'cheated a little' in deriving this formula, and the convergence is unproven for most lattice models, including Ising and O(n) interfaces, as the paper acknowledges. If this convergence fails, the martingale identification between SLE and CFT would break down.

Original abstract

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,523 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 This review provides an introduction to two dimensional growth processes. contribution 0.86 0.59 0.04 asserted 0.98
s1 Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. contribution 0.73 0.42 0.11 asserted 1.00
s2 It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. method 0.58 0.49 0.02 asserted 1.00
s3 SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. contribution 0.61 0.24 0.06 asserted 0.98
s4 A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. contribution 0.96 0.33 0.23 asserted 0.98
s5 Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. contribution 0.89 0.27 0.07 asserted 0.96
s6 This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject. gap 0.86 0.45 0.04 asserted 0.93

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.67.

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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Example of square and hexagonal lattice domains.
Figure 2. Left : the boundary of an hexagonal lattice domain wit
Figure 3. The definition of the percolation interface.
Figure 4. The percolation interface as a growth process.
Figure 5. Samples of the percolation interface for increasing
Figure 6. Samples of the harmonic navigator for increasing siz
Figure 7. Samples of the harmonic anti-navigator for increasi
Figure 8. Samples of the percolation navigator for increasing
Figure 9. Samples of the boundary harmonic navigator for incre
Figure 10. A loop-erased random walk with its shadow.
Figure 11. On the left: a large loop is about to be created. On the
Figure 12. A sample of the loop-erased random walk for the two bo
Figure 13. The logarithm of the average length of the loop-eras
Figure 14. A few self avoiding walks. Fig.14 shows a few samples. Producing a single clean sample of reasonable size starting from a walk far from equilibrium (like a straight segment) takes many Monte Carlo iterations. In fact it takes roughly the time needed to compute the fractal dimension with 1 percent error for our previous examples. However, once the large time regime is reached, one estimates that only a fraction of the number of iterations needed to thermalize is enough to get a new (almost) independent sample, so that a good numerical estimate of the fractal dimension of the self…
Figure 15. A configuration of the Ising model. 28
Figure 16. A sample for the critical Ising model. The bottom lin
Figure 17. FK-clusters configuration in the Potts models and th
Figure 18. An illustration of situations 1) and 2) for the case o
Figure 19. A DLA sample. extensively. There is now a consensus that the fractal dimension of 2d DLA clusters is Ddla ≃ 1.71. There is actually a debate on whether this dimension is geometry dependent but a recent study [127] seems to indicate that DLA clusters in a radial geometry and a channel geometry have identical fractal dimension. To add a new particle to the growing domain, a random walk has to wander around and the position at which it finally sticks is influenced by the whole domain. To rephrase this, for each new particle one has to solve the outer Laplace equation, a non-local…
Figure 20. An example of hull K in the upper hall plane together with the uniformizing map fK : H → H \ K and its invserse gK : H \ K → H. Consider for instance the hull formed by the segment {z = iy, y ∈ [0, a]}, a > 0, drawn between the origin 0 and point ia in the upper half plane. The uniformizing map g from H \ [0, ia] to H and its inverse are given by: g(z) = √ z 2 + a 2 , f(z) ≡ g −1 (z) = √ z 2 − a 2 The square root cut is specified by demanding that g(z) ∼ z at infinity so that g(x) is negative (positive) on the real axis for x negative (positive). It has a cut along the segmen
Figure 21. (a) The upper half plane cuted along a vertical slit.
Figure 22. Conformal invariance for change of domain. The meas
Figure 23. Conformal invariance for conditional probabiliti
Figure 24. The phases of SLE. The existence of these different phases may be grasped by looking at the motion of a point of the real axis. So let x ∈ R and consider the real process Yt ≡ ht(x)/ √ κ = (gt(x) − ξt)/ √ κ. By construction it is a Bessel process since it satisfies the stochastic equation dYt = (2/κ) Yt dt + dBt . A d-dimensional Bessel process is the process given by the modulus of Brownian motion in dimension d. If Rt denotes this modulus, it satisfies the stochastic equation dRt = (d−1)/2 Rt +dBt , see Appendix A. So the effective dimension for the process Yt is deff =
Figure 25. Two SLE samples for κ = 4 and κ = 6. will be given in the following Section 4.4. As for chordal SLE, they are de￾fined via a version of Loewner equation for a uniformizing map gt from which one reconstructs the curve. They are conformally invariant in the sense that they satisfy the Markov and identical increment properties. They differ by their global topological behavior but their local properties are identical. For instance, they have three different phases depending on the value of κ as for chordal SLE – with the same phase diagram – and their traces have identical fractal…
Figure 26. Three avatars of SLE: chordal, radial and dipolar.
Figure 27. A domain D cutted along portions of interfaces. Assuming the model to be at its critical point, we expect to be able to describe its continuum limit by a conformal field theory (CFT). At least for a wide class of observables O, the partition function and statistical expectation values become CFT correlation functions: ≺ O ≻D→ hOiD, b.c. hIiD, b.c. where h· · ·iD, b.c. refer to the CFT correlation functions in the domain D with specific boundary conditions (b.c.). We need to write the correlation function 68
Figure 28. A representation of the boundary hull state |Kti = Ght |ψ1;2i and of the map intertwining different formulations of the CFT. The state Ght |ψ1;2i may be interpreted as follows. Imagine defining the confor￾mal field theory in Ht via a radial quantization, so that the conformal Hilbert spaces are defined over curves topologically equivalent to half circles around the origin. Then, the SLE hulls manifest themselves as disturbances localized around the origin, and as such they generate states in the conformal Hilbert spaces. Since Ght intertwines the CFT in H and in Ht , these states ar
Figure 29. A representation of two configurations of growing curves forming arches whenever they touch themself. 82
Figure 30. Arch configurations for four SLE processes in an arbitrary domain. 86
Figure 31. The commutative diagram associated to the two alter
Figure 32. A SLE curve growing in the upper half plane minus a hul
Figure 33. Crossing probability in a equilateral triangle.
Figure 34. Crossing probability is identical to SLE swallowin
Figure 35. Uniformizing maps intertwining the ‘physical’
Figure 36. Cups formation in Laplacian growth. This behavior is quite generic. Conformal maps ft(w) such that their deriva￾tives are polynomials in w −1 are stable by the Laplacian growth dynamics. By construction, their zeroes are localized inside the unit disk. A singularity in the boundary curve occurs if one of these zeroes converges to the unit circle. The singularity is then a cusp ℓc y 2 ≃ x 3 as can be seen by expanding locally the conformal map around its singular point. Once regularized with an explicit ultraviolet cut-off, the processes are ex￾pected to be well defined for all time
Figure 37. Subcritical and critical algebraic curves. Black c
Figure 38. An exemple of cluster obtained by iterating conform
Figure 39. Iteration of conformal maps. To fully define the model one has to specify the choice of the parameter λn and θn at each iteration. Since λn codes for the linear size of the added bump and 132

Figures reproduced from arXiv:math-ph/0602049 by the authors.

References
How the literature uses it

18 arXiv papers cite this in 26 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 4 sentences without a confident label

Uses its method or tool 1 paper
Cites it as background 18 papers
  • CLEs are families of conformally invariant measures for random loops on domains of the plane [ref] , which emerged as generalizations of Schramm Loewner evolutions (SLEs) [ref] .

    Random loops and conformal field theory · Introduction · 0.97

  • Recently, a new picture of CFT has emerged thanks to the advent of Schramm-Loewner evolution (SLE) [ref] , and conformal loop ensembles (CLE) [ref] (for reviews on SLE and related aspects, see for instance [ref] ).

    Hypotrochoids in conformal restriction systems and Virasoro descendants · Introduction · 1.00

  • In the works by Bauer and Bernard [ref] as well as in the subsequent papers [ref] a nontrivial connection between $SLE$ and conformal field theories with central charge $c≤1$ was revealed.

    Connecting SLE and minisuperspace Liouville gravity · Introduction · 0.94

Discussion

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