OOded Schramm: From Circle Packing to SLE
Steffen Rohde ∗ July 12, 2010
Contents ∗ University of Washington, Supported in part by NSF Grant DMS-0800968. a r X i v : . [ m a t h . C V ] J u l Introduction
When I first met Oded Schramm in January 1991 at the University of California, San Diego, heintroduced himself as a “Circle Packer”. This modest description referred to his Ph.D. thesisaround the Koebe-Andreev-Thurston theorem and a discrete version of the Riemann mappingtheorem, explained below. In a series of highly original papers, some joint with Zhen-Xu He, hecreated powerful new tools out of thin air, and provided the field with elegant new ideas. At thetime of his deadly accident on September 1st, 2008, he was widely considered as one of the mostinnovative and influential probabilists of his time. Undoubtedly, he is best known for his inventionof what is now called the Schramm-Loewner Evolution (SLE), and for his subsequent collaborationwith Greg Lawler and Wendelin Werner that led to such celebrated results as a determinationof the intersection exponents of two-dimensional Brownian motion and a proof of Mandelbrot’sconjecture about the Hausdorff dimension of the Brownian frontier. But already his previous workbears witness to the brilliance of his mind, and many of his early papers contain both deep andbeautifully simple ideas that deserve better knowing.In this note, I will describe some highlights of his work in circle packings and the Koebeconjecture, as well as on SLE. As Oded has co-authored close to 20 papers related to circle packingsand more than 20 papers involving SLE, only a fraction can be discussed in detail here. Thetransition from circle packing to SLE was through a long sequence of influential papers concerningprobability on graphs, many of them written jointly with Itai Benjamini. I will present almost nowork from that period (some of these results are described elsewhere in this volume, for instance inChristophe Garban’s article on Noise Sensitivity). In that respect, the title of this note is perhapsmisleading.In order to avoid getting lost in technicalities, arguments will be sketched at best, and oftenideas of proofs will be illustrated by analogies only. In an attempt to present the evolution ofOded’s mathematics, I will describe his work in essentially chronological order.Oded was a truly exceptional person: not only was his clear and innovative way of thinking aninspiration to everyone who knew him, but also his caring, modest and relaxed attitude generated acomfortable atmosphere. As inappropriate as it might be, I have included some personal anecdotesas well as a few quotes from email exchanges with Oded, in order to at least hint at these sides ofOded that are not visible in the published literature.This note is not meant to be an overview article about circle packings or SLE. My primeconcern is to give a somewhat self-contained account of Oded’s contributions. Since SLE has beenfeatured in several excellent articles and even a book, but most of Oded’s work on circle packingis accessible only through his original papers, the first part is a bit more expository and containsmore background. The expert in either field will find nothing new, and will find a very incompletelist of references. My apology to everyone whose contribution is either unmentioned or, perhapseven worse, mentioned without proper reference.
Acknowledgement:
I would like to thank Mario Bonk, Jose Fern´andez, Jim Gill, Joan Lind,Don Marshall, Wendelin Werner and Michel Zinsmeister for helpful comments on a first draft. Iwould also like to thank Andrey Mishchenko for generating Figure 3, and Don Marshall for Figure6. 2
Circle Packing and the Koebe Conjecture
Oded Schramm was able to create, seemingly without effort, ingenious new ideas and methods.Indeed, he would be more likely to invent a new approach than to search the literature for anexisting one. In this way, in addition to proving wonderful new theorems, he rediscovered manyknown results, often with completely new proofs. We will see many examples throughout this note.Oded received his Ph.D. in 1990 under William Thurston’s direction at Princeton. His thesis,and the majority of his work until the mid 90’s, was concerned with the fascinating topic of circlepackings. Let us begin with some background and a very brief overview of some highlights of thisfield prior to Oded’s thesis. Other surveys are [Sa] and [Ste2].
According to the Riemann mapping theorem, every simply connected planar domain, except theplane itself, is conformally equivalent to a disc. The conformal map to the disc is unique, upto postcomposition with an automorphism of the disc (which is a M¨obius transformation). Thestandard proof exhibits the map as a solution of an extremal problem (among all maps of thedomain into the disc, maximize the derivative at a given point). The situation is quite differentfor multiply connected domains, partly due to the lack of a standard target domain. The standardproof can be modified to yield a conformal map onto a parallel slit domain (each complementarycomponent is a horizontal line segment or a point). Koebe showed that every finitely connected domain is conformally equivalent to a circle domain (every boundary component is a circle or apoint), in an essentially unique way. No proof similar to the standard proof of the Riemann mappingtheorem is known.
Theorem 2.1 ([Ko1]) . For every domain Ω ⊂ C with finitely many connected boundary compo-nents, there is a conformal map f onto a domain Ω (cid:48) ⊂ C all of whose boundary components arecircles or points. Both f and Ω (cid:48) are unique up to a M¨obius transformation. Koebe conjectured (p. 358 of [Ko1]) that the same is true for infinitely connected domains. Itlater turned out that uniqueness of the circle domain can fail (for instance, it fails whenever the setof point-components of the boundary has positive area, as a simple application of the measurableRiemann mapping theorem shows). But existence of a conformally equivalent circle domain is stillopen, and is known as Koebe’s conjecture or “Kreisnormierungsproblem”. It motivated a lot ofOded’s research.There is a close connection between Koebe’s theorem and circle packings. A circle packing P is a collection (finite or infinite) of closed discs D in the two dimensional plane C , or in the twodimensional sphere S , with disjoint interiors. Associated with a circle packing is its tangency graph or nerve G = ( V, E ), whose vertices correspond to the discs, and such that two vertices are joinedby an edge if and only if the corresponding discs are tangent. We will only consider packings whosetangency graph is connected.Conversely, the Koebe-Andreev-Thurston
Circle Packing Theorem guarantees the existence ofpackings with prescribed combinatorics. Loosely speaking, a planar graph is a graph that can bedrawn in the plane so that edges do not cross. Our graphs will not have double edges (two edgeswith the same endpoints) or loops (an edge whose endpoints coincide).3igure 1: A circle packing and its tangency graph.
Theorem 2.2 ([Ko2], [T], [A1]) . For every finite planar graph G , there is a circle packing in theplane with nerve G . The packing is unique (up to M¨obius transformations) if G is a triangulationof S . See the following sections for the history of this theorem, and sketches of proofs. In particular, inSection 2.3 we will indicate how the Circle Packing Theorem 2.2 can be obtained from the KoebeTheorem 2.1, and conversely that the Koebe theorem can be deduced from the Circle PackingTheorem. Every finite planar graph can be extended (by adding vertices and edges as in Figure3(c)) to a triangulation, hence packability of triangulations implies packability of finite planar graphs(there are many ways to extend a graph to a triangulation, and uniqueness of the packing is no longertrue). The situation is more complicated for infinite graphs. Oded wrote several papers dealing withthis case. Thurston conjectured that circle packings approximate conformal maps, in the followingsense: Consider the hexagonal packing H ε of circles of radius ε (a portion is visible in Fig. 2 andFig. 3(a)). Let Ω ⊂ C be a domain (a connected open set). Approximate Ω from the inside by acircle packing P ε of circles of Ω ∩ H ε , as in Fig. 2 and Fig. 3(a) (more precisely, take the connectedcomponent containing p of the union of those circles whose six neighbors are still contained inΩ). Complete the nerve of this packing by adding one vertex for each connected component ofthe complement to obtain a triangulation of the sphere (there are three new vertices v , v , v inFig. 3(c); the three copies of v are to be identified). By the Circle Packing Theorem, there isa circle packing P (cid:48) ε of the sphere with the same tangency graph (Figures 2 and 3(d) show thesepackings after stereographic projection from the sphere onto the plane; the circle corresponding to v was chosen as the upper hemisphere and became the outside of the large circle after projection).Notice that each of the complementary components now corresponds to one (“large”) circle of P (cid:48) ε ,and the circles in the boundary of P ε are tangent to these complementary circles. Now consider themap f ε that sends the centers of the circles of P ε to the corresponding centers in P (cid:48) ε , and extend it ina piecewise linear fashion. Rodin and Sullivan proved Thurston’s conjecture that f ε approximates4igure 2: A circle packing approximation to a Riemann map.the Riemann map, if Ω is simply connected (see Fig. 2): Theorem 2.3. [RSu] Let Ω be simply connected, p, q ∈ Ω , and P (cid:48) ε normalized such that the com-plementary circle is the unit circle, and such that the circle closest to p (resp. q ) corresponds toa circle containing (resp. some positive real number). Then the above maps f ε converge to theconformal map f : Ω → D that is normalized by f ( p ) = 0 and f ( q ) > , uniformly on compactsubsets of Ω as ε → . Their proof depends crucially on the non-trivial uniqueness of the hexagonal packing as theonly packing in the plane with nerve the triangular lattice. Oded found remarkable improvementsand generalizations of this theorem. See Section 2.6 for further discussion.
Despite their intrinsic beauty (see the book [Ste2] for stunning illustrations and an elementaryintroduction), circle packings are interesting because they provide a canonical and conformallynatural way to embed a planar graph into a surface. Thus they have applications to combinatorics(for instance the proof of Miller and Thurston [MT] of the Lipton-Tarjan separator theorem, seee.g. the slides of Oded’s circle packing talk on his memorial webpage), to differential geometry(for instance the construction of minimal surfaces by Bobenko, Hoffmann and Springborn [BHS]and their references), to geometric analysis (for instance, the Bonk-Kleiner [BK] quasisymmetricparametrization of Ahlfors 2-regular LLC topological spheres) to discrete probability theory (forinstance, through the work of Benjamini and Schramm on harmonic functions on graphs andrecurrence on random planar graphs [BS1],[BS2], [BS3]) and of course to complex analysis (discreteanalytic functions, conformal mapping). However, Oded’s work on circle packing did not followany “main-stream” in conformal geometry or geometric function theory. I believe he continued towork on them just because he liked it. His interest never wavered, and many of his numerous latecontributions to Wikipedia were about this topic.Existence and uniqueness are intimately connected. Nevertheless, for better readability I willdiscuss them in two separate sections. 5 a) (b) (c) (d)
Figure 3: A circle packing approximation of a triply connected domain, its nerve, its completionto a triangulation of S , and a combinatorially equivalent circle packing; (a)-(c) are from Oded’sthesis; thanks to Andrey Mishchenko for creating (d)6 .3 Existence of Packings Oded applied the highest standards to his proofs and was not satisfied with “ugly” proofs. Aswe shall see, he found four (!) different new existence proofs for circle packings with prescribedcombinatorics. Before discussing them, let us have a glance at previous proofs.The Circle Packing Theorem was first proved by Koebe [Ko2] in 1936. Koebe’s proof of existencewas based on his earlier result that every planar domain Ω with finitely many boundary components,say m , can be mapped conformally onto a circle domain . A simple iterative algorithm, due toKoebe, provides an infinite sequence Ω n of domains conformally equivalent to Ω and such that Ω n converges to a circle domain. To obtain Ω n +1 from Ω n , just apply the Riemann mapping theoremto the simply connected domain (in C ∪ {∞} ) containing Ω n whose boundary corresponds to the( n mod m ) − th boundary component of Ω. With the conformal equivalence of finitely connecteddomains and circle domains established, a circle packing realizing a given tangency pattern can beobtained as a limit of circle domains: Just construct a sequence of m -connected domains so that theboundary components approach each other according to the given tangency pattern. For instance,if the graph G = ( V, E ) is embedded in the plane by specifying simple curves γ e : [0 , → S , e ∈ E, then the complement Ω ε of the set (cid:91) e ∈ E γ e [0 , / − ε ] ∪ (cid:91) e ∈ E γ e [1 / ε, (cid:48) ε converge to the desired circle packing when ε → . Koebe’s theorem was nearly forgotten. In the late 1970’s, Thurston rediscovered the circlepacking theorem as an interpretation of a result of Andreev [A1], [A2] on convex polyhedra inhyperbolic space, and obtained uniqueness from Mostow’s rigidity theorem. He suggested an al-gorithm to compute a circle packing (see [RSu]) and conjectured Theorem 2.3, which started thefield of circle packing. Convergence of Thurston’s algorithm was proved in [dV1]. Other existenceproofs are based on a Perron family construction (see [Ste2]) and on a variational principle [dV2].Oded’s thesis [S1] was chiefly concerned with a generalization of the existence theorem topackings with prescribed convex shapes instead of discs, and to applications. A consequence ([S1],Proposition 8.1) of his “Monster packing theorem” is, roughly speaking, that the circle packingtheorem still holds if discs are replaced by smooth convex sets.
Theorem 2.4. ([S1], Proposition 8.1) For every triangulation G = ( V, E ) of the sphere, every a ∈ V , every choice of smooth strictly convex sets D v for v ∈ V \ { a } , and every smooth simpleclosed curve C , there is a packing P = { P v : v ∈ V } with nerve G , such that P a is the exterior of C and each P v , v ∈ V \ { a } is positively homothetic to D v . Sets A and B are positively homothetic if there is r > s ∈ C with A = rB + s. Strictconvexity (instead of just convexity) was only used to rule out that three of the prescribed setscould meet in one point (after dilation and translation), and thus his packing theorem applied inmuch more generality. Oded’s approach was topological in nature: Based on a cleverly constructed7igure 4: A packing of convex shapes in a Jordan domain, from Oded’s thesisspanning tree of G , he constructed what he called a “monster”. This refers to a certain | V | -dimensional space of configurations of sets homothetic to the given convex shapes, with tangenciesaccording to the tree, and certain non-intersection properties. Existence of a packing was thenobtained as a consequence of Brower’s fixed point theorem. Here is a poetic description, quotedfrom his thesis: One can just see the terrible monster swinging its arms in sheer rage, the tentacles causing afrightful hiss, as they rub against each other.
Applying Theorem 2.4 to the situation of Figure 3, with D v chosen as circles when v / ∈{ v , v , v } , and arbitrary convex sets D v j , Oded adopted the Rodin-Sullivan convergence proofto obtain a new proof of the following generalization of Koebe’s mapping theorem. The originalproof of Courant, Manel and Shiffman [CMS] employed a very different (variational) method. Theorem 2.5. ([S1], Theorem 9.1; [CMS]) For every n + 1 -connected domain Ω , every simplyconnected domain D ⊂ C and every choice of n convex sets D j , there are sets D (cid:48) j which arepositively homothetic to D j such that Ω is conformally equivalent to D \ ∪ n D (cid:48) j . Later [S7] he was able to dispose of the convexity assumption, and proved the packing theoremfor smoothly bounded but otherwise arbitrary shapes. As a consequence, he was able to generalizeTheorem 2.5 to arbitrary (not neccessarily convex) compact connected sets D j , thus rediscoveringa theorem due to Brandt [Br] and Harrington [Ha].Oded then developed a differentiable approach to the circle packing theorem. In [S3] he shows Theorem 2.6. ([S3],Theorem 1.1) Let P be a 3-dimensional convex polyhedron, and let K ⊂ R be a smooth strictly convex body. Then there exists a convex polyhedron Q ⊂ R combinatoriallyequivalent to P which midscribes K. Here “ Q midscribes K ” means that all edges of Q are tangent to ∂K. He also shows that thespace of such Q is a six-dimensional smooth manifold, if the boundary of K is smooth and has8ositive Gaussian curvature. For K = S , Theorem 2.6 has been stated by Koebe [Ko2] and provedby Thurston [T] using Andreev’s theorem [A1], [A2]. Oded notes that Thurston’s midscribabilityproof based on the circle packing theorem can be reversed, so that Theorem 2.6 yields a new proofof the Circle Packing Theorem (given a triangulation, just take K = S , Q the midscribing convexpolyhedron with the combinatorics of the packing, and for each vertex v ∈ V , let D v be the set ofpoints on S that are visible from v ).One defect of the continuity method in his thesis was that it did not provide a proof of uniqueness(see next section). In [S4] he presented a completely different approach to prove a far more generalpacking theorem, that had the added benefit of yielding uniqueness, too. A quote from [S4]: It is just about the most general packing theorem of this kind that one could hope for (it is moregeneral than I have ever hoped for).
A consequence of [S4] (Theorem 3.2 and Theorem 3.5) is
Theorem 2.7.
Let G be a planar graph, and for each vertex v ∈ V , let F v be a proper 3-manifoldof smooth topological disks in S , with the property that the pattern of intersection of any two setsin F v is topologically the pattern of intersection of two circles. Then there is a packing P whosenerve is G and which satisfies P v ∈ F v for v ∈ V . The requirement that F v is a 3-manifold requires specification of a topology on the space ofsubsets of S : Say that subsets A n ⊂ S converge to A if lim sup A n = lim inf A n = A and A c = int(lim sup A cn ) . An example is obtained by taking a smooth strictly convex set K in R andletting F be the family of intersections H ∩ ∂K, where H is any (affine) half-space intersecting theinterior of K. Specializing to K = S , F is the familiy of circles and the choice F v = F for all v reduces to the circle packing theorem.The proof of Theorem 2.7 is based on his incompatibility theorem , described in the next section.It provides uniqueness of the packing (given some normalization), which is key to proving existence,using continuity and topology (in particular invariance of domains). I was always impressed by the flexibility of Oded’s mind, in particular his ability to let go of apromising idea. If an idea did not yield a desired result, it did not take long for him to come upwith a completely different, and in many cases more beautiful, approach. He once told me thatif he did not make progress within three days of thinking about a problem, he would move on todifferent problems.Following Koebe and Schottky, uniqueness of finitely connected circle domains (up to M¨obiusimages) is not hard to show, using the reflection principle: If two circle domains are conformallyequivalent, the conformal map can be extended by reflection across each of the boundary circles, toobtain a conformal map between larger domains (that are still circle domains). Continuing in thisfashion, one obtains a conformal map between complements of limit sets of reflection groups. Asthey are Cantor sets of area zero, the map extends to a conformal map of the whole sphere, henceis a M¨obius transformation. Uniqueness of the (finite) circle packing can be proved in a similar9ashion. To date, the strongest rigidity result whose proof is based on this method is the followingtheorem of He and Schramm. See [Bo] for the related rigidity of Sierpinski carpets.
Theorem 2.8 ([HS2], Theorem A) . If Ω is a circle domain whose boundary has σ − finite length,then Ω is rigid (any conformal map to another circle domain is M¨obius). For finite packings, there are several technically simpler proofs. The shortest and most ele-mentary of them is deferred to the end of this section, since I believe it has been discovered last.Rigidity of infinite packings lies deeper. The rigidity of the hexagonal packing, crucial in the proofof the Rodin-Sullivan theorem as elaborated in Section 2.6 below, was originally obtained fromdeep results of Sullivan’s concerning hyperbolic geometry. He’s thesis [He] gave a quantitative andsimpler proof, still using the above reflection group arguments and the theory of quasiconformalmaps. In one of his first papers [S2], Oded gave an elegant combinatorial proof that at the sametime was more general:
Theorem 2.9 ([S2], Theorem 1.1) . Let G be an infinite, planar triangulation and P a circle packingon the sphere S with nerve G. If S \ carrier( P ) is at most countable, then P is rigid (any othercircle packing with the same combinatorics is M¨obius equivalent). The carrier of a packing { D v : v ∈ V ( G ) } is the union of the (closed) discs D v and the“interstices” (bounded by three mutually touching circles) in the complement of the packing. Therigidity of the hexagonal packing follows immediately, since its carrier is the whole plane.The ingenious new tool is his Incompatibility Theorem , a combinatorial analog to the confor-mal modulus of a quadrilateral. To fully appreciate it, lets first look at its classical continuouscounterpart, and defer the statement of the Theorem to Section 2.4.2 below.
If you conformally map a 3x1-rectangle to a disc, such that the center maps to the center, whatfraction of the circle does the image of one of the two short sides occupy? Despite having knownthe effect of “crowding” in numerical conformal mapping, I was surprised to learn of the numericalvalue of 0 . ... from Don Marshall (see [MS].) Of course, the precise value can be easily computedas an elliptic integral, but if asked for a rough guess, most answers are around 1/10 (the uniformmeasure with respect to length would give 1/8). Oded’s answer, after a moments thought (during atennis match in the early 90’s), was 1/64, reasoning that this is the probability of a planar randomwalker to take each of his first three steps “to the right”.An important classical conformal invariant, masterfully employed by Oded in many of his papers,is the modulus of a quadrilateral. Let Ω be a simply connected domain in the plane that is boundedby a simple closed curve, and let p , p , p and p be four consecutive points on ∂ Ω . Then thereis a unique
M > f : Ω → [0 , M ] × [0 ,
1] and such that f takes the p j to the four corners with f ( p ) = 0 (by a classical theorem of Caratheodory, f extendshomeomorphically to the boundary of the domains). There are several quite different instructiveproofs of uniqueness of M . Each of the following three techniques has a counterpart in the circlepacking world that has been employed by Oded. Suppose we are given two rectangles and aconformal map f between them taking corners to corners.10ne method to prove uniqueness is to repeatedly reflect f across the sides of the rectangles.The resulting extention is a conformal map of the plane, hence linear, and it follows that the aspectratio is unchanged. This is similar to the aforementioned Schottky group argument.A second method is to explicitly define a quantity λ depending on a configuration (Ω , p , ..., p )in such a way that it is conformally invariant and such that one can compute λ for the rectangle[0 , M ] × [0 , extremal length of the family Γ of all rectifiable curves γ joining two opposite “sides” [ p , p
2] and [ p , p
4] of Ω . The extremal length of a curve family Γ isdefined as λ (Γ) = sup ρ (inf γ (cid:82) γ ρ | dz | ) (cid:82) C ρ dxdy , (1)where the supremum is over all “metrics” (measurable functions) ρ : C → [0 , ∞ ). For the family ofcurves joining the horizontal sides in the rectangle [0 , M ] × [0 , λ (Γ) = M. This simple idea is actually one of the most powerful tools of geometric function theory. See e.g.[Po2] or [GM] for references, properties and applications.Discrete versions of extremal length (or the “conformal modulus” 1 /λ ) have been around sincethe work of Duffin [Duf]. In conformal geometry, they have been very succesfully employed be-ginning with the groundbreaking paper [Can]. Cannon’s extremal length on a graph G = ( V, E )is obtained from (1) by viewing non-negative functions ρ : V → [0 , ∞ ) as metrics on G , definingthe length of a “curve” γ ⊂ V as the sum (cid:80) v ∈ γ ρ ( v ), and the “area” of the graph as (cid:80) ρ ( v ) . See [CFP1] for an account of Cannon’s discrete Riemann mapping theorem, and for instance thepapers [HK] and [BK] concerning applications to quasiconformal geometry. Oded’s applications tosquare packings and transboundary extremal length are briefly discussed in Section 2.7 below.A third and very different method is topological in nature and is one of the key ideas in [HS1].Suppose we are given two rectangles Ω , Ω (cid:48) with different aspect ratio and overlapping as in Fig. 5,and a conformal map f between them mapping corners to corners. Then the difference f ( z ) − z is (cid:54) = 0 on the boundary ∂ Ω. Traversing ∂ Ω in the positive direction, inspection of Fig. 5 shows thatthe image curve under f ( z ) − z winds around 0 in the negative direction. But a negative windingis impossible for analytic functions (by the argument principle, the winding number counts thenumber of preimages of 0). Again consider the overlapping rectangles Ω , Ω (cid:48) of Fig. 5, and two combinatorially equivalent pack-ings P, P (cid:48) whose nerves triangulate the rectangles, as in Fig. 6. Assume for simplicity that the sets D v and D (cid:48) v of the packings are closed topological discs (except for the four sides D , ...D , D (cid:48) , ..., D (cid:48) of the rectangles, which are considered to be sets of the packing). Intuitively, two topological discs D and D (cid:48) are called incompatible if they intersect as in Fig. 5. More formally, say that D cuts D (cid:48) if there are two points in D (cid:48) \ interior( D ) that cannot be connected by a curve in interior( D (cid:48) \ D ).Then Oded calls D and D (cid:48) incompatible if D cuts D (cid:48) or D (cid:48) cuts D. As he notes, the motivationfor the definition comes from the simple but very important observation that the possible patternsof intersection of two circles are very special, topologically.
Indeed, any two circles are compatible.
Theorem 2.10 ([S2], Theorem 3.1) . There is a vertex v for which D v and D (cid:48) v are incompatible. M and hence are similar: if they could, justplace the two packings on top of each other as in Fig. 6 and obtain two incompatible circles, acontradiction. In the same vein, it is not difficult to reduce the proof of the rigidity Theorem 2.9to an application of the incompatibility theorem. To end this section, here is a beautifully simple proof of the rigidity of finite circle packings whosenerve triangulates S . I copied it from the wikipedia (search for circle packing theorem), and believeit is due to Oded. As before, stereographically project the packing to obtain a packing of discs inthe plane. This time, assume that the north pole belongs to the complement of the discs, so thatthe planar packing will consist of three “outer” circles and the remaining circles contained in theinterstice between them. “There is also a more elementary proof based on the maximum principle, which we now sketch.The key observation here is that if you look at the triangle formed by connecting the centers ofthree mutually tangent circles, then the angle formed at the center of one of the circles is monotonedecreasing in its radius and monotone increasing in the two other radii. Consider two packingscorresponding to G. First apply reflections and M¨obius transformations to make the outer circlesin these two packings correspond to each other and have the same radii. Next, consider a vertex vwhere the ratio between the corresponding radius in the one packing and the corresponding radiusin the other packing is maximized. Since the angle sum formed at the center of the correspondingcircles is the same (360 degrees) in both packings, it follows from the above observation that theradius ratio is the same at all the neighbors of v as well. Since G is connected, we conclude theradii in the two packings are the same, which proves uniqueness.” Koebe’s 1908 conjecture [Ko1] that every planar domain can be mapped conformally onto a circledomain is still open, despite considerable effort by Koebe and others. Important contributionswere made by Gr¨otzsch, Strebel, Sibner and others. One difficulty is the aforementioned lack ofuniqueness. Another problem is that Theorem 2.5 is not true in the infinitely connected case, as thefollowing example from [S6] illustrates: If K = { x + iy : x = 0 , ± , ± , ± , ..., y ∈ [ − , } , and if D = ˆ C \ K , then there is no conformal map f of D , normalized by f ( z ) − z → z → ∞ , such thatthe component { iy : y ∈ [ − , } of ∂D corresponds to a horizontal line segment (or a point) whilethe other complementary components of f ( D ) are vertical line segments. The same example alsoillustrates the fundamental continuity problem: There is a circle domain D (cid:48) conformally equivalentto D, but the boundary component corresponding to { iy : y ∈ [ − , } is just a point, so that theconformal map from D (cid:48) to D cannot be extended to the boundary.The first joint paper of He and Schramm provided a breakthrough:13 heorem 2.11 ([HS1]) . If Ω has at most countably many boundary components, then Ω is confor-mally equivalent to a circle domain Ω (cid:48) , and Ω (cid:48) is unique up to M¨obius transformation. Essentially, this result is still the strongest to date. Oded later [S6] gave a conceptually differentand simpler proof based on his transboundary extremal length, which also applies to certain classesof domains with uncountably many boundary components.The proof in [HS1] used transfinite induction and was based on the topological concept of the fixed-point index . I will illustrate the beautiful idea by sketching their proof of uniqueness. As itturned out, this argument for uniqueness had been given earlier by Strebel [Str]. The simple butcrucial idea is to use the following (see [HS1], Lemma 2.2): If f is a fixed-point free orientationpreserving homeomorphism between two circles C (cid:48) and C (cid:48)(cid:48) , then the winding number of the curve f ( z ) − z, z ∈ C (cid:48) , around 0 is non-negative (recall Fig. 5 for a situation where the winding numberis negative). Let f : Ω (cid:48) → Ω (cid:48)(cid:48) be a conformal map and assume for simplicity that f extendscontinuously to the boundary (in case of finitely many boundary components this is immediatefrom the reflection principle, but in the countable case this step is non-trivial), and that f hasno fixed points on the boundary. Composing with M¨obius transformations, we may assume that ∞ ∈ Ω (cid:48) and that f ( z ) = z + a /z + a /z + · · · . We want to show that f is the identity. If not,denote a j the first non-zero Taylor coefficient, then f ( z ) − z has winding number − j as z traversesa large circle | z | = R, because f ( z ) − z behaves like a j z − j . Moreover, each circular boundarycomponent maps to a circular component. These boundary components are oriented negatively (tokeep the domain to the left) and thus, by the above crucial idea, contribute a non-positive numberto the winding of f ( z ) − z, z ∈ ∂ (Ω ∩ {| z | ≤ R } around 0. Hence the total winding number isnegative, contradicting their generalization of the argument principle (the winding number countsthe number of zeroes of f ( z ) − z. ) Of course, I have swept most details under the rug, most notablythe proof of continuity based on a powerful generalization of Schwarz’ Lemma to circle domains(Theorem 0.6 in [HS1]).Combining the fixed-point index method of [HS1] with an analysis of quasiconformal deforma-tions using the reflection group approach and Sullivan’s rigidity theorems, He and Schramm [HS4]improve Theorem 2.11 to domains Ω for which all boundary components are circles or points ex-cept those in a countable and closed family. They also obtain the following generalization of theRiemann mapping theorem. Let A ⊆ C be simply connected. Theorem 2.12 ([HS4],[HS5]) . If Ω ⊂ A is a relative circle domain (each connected componentof A \ Ω is a point or a closed disc), then there is a relative circle domain Ω ∗ in D conformallyequivalent to Ω , and so that ∂A corresponds to ∂ D . Conversely, if Ω ∗ is a relative circle domain in D , there is such Ω ⊂ A. The converse direction is the main result of [HS5].
Let us return to the setting of the Rodin-Sullivan Theorem 2.3 about convergence of the discretemap f ε to the conformal map f. Consider the piecewiese linear extension of f ε from the carrier of P to the carrier of P (cid:48) that maps equilateral triangles to the corresponding triangles (formed bythe centers of P (cid:48) ). By the elementary “Ring Lemma” of [RSu], the angles of these triangles are14ounded away from 0 and π (so that f ε is quasiconformal with dilation uniformly bounded above).At the heart of the Rodin-Sullivan proof is the uniqueness of the hexagonal packing as the onlypacking in the plane with nerve the triangular lattice (see the discussion in Section 2.4). It rathereasily implies that tangent circles centered in a compact set of Ω correspond to tangent circles in D whose radii are asymptotically equal as ε → . Hence the triangles in P (cid:48) are nearly equilateralswhen ε is small (the angles tend to π/ f ε is nearly angle preserving in each triangle. Nowthe theory of quasiconformal maps readily yields equicontinuity of the family of maps f ε , and showsthat every subsequential limit lim f ε j is a conformal map. The theorem follows from uniqueness ofnormalized conformal maps.He’s thesis [He] provided a quantitative estimate for the rate of convergence of the angles (thedifference to π/ O ( ε )). This estimate was known to imply convergence of the ratio of corre-sponding radii rad(D (cid:48) ) / rad(D) to the absolute value | f (cid:48) | of the derivative. A probabilistic proof of C (locally uniform) convergence of circle packings was given by Stephenson [Ste1]. Convergence of f ε to f for packings other than the hexagonal was proved in [HR], under the assumption of boundedvalency of the graph. In [DHR], the quality of convergence was improved to convergence in C (that is, convergence of first and second derivatives; strictly speaking, instead of f ε they consideredthe “piecewise M¨obius” map that sends interstices between triples of mutually tangent circles tothe corresponding interstices). He and Schramm [HS6] found an elementary new convergence proof,based on the topological ideas discussed above and thus avoiding quasiconformal maps. Their proofalso gave convergence up to C , and worked in a more general setting. In particular, it does notneed the assumption of uniformly bounded degree of [HR].In the remarkable paper [HS8], He and Schramm proved C ∞ -convergence of hexagonal diskpackings to the Riemann map: Theorem 2.13 ([HS8], Theorem 1.1) . The discrete functions f ε : V ε → D converge in C ∞ to theRiemann mapping f : Ω → D , in the sense that the discrete partial derivatives of f ε of any orderconverge locally uniformly to the corresponding partial derivatives of f . The discrete first-order derivatives for v ∈ V ε are ∂ ε,k f ε ( v ) = ε − ( f ε ( v + ε ω k ) − f ε ( v )) , where k ∈ , , ..., ω = (1 + i √ / − th root of unity. In particular, it follows that( ∂ ε, ) k f ε converges to the k − th derivative f ( k ) locally uniformly on G .The Schwarzian derivative S ( f )( z ) = f (cid:48)(cid:48)(cid:48) ( z ) f (cid:48) ( z ) − f (cid:48)(cid:48) ( z ) f (cid:48) ( z ) (2)of a locally univalent analytic function measures the deviation of f from a M¨obius transformation,in particular S ( f ) ≡ f is M¨obius. A key idea in the proof is to define a discrete analog ofthe Schwarzian derivative, to compute the (discrete) Laplacian of this Schwarzian, and to employa regularity theorem for discrete elliptic equations to obtain boundedness of all partials of theSchwarzian. The definition of the discrete Schwarzian is the circle packing analog of an invariantthat Oded so masterfully employed in his earlier work [S8] on circle patterns with the combinatoricsof the square grid. 15 .7 Other topics Oded’s approach to both mathematics and to life was extraordinarily innovative and unacceptingof conventions. Notions that most people take for granted without even thinking about, he wouldopen-mindedly question, often coming up with amazing alternative solutions. For example, I wouldnot even think about camping on the foot of a glacier without a sleeping bag. Climbing littleTahoma peak with Oded, he proved to me that even this idea can be pursued. It was perhaps oneof his less successful innovations, though.In the lovely paper [S5], Oded shows that for each triangulation G of a quadrilateral, there isa packing of a rectangle R by (horizontal) squares with the combinatorics of G (a square mightdegenerate to a point, as in Figure 7). (a) (b) Figure 7: A triangulation and the associated square packing. Thanks to David Wilson for providingthis figure from [S5]The packing is actually a tiling: Indeed, Oded points out the following simple observation.
Let P a , P b , P c be three rectangles whose edges are parallel to the coordinate axis. Suppose that theintersection of every two of these rectangles is nonempty. Then P a ∩ P b ∩ P c (cid:54) = ∅ . The same tiling theorem was obtained independently by Cannon, Floyd and Parry in [CFP1]. Bothemploy Cannon’s discrete extremal length (see Section 2.4.1) and obtain the side lengths s ( v ) of thesquares as the weights ρ ( v ) of the extremal metric (corresponding to the family of “combinatorialcurves” joining two opposite sides of the quadrilateral). It is quite different from the classical squarepackings of Brooks, Smith, Stone and Tutte [BSST], in particular, since the metrics considered herelive on the vertices rather than the edges of the graph.A very similar idea is exploited in the important paper [S6]. The classical setting of extremallength (recall (1)) is a family Γ of curves contained in a domain Ω . Invariance λ ( f (Γ)) = λ (Γ)16nder conformal maps of Ω is almost trivial (just pull back metrics from f (Ω)). Oded’s notion of transboundary extremal length λ Ω (Γ) applies to curve families Γ that are not necessarily containedin Ω . The metrics are now replaced by generalized metrics ρ that, roughly speaking, also assignlength to complementary components. The length (cid:82) γ ρ | dz | is replaced by (cid:82) γ ∩ Ω ρ | dz | + (cid:80) p ρ ( p ) if γ is not contained in Ω , where the sum is over all boundary components of Ω that γ meets. Thenthe definition is λ Ω (Γ) = sup ρ (inf γ (cid:82) γ ∩ Ω ρ | dz | + (cid:80) p ρ ( p )) / (cid:82) C ρ dxdy , and conformal invariance isagain immediate. Using this innocent looking extension, Oded provides an elegant self-containedproof of the countable Koebe conjecture, and moreover is able to deal with the case of domains forwhich the complementary component satisfy a certain fatness condition (area( A ∩ B ( x, r )) ≥ cr for each component A , each x ∈ A and each disc B ( x, r ) that does not contain A ).Circle packings corresponding to infinite graphs G can be obtained by taking Hausdorff limitsof packings corresponding to finite subgraphs, but where do they “live”? Beardon and Stephenson[BSt1],[BSt2] have shown, under the assumption that the degrees of the vertices are uniformlybounded, that the carrier of such a packing is either the plane (call this case parabolic ), or that itcan be chosen to be the disc ( hyperbolic ). They also showed that both cases are mutually exclusive,and that the packing is hyperbolic if each degree is at least seven. The uniform boundednessassumption was later removed by He and Schramm [HS1], and they proved in general that the type of a packing is unique (that is, there is no infinite graph that packs both the disc and the plane). Inthe impressive paper [HS3], they characterize the type in terms of the discrete extremal length, anduse it to show that the packing is parabolic if simple random walk on G is recurrent. They conclude(Theorem 10.1) that a packing is parabolic if at most finitely many vertices have degree greaterthan 6 (notice that every vertex of the hexagonal packing has degree 6). This paper contains theirearlier result [HS3b] that a packing is hyperbolic if the lower average degree is greater than 6. Bydefinition, the lower average degree islav( G ) = sup W inf W ⊃ W | W | (cid:88) v ∈ W deg( v ) . In the case that the degrees of the vertices are uniformly bounded, they also show that transienceimplies hyperbolicity. Jointly with Itai Benjamini, this line of investigation was carried further in[BS1] and [BS2], by applying circle- and square packings to constructions of harmonic functions ongraphs. Another nice application of circle packings is the recurrence of (weak) limits of randomplanar graphs with bounded degree, [BS2].I have always admired Oded’s ability to find a good modification of a difficult problem thatturns it into a tractable problem while keeping its essential features. One of the many examples ishis work on discrete analytic function [S8]. Since circle packings can be viewed as discrete analogs ofconformal maps, it is natural to ask for the analogs of analytic functions, thus giving up injectivity(disjointness of the discs). See [Ste2] for the state of the art and beautiful illustrations. Peter Doyledescribed collections of discs that are tangent according to the hexagonal pattern that are analogsof the exponential function. He conjectured that these would be the only “entire” circle packingimmersions. While Oded was not able to resolve this conjecture, he did find that collections ofoverlapping discs based on the square grid seem better suited for the problem, and constructedthe analog of the error function (cid:82) e − z dz in this setting. Along the way, he introduced M¨obiusinvariants that are discrete analogs of the Schwarzian derivative and became instrumental in hislater work [HS8]. 17 a) (b) Figure 8: The square grid and the √ i SG erf pattern, from [S8]
There are several excellent lecture notes, overview articles, and a textbook on SLE [La3], mostlyby and for probabilists or theoretical physicists, see [BB3], [Car2], [Dup], [GK], [KN], [S11], [W1],[W2] and the references therein. It is not my intention to provide another streamlined introductionto the area. Instead, I would like to give a somewhat historic account with an emphasis on Oded’scontributions, highlighting some of the mathematical challenges he faced.
It is perhaps appropriate to very briefly describe the state of knowledge related to conformallyinvariant scaling limits prior to Oded’s discovery of SLE, and to describe some of the results thatwere instrumental in his work. Oded’s own historical narrative is Section 1.2 in [S11].Two-dimensional lattice processes such as the Self-Avoiding Walk (SAW), the Ising model, per-colation, and diffusion limited aggregation (DLA), to name just a few, have been intensively studiedby physicists and by probabilists for a long time. See Figure 21 for some pictures, and the afore-mentioned articles for descriptions of the models. In the physics community, many problems suchas finding the Hausdorff dimension of scaling limits of these sets were considered well-understood.The implicit assumption of conformal invariance of the scaling limit allowed the use of the powerfulmachinery of conformal field theory and led to results such as Cardy’s formula for the crossingprobability of critical percolation [Car1]. On the mathematical side, progress was much slower, oneof the hurdles being that in most cases the existence of a scaling limit was unknown. Even findingsuitable definitions of the concept of scaling limit was a nontrivial task.In the late 1980’s, Christian Pommerenke told me how compositions of (random) conformalmaps onto slitted discs could be viewed as a variant of the Witten-Sander model for DLA [WS].At the same time, Richard Rochberg and his son David were working on this setup. It seems thatthe only trace of this is a talk given by Rochberg at the March 1990 AMS Regional meeting inManhattan, Kansas, titled “Stochastic Loewner Equation”. Their model is similar to an approach18o Laplacian growth proposed by Hastings and Levitov [HL], and is quite different from what isnow called Stochastic Loewner Evolution or Schramm-Loewner Evolution SLE. At that time, otheranalysts such as Lennart Carleson, Peter Jones and Nick Makarov worked with similar ideas, seee.g. [CM]. Oded was at best dimly aware of these activities, and was not really interested instochastic processes such as DLA until much later.Greg Lawler’s invention [La] of the Loop Erased Random Walk (LERW) provided the mathe-matics community with a process that shared some features with the Self-Avoiding Walk, but atthe same time was more tractable, partly due to its Markovian property. Pemantle’s work [ ? ] andWilson’s algorithm provided a link between Uniform Spanning Trees (UST) and LERW’s. Inten-sive research on the UST [Ly] culminated in the paper [BLPS] by Benjamini, Lyons, Peres andSchramm. The deep work of Rick Kenyon [Ke1] combined powerful combinatorics and discretecomplex analysis and exhibited conformal invariance properties of the LERW. He was also able todetermine its expected length. Oded told me in 1997 about his idea to exploit conformal invariance in order to study the LERW.The streamlined way to present SLE in courses or texts, beginning with a crashcourse on theLoewner equation followed by a crash course on stochastic calculus (or the other way round) is, ofcourse, not quite representative of its emergence. In a 2006 email exchange with Yuval Peres andmyself about the history of SLE, Oded wrote:
Up to the time when I started thinking about SLE, I did not really know what Loewner’s equationwas, or what was the idea behind it, though I did know that it was a tool which was important forthe coefficients problem and that it involved slit mappings and a differentiation in the space ofconformal maps. I kind of rediscovered it in the context of SLE and then made the connection.
Loewner ([Lo]; see also [Dur],[Po2] or [La3]) introduced his differential equation as a tool in hisattempt to prove the Bieberbach conjecture | a n | ≤ n concerning the Taylor coefficients of normalizedconformal maps f ( z ) = z + (cid:80) ∞ n =2 a n z n of the unit disc. It was also instrumental in the final solutionby de Branges in 1984.Let γ be a simple path that is contained in D except for one endpoint on ∂ D . More precisely,let γ : [0 , τ ] → D be continuous and injective with γ (0) ∈ ∂ D and γ ( τ ) = 0, such that γ (0 , τ ] ⊂ D . Denote G t = D \ γ [0 , t ] so that G = D . Then, for each 0 ≤ t < τ , there is a unique conformalmap g t : G t → D that is normalized by g t (0) = 0 and g (cid:48) t (0) > . By Schwarz’ Lemma, g (cid:48) t (0) strictlyincreases, g (cid:48) (0) = 1 , and it is not hard to see that g (cid:48) t (0) → ∞ as t → τ. Hence we can reparametrize γ so that τ = ∞ and g (cid:48) t (0) = e t . Loewner’s theorem says that ∂∂t g t ( z ) = g t ( z ) ζ t + g t ( z ) ζ t − g t ( z ) (3)for all t ≥ z ∈ G t , where the “driving term” ζ t = g t ( γ ( t )) ∈ ∂ D g t is only defined in G t , but it can be shown that g t extends to γ ( t )).A simple but crucial observation is that the driving term ζ T of the curve γ T = g T ( γ ) (moreprecisely, the parametrized curve γ T ( s ) := g T ( γ ( T + s ))) is given by ζ Ts = ζ T + s . Thus “conformallypulling down” a portion of γ corresponds to shifting the driving term. Intuitively, one can think ofthe Loewner equation as describing a conformal map to a slitted disc as a composition of conformalmaps onto infinitesimally slitted discs with slit at ζ t , plus the statement that the conformal maponto such a disc is z (cid:55)→ z + z ζ t + zζ t − z ∆ t up to first order in ∆ t .Thus the Loewner equation associates with each simple curve γ ⊂ D a continuous function ζ t with values in ∂ D . Conversely, it is not hard to show that the solution g t ( z ) to the initial valueproblem (3), g ( z ) = z, forms a family of conformal maps of simply connected domains G t onto D . In fact, G t is the set of those points ζ ∈ D for which the solution is well-defined on the interval[0 , t ]. It easily follows that G t increases in t, and that z ∈ G t unless g s ( z ) = ζ s for some s ≤ t. Thecomplement K t = D \ G t is called the hull of ζ. In our original setup of a slit disc, we simply recover the curve, K t = γ [0 , t ] . It has been known since Kufarev [Ku] that smooth functions ζ generate smooth curves γ, butthat there also exist continuous functions ζ for which the associated hull is not a simple arc in D . Kufarev’s example simply is the computation that a circular chord γ of the unit circle hascontinuous driving term. In fact, it can be topologically wild (not locally connected), see [MR].My own interest in the Loewner equation originated when Oded asked me which driving termsgenerate curves. 20 .2.2 The scaling limit of LERW The LERW is obtained from simple random walk by erasing loops chronologically. The main resultFigure 10: A loop erased random walk in the disc, from [S9].of Oded’s celebrated paper [S9] was a conditional theorem: Assuming the existence and conformalinvariance of the scaling limit of LERW, he showed that the Loewner driving term of the resulting(random) limiting curve is a Brownian motion on the unit circle, ζ t = e iB t . To make this rigorous,he first gave the following definition of the notion of scaling limit: Let D (cid:40) C be a domain, fix a ∈ D , and for δ >
0, consider the LERW on the graph δ Z ∩ D , started at a point closest to a and stopped when reaching ∂D. Viewing the path of the LERW as a random subset of thesphere S = C ∪ {∞} , its distribution is a discrete measure µ δ on the space of compact subsetsof S . Equipped with the Hausdorff distance, the space of compact subsets of S is a compactmetric space, and so is the space of its Borel measures. The existence of subsequential weak limits µ = lim j µ δ j follows at once. If the limit measure µ = lim δ → µ δ exists, it is called the scaling limitof LERW from a to ∂D . Theorem 3.1 ([S9], Theorem 1.1) . If each connected component of ∂D has positive diameter, thenevery subsequential scaling limit measure µ of the LERW from a to ∂D is supported on simplepaths. In other words, the measure of the set of non-simple curves is zero. This theorem is interestingin its own right. It has been known previously that, loosely speaking and under mild assumptions,random curves have uniform continuity properties that imply their (subsequential) scaling limitsto be supported on continuous curves [AB]. However, the fact that the loop erased paths are simple curves does not directly imply that the limiting objects have no loops. Indeed, the limitsof other discrete random simple curves such as the critical percolation interface or the uniformspanning tree Peano path are not simple. The proof uses estimates for the probability distributionof “bottlenecks”, based on harmonic measure estimates and Wilson’s algorithm, and a topologicalcharacterization of simple curves.Next, Oded formulated the conjecture of existence and conformal invariance of the scaling limitas follows. 21 onjecture ([S9],1.2) Let D (cid:36) C be a simply connected domain in C , and let a ∈ D . Then thescaling limit of LERW from a to ∂D exists. Moreover, suppose that f : D → D (cid:48) is a conformalhomeomorphism onto a domain D (cid:48) ⊂ C . Then f ∗ µ a,D = µ f ( a ) ,D (cid:48) , where µ a,D is the scaling limitmeasure of LERW from a to ∂D , and µ f ( a ) ,D (cid:48) is the scaling limit measure of LERW from f ( a ) to ∂D (cid:48) . The most important and exciting result of [S9] was the insight that this conjecture impliedan explicit construction of the limit in terms of the Loewner equation. By Theorem 3.1, theconjectural scaling limit µ induces a measure on the space of continuous real-valued functions ˆ ζ t via the correspondence γ (cid:55)→ ζ = e i ˆ ζ of the Loewner equation. Oded showed that the law of ˆ ζ isthat of a time-changed Brownian motion, B t : Theorem 3.2 ([S9], Theorem 1.3) . Assuming the above conjecture, the scaling limit µ is equal tothe law of the hulls K associated with the driving term ζ = e iB t , where B t , t ≥ is a Brownianmotion started at a uniform random point in [0 , π ) . In his characteristic way, Oded pointed out the simple idea behind the theorem. From his paper:
At the heart of the proof of Theorem 3.2 lies the following simple combinatorial fact aboutLERW. Conditioned on a subarc β (cid:48) of the LERW β from 0 to ∂D , which extends from some point q ∈ β to ∂D , the distribution of β \ β (cid:48) is the same as that of LERW from 0 to ∂ ( D − β (cid:48) ) , conditionedto hit q . When we take the scaling limit of this property, and apply the conformal map from D − β (cid:48) to D , this translates into the Markov property and stationarity of the associated L¨owner parameter ζ . He also notes that “the passage to the scaling limit is quite delicate”. The translation into theMarkov property and stationarity is by means of the aforementioned principle that “conformallypulling down” a portion γ (cid:48) of γ corresponds to shifting the driving term. Thus ˆ ζ is a continuousprocess with stationary and independent increments. Now the theory of Levy processes (and thesymmetry of LERW under reflection) implies that ˆ ζ t has the law of √ κB t for some κ > B . It remained to determine the constant κ. To this end, Oded givesthe following
Definition.
The (radial) stochastic Loewner evolution
SLE κ with parameter κ > is the randomprocess of conformal maps g t generated by the Loewner equation driven by ζ t = e i √ κB t . In Section 7 of [S9] he actually defined SLE as a process of random paths generated by the Loewnerequation, and therefore had to restrict to those values of κ for which the resulting hulls are simplecurves; he conjectured that this is the interval [0 , θ κ ( t ) =arg γ ( t ), t ≥
0, denotes the continuously defined argument along the curve, then he computed thevariance E [ θ κ ( t ) ] = ( κ + o (1)) log t. It follows that the winding number of the portion of the SLE path until its first hitting of the circleof radius ε centered at 0 has variance ( κ + o (1)) log(1 /ε ) . On the other hand, Kenyon’s work [Ke2]implies that the variance of the winding number of LERW in D ∩ ε Z is (2 + o (1)) log(1 /ε ) , andafter some work the conclusion κ = 2 follows. 22ded went on to compute what he called the critical value for SLE: He proved that for κ > , almost surely SLE will not generate simple paths, and conjectured that it will for κ ∈ [0 , This must all be quite standard, to people with the right background. But not for me.
The classical (radial) Loewner equation is well-suited for curves that join an interior point to a boundary point , such as curves generated by the LERW. Other processes generate curves joining two boundary points . Oded realized how important the “correct” normalizations are in dealingwith conformal maps and in particular the Loewner equation, and found the appropriate variantof the Loewner equation (it turned out later that this version has been described earlier, beginningwith N.V. Popova [Pop1],[Pop2]; I would like to thank Alexander Vasiliev for this reference). Hedescribes this in another email in January 1999:Figure 11: Percolation in a domain, from [S11].
I have a mathematical querry. Before the question itself, here’s the motivation. For the LERWscaling limit, the natural object is a probability measure on the set of paths from a point in thedomain to the boundary. In other settings, the natural object is a probability measure joining twopoints on the boundary of a domain. Consider, for example, percolation in the unit disk. Let ( γ, β ) be a partition of the boundary of the disk to two arcs, disjoint except for the endpoints. Let K be theunion of all percolation clusters inside the disk that are connected to γ . Then the outer boundary of K is a path, α , joining the two endpoints of γ . The scaling limit of α is conjecturally conformallyinvariant (but not a simple path). Assuming conformal invariance, I’m optimistic that the scalinglimit can be represented by a Loewner-like Brownian evolution. The first step for this seems to bethe following variation on Loewner’s theorem: Thm:
Let α : [0 , ∞ ) → C be a continuous simple path such that α (0) = 0 , lim t →∞ α ( t ) = ∞ and Im( α ( t )) > when t > . Let f t be the conformal map from the upper half plane to the upper half lane minus { α ( s ) : 0 ≤ s ≤ t } , which is normalized by f t ( z ) = z + O (1 /z ) near infinity. Bychange of parameterization of α (and perhaps changing its interval of definition), we may assumethat f t ( z ) = z + t/z + O (1) /z near infinity. Set g ( w, t ) = f − t ( w ) . Then { ∂g/∂t } = 1 / ( g − k ( t )) ,where k ( t ) = g ( α ( t ) , t ) . In the situation of percolation and related conformal invariance models, one should expect k ( t ) = cBM ( t ) , where BM is on the real line. Have you seen this theorem? The proof should not be difficult.It can either be derived from Loewner’s theorem, or by adapting the proof. (a) (b) Figure 12: The percolation exploration pathIn order to coincide with the normalization of the radial Loewner equation, he later slightlyadjusted the parametrization of the path α so that f t ( z ) = z + 2 t/z + O (1) /z and ∂∂t g t ( z ) = 2 g t ( z ) − W t . (4)With this normalization and assuming conformal invariance of the percolation scaling limit, heshowed that the (non-simple) limit curves would satisfy the chordal Loewner equation (4) with W t = √ B t . The value 6 can be found as the only κ such that the random sets generated by √ κB t satisfy Cardy’s formula, or the locality property discussed below. This led to the definitionof chordal SLE κ as the random process of conformal maps g t : H \ K t → H generated by the Loewner equation (4) with driving function W ( t ) = √ κB t , where B is a standardBrownian motion. 24he hull K t is the set of those points z for which g s ( z ) = W s for some s ≤ t so that (4) becomesundefined. Since g t is determined by K t , one has the equivalent Definition:
Chordal
SLE κ is the process of random hulls ( K t , t ≥
0) generated by the Loewnerequation (4) with W t = √ κB t .In the same paper, Oded also defines and analyzes subsequential scaling limits of the uniformspanning tree. He ends the paper by speculating (that is, stating without giving detailed proofs)about the Loewner driving term of the UST Peano curve in the upper half plane H . He finds that,again assuming existence and conformal invariance of the limit, this random space filling curve is SLE . At the end of the introduction, he summarizes the findings of the paper as follows: The emerging picture is that different values of κ in the differential equation (3) or (4) producepaths which are scaling limits of naturally defined processes, and that these paths can be space-filling,or simple paths, or neither, depending on the parameter κ . Figure 13: The UST Peano curve, from [RoS].
Two exciting developments took place shortly after the introduction of SLE, namely the veryproductive collaboration of Greg Lawler, Oded Schramm and Wendelin Werner, and the surprisingproof of existence and conformal invariance of the percolation scaling limit by Stas Smirnov. I willbegin describing the former, and defer the latter to Section 3.3.4.
In the important papers [LW1] and [LW2], Lawler and Werner discovered that Brownian excursionshave a certain restriction property (explained below), and that intersection exponents of conformallyinvariant processes with this property are closely related to those of Brownian motion. What wasmissing was a way to compute exponents of some conformally invariant process. Lawler, Schramm25nd Werner discovered that SLE provided such a process to which the universality arguments ofLawler and Werner [LW2] applied. The following email from Oded describes the crucial property:
I don’t remember if I’ve mentioned to you the restriction property for SLE(6) that Greg, Wen-delin and I have proved. It says that (up to time parameterization) the law of SLE(6) in an arbitrarydomain D starting from a point p on the boundary and stopped when it exits a small ball B aroundp, does not depend on the shape of the domain outside B (provided that D-B is connected, say).Thus, SLE(6) is purely a local process, like BM. This is not true for κ (cid:54) = 6 . For example, SLE(6)in the disk (with Loewner’s original equation) is the same as SLE(6) in the half plane, with myvariation on Loewner’s equation. This seems to say that SLE(6) is a very special process. The above property is trivial for the discrete critical percolation exploration path, since thepath can be grown “dynamically” by deciding the color of a hexagon only when the path meets itand needs to decide whether to turn “right” or “left”. Hence, in light of the conjectured scalinglimit, locality for chordal
SLE was not unexpected. The coincidence of chordal and radial SLE ,discussed below, was more surprising.Here is a precise statement of the locality property. Let D = H \ A be a simply connectedsubdomain of H such that A is bounded and also bounded away from 0 . Denote g A the conformalmap from D onto H with the hydrodynamic normalization ( g A ( z ) − z → z → ∞ ), and setΦ A = g A − g A (0) . Then SLE in D from 0 to ∞ is defined as the preimage of SLE in H from 0 to ∞ under Φ A . The following expresses the fact that SLE in D is a time-change of SLE in H , up tothe time that the process hits A . Let T = inf { t : K t ∩ A (cid:54) = ∅} and ˜ T = inf { t : K t ∩ Φ A ( ∂A ) (cid:54) = ∅} . Theorem 3.3 ([LSW2], Theorem 2.2) . For κ = 6 , the processes (Φ A ( K t ) , t < T ) and ( K t , t < ˜ T ) have the same law, up to re-parametrization of time. The equivalence of chordal and radial
SLE was established in [LSW3], Theorem 4.1: Definethe hulls K t of chordal SLE κ in D from 1 to − SLE κ in H , under theconformal map f ( z ) = ( i − z ) / ( i + z ). Thus K t are hulls growing from 1 towards -1 in D . Denote T ≤ ∞ the first time when K t contains 0 . Similarly, denote (cid:101) K t the chordal SLE κ hulls in D , startedat 1 , and denote (cid:101) T the first time when (cid:101) K t contains − . Then, for κ = 6 , the laws of ( K t , t < T )and ( (cid:101) K t , t < (cid:101) T ) are the same, up to a random time change s = s ( t ). The proof and Girsanov’stheorem also show that for all values of κ, the laws of K t and (cid:101) K s ( t ) are equivalent (in the sense ofabsolute continuity of measures), if t and (cid:101) t are bounded away from T and (cid:101) T .
See Proposition 4.2in [LSW3].The first proof of the locality Theorem 3.3 was rather long and technical, based on an analysis ofthe Loewner driving function of a curve under continuous deformation of the surrounding domain.A different and simpler proof was found later (see Proposition 5.1 in [LSW8]), by analyzing ˜ W t := h t ( W t ) , where h t := ˜ g t ◦ g A ◦ g − t and ˜ g t = g g t ( D \ K t ) is the normalized conformal map of g t ( D \ K t )to H . Writing ˜ g t ( z ) = z + a t z + O ( 1 z ) , computation shows that ∂ t ˜ g t ( z ) = ∂ t a t ˜ g t ( z ) − ˜ W t = 2 h (cid:48) t ( W t ) ˜ g t ( z ) − ˜ W t . W t = √ κB t , computation using Ito’s formula shows d ˜ W t = h (cid:48) t ( W t ) dW t + (cid:0) ( κ/ − (cid:1) h (cid:48)(cid:48) t ( W t ) dt. (5)Thus ˜ W t is a local martingale if (and only if) κ = 6 , and a time change shows that (˜ g t , t ≥
0) is
SLE . The locality of
SLE has been used to determine the so-called intersection exponents of 2-dimensionalBrownian motion, and to compute the Hausdorff dimensions of various sets associated with itstrace. These results established Schramm’s SLE and the Lawler-Werner universality arguments asa fundamental and powerful new tool.If B t and B t are two independent planar Brownian motions started at two different points B (cid:54) = B , it easily follows from the subadditivity of t (cid:55)→ log P [ B [0 , t ] ∩ B [0 , t ] = ∅ ] that there is anumber ζ > P [ B [0 , t ] ∩ B [0 , t ] = ∅ ] = (cid:18) t (cid:19) ζ + o (1) . Similarly, the half-plane exponent ˜ ζ of the event that two independent motions do not intersectand stay in a halfplane is given by P [ B [0 , t ] ∩ B [0 , t ] = ∅ and B j [0 , t ] ⊂ H , j = 1 ,
2] = (cid:18) t (cid:19) ˜ ζ + o (1) . More generally, one considers exponents ζ p for the probability of the event that p independentmotions are mutually disjoint, ζ ( j, k ) for the event that two packs of Brownian motions B ∪· · ·∪ B j and B j +1 ∪ · · · ∪ B j + k are disjoint, and the corresponding half-plane exponents ˜ ζ p and ˜ ζ ( j, k ). So ζ = ζ (1 , disconnection exponent η j for the event that the union of j Brownian motions, started at 1, does not disconnect 0 from ∞ before time t. These and other intersection exponents have been studied intensively, and values such as ζ = 5 / ζ ( j, k ) for positive real k > Theorem 3.4.
For all integers j ≥ and all real numbers k ≥ ,ζ ( j, k ) = √ j + 1 + √ k + 1 − − , ζ n = 4 n − , ˜ ζ ( j, k ) = √ j + 1 + √ k + 1 − − , ˜ ζ n = 2 n + n , nd η k = ζ ( k,
0) = ( √ k + 1 − − . In particular, ζ = 58 , ˜ ζ = 53 , η = 14 , η = 23 . The proofs are technical masterpieces combining a variety of different methods. A very roughdescription is as follows: First, half-plane intersection exponents of
SLE are computed, based onestimates for the crossing probability of (long) rectangles. This is done by establishing a versionof Cardy’s formula. Then, the universality ideas of [LW2] are employed to pass from SLE toBrownian motion. Finally, to cover the case k <
1, real analyticity of the exponent is shown byrecognizing e − ζ ( j,k ) as the leading eigenvalue of an operator T k on a space of functions on pairs ofpaths.For a fixed time t, the Brownian frontier is the boundary of the unbounded connected componentof the complement of B [0 , t ], and the set of cut points is the set of those points p for which B [0 , t ] \{ p } is disconnected. The set of pioneer points is the union of the frontiers over all t >
0. Mandelbrot[M] observed that the Brownian frontier looks like a long self-avoiding walk. Since the Hausdorffdimension of the self-avoiding walk was predicted by physicists to have Hausdorff dimension 4/3,he conjectured that the Hausdorff dimension of the Brownian frontier is 4/3. Greg Lawler hadshown in a series of papers (see [La2]) how the intersection exponents are related to the Hausdorffdimension of subsets of the Brownian path. He found the values 2 − ζ , 2 − η , and 2 − η for thedimension of the Brownian frontier, the set of cut points, and the set of pioneer points. This actuallyrequired his stronger estimates of the intersection probabilites up to constant factors, rather thanup to (1 /t ) o (1) . Simpler proofs of those estimates are the content of [LSW6]. In combination withTheorem 3.4, this proved Mandelbrot’s conjecture. Theorem 3.5 ([LSW3], [LSW7]) . The Hausdorff dimension of the frontier, the set of cut points,and the set of pioneer points of 2-dimensional Brownian motion is / , / and / almost surely. Figure 14: A Brownian path, with part of the frontier highlighted28o put this result in perspective, notice that it is rather difficult to show even that the dimensionof the Brownian frontier is more than one [BJPP], and that the set of cutpoints is non-empty, [Bu].It should also be mentioned that by work of Lawler, the intersection exponents for simple randomwalk are the same as for Brownian motion, so that Theorem 3.4 also shows, for instance, that P [ S [0 , n ] ∩ S [0 , n ] = ∅ ] = (cid:18) n (cid:19) + o (1) if S and S are two independent planar simple random walks started at different points.Meanwhile, there is a more elegant approach to these dimension results, also due to Lawler,Schramm and Werner, see Section 3.3.5 I have been fortunate to collaborate with Oded on several projects over the past two decades.Sometimes this meant just trying to catch up with his fast output, and watching in awe how oneclever idea replaced another. But perhaps even more impressively, Oded had an amazing abilityand willingness to listen, and to think along. Sometimes, when I failed in an attempt to articulatea vague idea and was about to give up a faint line of thought, he surprised me by completelyunderstanding what I tried to express, and by continuing the thought, almost like mind reading.The definition of
SLE κ as a family of conformal maps g t through a stochastic differentialequation does not shed much light upon the structure of the hulls K t = g − t ( H ) . By Section 3.3.1,it is enough to consider chordal SLE. We say that the hull ( K t ) t ≥ is generated by a curve γ if γ : [0 , ∞ ) → H is continuous and if K t is obtained from γ by “filling in the holes” of γ [0 , t ] (moreprecisely, K t is the complement in H of the unbounded connected component of H \ γ [0 , t ]). Sincecontinuity of the driving term W t is equivalent to the requirement that the increments K t + ε \ K t have“small diameter within D t ” (more precisely, there is a set S ⊂ D t of small diameter that disconnects K t + ε \ K t from ∞ within D t , see [LSW2], Theorem 2.6), such a curve cannot cross itself, but it canhave double points and “bounce off” itself ([Po1]). There are examples of continuous W for which K t is not locally connected, and such sets cannot be generated by curves [MR]. Fortunately, thisdoes not happen for SLE: Theorem 3.6 ([RoS], Theorem 5.1; [LSW9], Theorem 4.7) . For each κ > , the hulls K t aregenerated by a curve, almost surely. It follows that, a.s., the conformal maps f t = g − t extend continuously to the closed half space H , and γ ( t ) = f t ( W t ) . For κ (cid:54) = 8, the proof hinges on estimates for the derivative expectations E [ | f (cid:48) t ( z ) | p ]. For κ = 8, the only known proof is by exploiting the fact that SLE is the scaling limitof UST, and that the UST scaling limit is a continuous curve a.s., [LSW9].As Oded already noticed in [S9], the SLE κ trace has different phases, depending on the valueof κ. Theorem 3.7 ([RoS]) . For κ ≤ , the SLE trace γ is a simple curve in H ∪ { } , almost surely. It“swallows” points (for fixed z ∈ H \ { } , a.s. z ∈ K t for large t , but z / ∈ γ [0 , ∞ ) ) if < κ < ,and it is space-filling ( γ [0 , ∞ ) = H ) if κ ≥ . For all κ, the trace is transient a.s.: | γ ( t ) | → ∞ as t → ∞ . κ κγ γ (t ) (t )0 < 4 < < 8 > 8 K K t t< 4
Figure 15: The three phases of SLE; picture courtesy of Michel Bauer and Denis Bernard [BB1]Let us explain the phase transition at κ = 4, already observed and conjectured in [S9] (in theradial case). Let x > X t = g t ( x ) − W t , where W t = √ κB t . Then dX t = 2 X t dt − √ κdB t is an Ito diffusion and can be easily analyzed using stochastic calculus. In fact, X t is a Besselprocess of dimension 1 + 4 /κ . Thus X t > t if and only if κ ≤ . In this range, we obtain K t ∩ R = { } for all t, and it is an easy consequence of Theorem 3.6 that γ is simple (if r < s < t are such that γ ( r ) = γ ( t ) (cid:54) = γ ( s ), then the curve g s ( γ [ s, t ]) has the law of SLE κ shifted by g s ( γ ( s )),but has two points on R ).The other phase transition can be seen by examining the SLE-version of Cardy’s formula: If X = inf (cid:0) [1 , ∞ ) ∩ γ [0 , ∞ ) (cid:1) denotes the first intersection of the SLE trace with the interval [1 , ∞ ) , then a.s. X = 1 if κ ≥
8, whereas for κ ∈ (4 , P [ X ≥ s ] = 4 ( κ − /κ √ π F (1 − /κ, − /κ, − /κ, /s ) s (4 − κ ) /κ Γ(2 − /κ ) Γ(4 /κ − / , (6)where F denotes the hypergeometric function. At the corresponding time where γ ( t ) = X, thenontrivial interval [1 , X ] gets “swallowed” by K at once. The proof of Cardy’s formula in [RoS] issimilar to the more elaborate Theorem 3.2 in [LSW2] and based on computing exit probabilities ofa renormalized version of g t , Y t = g t (1) − W t g t ( s ) − W t ∈ (0 , . At the exit time T , we have Y T = 0 or 1 according to whether X < s or > s. Now Cardy’s formulacan be obtained using standard methods of stochastic calculus.For a simply connected domain D (cid:54) = C and boundary points p, q , chordal SLE from p to q in D is defined as the image of SLE in H under a conformal map of H onto D that takes 0 and ∞ to p and q . Since the conformal map between H and D generally does not extend to H , the continuityof the SLE trace in D does not follow from Theorem 3.6. However, using Theorem 3.9 below andgeneral properties of conformal maps, it can be shown to still hold true, [GRS]. Another naturalquestion is whether SLE is reversible , namely if SLE in D from p to q has the same law as SLEfrom q to p. This question was recently answered positively for κ ≤ κ ≥ < κ < . heorem 3.8 ([Z1]) . For each κ < , SLE κ is reversible, and for κ ≥ it is not reversible. The aforementioned derivative expectations E [ | f (cid:48) t ( z ) | p ] also led to upper bounds for the dimen-sions of the trace and the frontier. The technically more difficult lower bounds were proved byVincent Beffara [Be] for the trace.For κ >
4, notice that the outer boundary of K t is a simple curve joining two points on the realline. There is a relation between SLE κ and SLE /κ , first derived by Duplantier with mathemati-cally non-rigorous methods, and recently proved in the papers of Zhang [Z2] and Dubedat [Dub3].Roughly speaking, Duplantier duality says that this curve is SLE /κ between the two points. Aprecise formulation is based on a generalization of SLE , the so-called
SLE ( κ, ρ ) introduced in[LSW8]. As a consequence, the dimension of the frontier can thus be obtained from the dimensionof the dual SLE.Based on a clever construction of a certain martingale, in [SZ] Oded and Wang Zhou determinedthe size of the intersection of the trace with the real line. The same result was found independentlyand with a different method by Alberts and Sheffield [AlSh]. Summarizing: Theorem 3.9.
For κ ≤ , dim γ [0 , t] = 1 + κ . For κ > , dim ∂ K t = 1 + 2 κ . For < κ < , dim γ [0 , t] ∩ R = 2 − κ . The paper [SZ] also examined the question how the SLE trace tends to infinity. Oded and Zhoushowed that for κ <
4, almost surely γ eventually stays above the graph of the function x (cid:55)→ x (log x ) − β , where β = 1 / (8 /κ − . In [LSW2], Lawler, Schramm and Werner wrote that ... at present, a proof of the conjecture that
SLE is the scaling limit of critical percolationcluster boundaries seems out of reach... Smirnov’s proof [Sm1] of this conjecture came as a surprise. More precisely, he proved conver-gence of the critical site percolation exploration path on the triangular lattice (see Figure 12(b))to
SLE . See also [CN] and [Sm2]. This result was the first instance of a statistical physics modelproved to converge to an SLE. The key to Smirnov’s theorem is a version of Cardy’s formula.Lennart Carleson realized that Cardy’s formula assumes a very simple form when viewed in theappropriate geometry: When κ = 6 , the right hand side f ( s ) of (6) is a conformal map of theupper half plane onto an equilateral triangle ABC such that 0 , ∞ correspond to A, B and C. Since
SLE in ABC from A to B has the same law as the image of SLE in H from 0 to ∞ , the first point X (cid:48) of intersection with BC has the law of f ( X ) . It follows that X (cid:48) is uniformlydistributed on BC. (A similar statement is true for all 4 < κ <
8, where “equilateral” is replacedby “isosceles”, and the angle of the triangle depends on κ , [Dub1]). Smirnov proved that the law ofa corresponding observable on the lattice converges to a harmonic function, as the lattice size tends31o zero. And he was able to identify the limit, through its boundary values. The proof makes useof the symmetries of the triangular lattice, and does not work on other lattices such as the squaregrid, where convergence is still unknown.The next result concerning convergence to SLE was obtained by the usual suspects Lawler,Schramm and Werner [LSW9]. They proved Oded’s original Conjecture 3.2.2 about convergence ofLERW to SLE , and the dual result (also conjectured in [S9]) that the UST converges to SLE ,see Figures 10 and 13.The harmonic explorer is a (random) interface defined as follows: Given a planar simply con-nected domain with two marked boundary points that partition the boundary into black and whitehexagons, color all hexagons in the interior of the domain grey, see Figure 16 (a). The (growing)Figure 16: Definition of the Harmonic explorer path, from [SS1].interface γ starts at one of the marked boundary points and keeps the black hexagons on its leftand the white hexagons on its right. It is (uniquely) determined (by turning left at white hegagonsand right at black) until a grey hexagon is met. When it meets a grey hexagon h (marked by ? inFigure 16) the (random) color of h is determined as follows. A random walk on the set of hexagonsis started, beginning with the hexagon h . The walk stops as soon as it meets a white or blackhexagon, and h assumes that color. Continuing in this fashion, γ will eventually reach the otherboundary point. In [SS1], Oded and Scott Sheffield showed (distributional) convergence of γ to SLE . The overall strategy is again to directly analyze the Loewner driving term of the discretepath. The crucial property of SLE is that, conditioned on the SLE trace γ [0 , t ], the probabilitythat a point z ∈ H will end up on the left of γ [0 , ∞ ) is a harmonic function of z (it is equal to theargument of g t ( z ) − W t , divided by π ).Other processes are believed to converge to SLE , too, in particular Rick Kenyon’s doubledomino path, and the q − state Pott’s model with q = 4 . The self-avoiding walk, first proposed in 1949 as a simple model for the structure of polymers, hasplayed an important role in the development of SLE, in several ways: First, Lawler’s invention of theLERW was partly motivated by the desire to create a model that is simpler than SAW. Second, the32igure 17: Harmonic explorer path, from [S11].apparent similarity to the Brownian frontier motivated Mandelbrot’s conjecture. Third, and mostsignificantly, the SAW is conjectured to converge to
SLE / . See [LSW10] for precise formulations,and a proof of this conjecture assuming existence and conformal invariance of the scaling limit, and[K] for strong numerical evidence. However, still very little is known rigorously about the SAW.
SAW in half plane - 1,000,000 steps
Figure 18: half-plane SAW, picture courtesy of Tom Kennedy.Another famous classical model is the Ising model for ferromagnetism. Stas Smirnov [Sm3] hasrecently obtained another breakthrough concerning convergence of lattice models to SLE. He foundobservables for the Ising model at criticality and was able to prove their conformal invariance inthe scaling limit. As a consequence, he obtained
SLE in the limit. Quoting from [Sm2]: Theorem.
As the lattice step goes to zero, interfaces in Ising and Ising random cluster modelson the square lattice at critical temperature converge to SLE(3) and SLE(16/3) correspondingly.
The elegant and important paper [LSW8] is a culmination of the universality arguments that havebeen initiated in [LW2] and developed in the subsequent collaboration of Lawler, Schramm andWerner. In the setting of random sets joining two boundary points of a simply connected domain,[LSW8] gives a complete characterization of laws satisfying the conformal restriction property, andvarious constructions of them.Roughly speaking, a family of random sets K joining 0 and ∞ in H satisfies conformal restriction,if for every reasonable subdomain D = H \ A of H , the law of K conditioned on K ⊂ D is the sameas the law of g ( K ) , where g is a conformal map from D to H fixing 0 and ∞ . More precisely, thesets K are supposed to be connected, have connected complement, and are such that H \ K hastwo connected components. The subdomain D is reasonable if it is simply connected and contains(relative) neighborhoods of 0 and ∞ .An equivalent definition is to consider, for each simply connected domain D and each pair ofboundary points a, b , a law P D,a,b on subsets of D joining a and b. Then the two required propertiesare conformal invariance, namely g ∗ P D,a,b = P g ( D ) ,g ( a ) ,g ( b ) for conformal maps g of D , and “restriction”: For reasonable D (cid:48) ⊂ D, the law P D,a,b of K , whenrestricted to K ⊂ D (cid:48) , equals P D (cid:48) ,a,b . The remarkable main result is that there is a unique one-parameter family of such measures. Theorem 3.10. P = P H , , ∞ is a conformal restriction measure if and only if there is α > suchthat P [ K ⊂ D ] = g (cid:48) D (0) α (7) for every reasonable D ⊂ H . For each α ≥ there is a conformal restriction measure P α . Further-more, α = is the smallest α for which there is a restriction measure, P / is the only restrictionmeasure supported on simple curves, and P / is SLE / .
34n important observation, due to Balint Virag [V], is that P is the law of Brownian excursionsfrom 0 to ∞ in H (roughly, Brownian motion started at 0 and conditioned to “stay in H ” for alltime). An elegant application goes as follows. If K and K are independent samples from P α and P α , then (7) implies that K ∪ K has the law of P α + α (after the “loops” of the union havebeen filled in ). By uniqueness, it follows that the law of the union of 5 independent Brownianexcursions in H (plus loops) is the same as that of 8 copies of SLE / (with loops added). Inparticular, the frontiers are the same and thus have Hausdorff dimension 4 / SLE , stopped whenreaching the boundary of a disc D , is the same as the law of a planar Brownian motion (with thebounded complementary components added), stopped upon leaving D. The proof that
SLE / is P / is based on the following computation. Using the same notationas (5), one can show d h (cid:48) t ( W t ) = h (cid:48)(cid:48) t ( W t ) dW t + (cid:16) h (cid:48)(cid:48) t ( W t ) h (cid:48) t ( W t ) + (cid:0) κ − (cid:1) h (cid:48)(cid:48)(cid:48) t ( W t ) (cid:17) dt. (8)For κ = , it follows that d h (cid:48) t ( W t ) / = 58 h (cid:48)(cid:48) t ( W t ) h (cid:48) t ( W t ) / dW t so that h (cid:48) t ( W t ) / is a local martingale. Writing as before T = inf { t : K t ∩ A (cid:54) = ∅} , it is not hardto show that h (cid:48) t ( W t ) tends to 0 as t → T if K ∩ A (cid:54) = ∅ (the case T < ∞ ), and lim t → T h (cid:48) t ( W t ) → P [ K ⊂ D ] = P [ T = ∞ ] = E [ h (cid:48) T ( W T ) / ] = E [ h (cid:48) ( W ) / ] = g (cid:48) D (0) / . For values greater than 5 / , there are several constructions of the restriction measures describedin [LSW8]. One is by adding “Brownian bubbles” to SLE-traces. There are other versions of the Loewner equation. The “whole plane” equation was developed andused in [LSW3] to deal with hulls K t that are growing in the plane rather than a disc or half-plane.“Di-polar SLE” was introduced in [BB2], see also [BBH]. An important generalization of SLEare the SLE ( κ, ρ ) and variations, first introduced in [LSW8]. An elegant and unified treatmentof all these variants is in [SWi]. Also, defining SLE in multiply connected domains creates a newdifficulty that is not present in the simply connected case, since a slit multiply connected domainis not conformally equivalent to the unslit domain. See [Z],[BF1],[BF2].Since SLE is amenable to computations, the convergence of discrete processes to SLE can beused to obtain results about the original process. In this fashion, Oded [S10] obtained the limitingprobability, as the lattice size tends to zero in critical site percolation on the triangular lattice inthe disc D , that the union of a given arc A ⊂ ∂ D and a percolation cluster surrounds 0. In [LSW4],Lawler, Schramm and Werner showed that the probability of the event 0 ↔ C R that the percolationcluster containing the origin reaches the circle of radius R behaves like R − / ,P [0 ↔ C R ] = R − / o (1) R → ∞ . See also [SW] for related exponents.Because of space, in this note we have ignored the mathematically nutritious “Brownian loopsoup” [LW3] and its relation to restriction measures, as well as the growing literature around theimportant Conformal Loop Ensemble
CLE κ introduced by Scott Sheffield [Sh2]. See [W3] and[SSW].There are deep and exciting connections between the Gaussian Free Field and SLE, as exploredby Oded and Scott Sheffield. The GFF has made its first appearance in this area in Rick Kenyon’swork on the height of domino tilings [Ke3]. See [Sh1] for definitions and properties. Here is avery brief description of their work. Let D ⊂ C be a domain bounded by a simple closed curvethat is partitioned into two arcs by two marked boundary points. Approximate D by a portion G = ( V, E ) of the triangular grid as before (see Figure 20, where again vertices are represented byhexagons), and denote ∂V the boundary vertices. Fix a constant λ , and let h = h ε be an instanceof the Discrete Gaussian Free Field, with boundary values ± λ on the two boundary arcs. Thismeans that h ( v ) , v ∈ V \ ∂V , is a (cid:93) ( V \ ∂V )-dimensional Gaussian random variable whose densityis proportional to exp( − (cid:80) ( u,v ) ∈ E ( h ( v ) − h ( u )) / . Extend h in a piecewise linear fashion fromthe vertices to the triangles. The main result of the deep and very long paper [SS2] is, roughlyspeaking, the following. If λ = 3 − / (cid:112) π/ , then the level curve γ (cid:15) of level h = 0, joining the twomarked boundary points, converges to SLE as ε → . Other values of λ lead to variants of SLE . See also [SS3] and [Dub4].Figure 20: Level set of the Discrete GFF, from [SS2].Finally, there are several collaborations of Oded being written at the moment. For instance,there are deep results of Christophe Garban, Gabor Pete and Oded concerning near-critical per-colation and its scaling limit, which is different from
SLE , see [GPS]. The nice paper [ShWi]36escribes Oded’s (unpublished) proof of Watt’s formula for double crossings in critical percolation,and provides insight into Oded’s masterful use of Mathematica. Watt’s formula was first provedrigorously by Dubedat [Dub2]. Many of Oded’s papers contain open problems, some (such as [S6]) even propose a direction to tacklethem. His ICM talk [S11] contains a large number of problems around SLE, and has provided thefield with a sense of direction. Some additional SLE-related problems are in [RoS]. Several of hisproblems have been solved since their publication.As already mentioned, the convergence of the Ising interface to
SLE , Problem 2.5 in [S11],was proved by Smirnov. The reversability of the chordal SLE path, Problem 7.3 of [S11], hasbeen established for κ ≤ However, there is still no mathematical understanding of the KPZ formula. In fact, the author’sunderstanding of KPZ is too weak to even state a concrete problem.
The Problem 4.1 in [S11], to show the existence of the (weak) Gromov-Hausdorff scaling limitof the graph metric on random triangulations of the sphere, was solved in the impressive work of LeGall [LG]. Le Gall and Paulin showed [LGP] that the limiting space is a topological sphere, almostsurely. Duplantier and Sheffield [DS] described a random measure (a scaling limit of the measure e ch ε dxdy where h ε is the Gaussian free field, averaged over circles of radius ε ) which exhibits aKPZ-like relation. They conjecture a precise relation between a scaling limit of [AnS] and theirrandom continuous space, and discuss connections to SLE. Following Duplantier and Sheffield,simpler random metric spaces exhibiting KPZ were considered in [BS4] and [RV]. We have seen how Oded shaped the field of circle packings, and how he developed a deep un-derstanding of discrete approximations to conformal maps. His results on the Koebe conjectureare still the best to date. We have also seen how Oded’s discovery of SLE led to a powerful newtool in probability theory and in mathematical physics. In fact, it has changed the way physicistsand mathematicians think about critical lattice interfaces, and has led to very fruitful interactionsacross disciplines. The number of mathematicians and physicists working with SLE is increasingfast, and the last few years have seen a number of exciting developments. Oded has already es-tablished his place in the history of mathematics. I have no doubt that we will see many morewonderful developments directly or indirectly related to Oded’s work, thus keeping his spirit alivethrough the work of his fellow mathematicians, coauthors, and friends.37 a) LERW, κ = 2 SAW in half plane - 1,000,000 steps (b) SAW, κ = ?(c) Ising, κ = 3 (d) Harmonic explorer, κ = 4(e) Percolation, κ = 6 (f) UST, κ = 8 Figure 21: Various random curves converging to SLE’s38 eferences [AB] M. Aizenman, A. Burchard, H¨older regularity and dimension bounds for random curves,
Duke Math. J. (1999), 419–453.[AlSh] T. Alberts, S. Sheffield, Hausdorff Dimension of the SLE curve intersected with the real line, Electron. J.Probab. (2008), 1166–1188.[A1] E. Andreev, Convex polyhedra in Lobacevskii space, Math. USSR Sbornik (1970), 413–440.[A2] E. Andreev, Convex polyhedra of finite volume in Lobacevskii space, Math. USSR Sbornik (1970), 255-259.[AnS] O. Angel, O. Schramm, Uniform Infinite Planar Triangulations, Comm.Math.Phys. (2003), 191–213.[BB1] M. Bauer, D. Bernard, Conformal Field Theories of Stochastic Loewner Evolutions,
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