Random, conformally invariant scaling limits in two dimensions
Schramm, Oded
Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been und erstood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
 So it's my privilege to introduce this final plenary speaker who is older, Sram from Microsoft in the United States. So Sram has a background in complex analysis. He was a student of Thurston and he worked early on on packing problems and the proof of the Riemann mapping theorem that way. And then he was interested in graphs and in probability aspects of that. But the real jackpot he hit by having the idea that one should study the Lerner equation not with only smooth coefficients as was Lerner's original idea but also in the general case, in more general cases. And that the choice of the Brownian motion as the leading function in the differential equation was going to produce interesting results. And he was indeed completely right. It was later supplemented by his work together with Lawler and Werner who was one of the recipients of the Fields Medal this year. And as you know, they were able to prove, for example, the Mandelbeut conjecture about the dimension of the outer boundary of the Brownian motion, that the outer dimension is four-third. And if you had mentioned to me ten years ago that somebody was going to prove this conjecture, I would have said that you were out of your mind that there is no chance that this can ever be proved in a strictly mathematical sense. So I think we have witnessed here one of the most interesting development and it would be a great opportunity for you, if you don't know it already, to learn about this in the lecture that Odechram is going to deliver. I cannot help making a personal comment about the method. So it's all based on the deformation of a domain in the complex plane according to the learner scheme. And in this way it is a flow where time plays an essential role. And it's really remarkable to think about that. The proof of the, it started with the Biebebach conjecture which is based precisely on a deformation also of a domain into what was supposed to be the extrema case. And in this case you could get monotonicity. And of course one can think about Perlman's proof of the Frank-Array conjecture as based on completely the same idea that you have a difference equation according to which you deform a domain and then you, there the great difficulties are to understand the singularities. So I think this method of deformation of domains in the complex is one of the greatest achievements of our time in analysis. And I think there are prospects for very interesting developments in the future also. Personally I've been involved in trying to do what is called the DLA. And there is a healer show flow which one doesn't really understand which is similar to that. So I think you will see even a continuation of this method. But right now we look forward to order some lecture on conformal invariance scaling limits. Thank you. Thank you very much, Lena, to a great honor of being here. Okay, well most speakers had some slides showing a plan of the talk and this is my plan. We're going to look at random systems such as the one you see on the slide. This one has a very simple description. All the hexagons are independently white or black with probability one-half. And we're going to study such systems. And the way we're going to learn information about that is by studying random curves that appear in these systems such as the red curve here. Percolation is going to be the primary example but not the only one. So this is called percolation. There are two closely related talks, actually very closely related talks. I'd like to mention the talk by Stanislav Smirnov yesterday and Wendelin Werner's talk later today. And some things which I will omit for lack of time, you can probably learn from Wendelin Werner's talk. Let's start with something rather simple. Let's discuss one-dimensional Brownian motion. One way to think of one-dimensional Brownian motion is a limit of simple random walk. Here you see a simple random walk where the x-axis is the time axis and the y-axis is the spatial coordinate and at specific time units on an intergeal lattice, the walk with probability one-half goes up or down regardless of what has happened in the past. When you get such a graph, the shape, these rectangular steps are not drawn with the height difference equal to the time difference because we're going to take a limit as the time and space scales and get a Brownian motion. And you have to scale time and space differently if you scale space by delta, you have to scale time by delta squared. So this is some definition and a refresher on Brownian motion. Here are some properties Brownian motion has. It is a continuous path with probability one. It satisfies the Markov property which means that if you want to know, if you're a time t and you want to predict something about the future, not with certainty but with some reliability, well, the only information that's relevant for you that's in the past is your current position. And Brownian motion satisfies Brownian scaling. Delta times the Brownian motion is the same in law, has the same distribution as the Brownian motion with time scaled by delta squared. Let's go on to two dimensions. The rest of the talk will be entirely two-dimensional. When you take two-dimensional Brownian motion, you have two natural ways to define it. Since we define Brownian motion as the limit of simple random walk, you can also define two-dimensional Brownian motion as a limit of simple random walk on the square grid. Alternatively, you can take two independent Brownian motion, one in the x-coordinate, one in the y-coordinate, and therefore get a two-dimensional curve. And these two definitions are equivalent. Both definitions suggest that Brownian motion will actually, well, both definitions use the coordinates, the x and y coordinates, but a nice property of two-dimensional Brownian motion and higher-dimensional Brownian motion is that Brownian motion, in fact, doesn't depend on the coordinate system, only depends on the inner product, namely, its rotationally invariant. In two dimensions, it has a higher form of symmetry. It's conformally invariant. That means the following. You have here a domain, and you have a base point, and you start Brownian motion from the base point, you stop when you hit the boundary, and you consider an analytic map or a conformal map to another domain. You start in the other domain a Brownian motion from the image of the base point. Well, let me put it this way. The image of this Brownian motion in this domain is the same as a Brownian motion starting at the image of the base point, except that the time parameterization changes. You should keep this in mind because later on we will discuss various conformal invariants of different models. I will be rather loose, and I will not say what conformal invariance means, but you should think of something in this flavor. Maybe I should say a simple effect is just that the hitting point on the boundary is conformally invariant, but this conformal invariance means much more. That means the whole distribution of the curve is conformally invariant. Let's move on to a more difficult topic, which is percolation. Here you have a mesh of hexagons. This is the most successful two-dimensional percolation model right now. If you're doing a Bernoulli-P percolation, each hexagon is white or open with probability P independently. Then, well, so far it's just a system of independent bits, but once you start studying connected components of white or red or black regions, then you're doing percolation. There are various alternative models. There's a large variety of alternative models of percolation. One alternative model is you work on a grid and you flip a coin for each edge to decide if you keep it or throw it away. Now that's percolation. Now we will discuss critical percolation. It turns out, and that's pretty easy to see, though I will not prove it to you, that there is some number between 0 and 1. See, we have this P as our parameter, which we can decide. P is the probability for each hexagon to be white. We can play with this parameter. There is some specific value that is crucial, and it's called P sub C, the critical value of P, and it can be defined in various different ways, has been defined in various different ways, but the simplest definition mathematically is by saying that if P is larger than that value, you will have an infinite white component. If P is smaller than this value, then the probability for the existence of an infinite white component is going to be 0. This is a good model for a phase transition because the large-scale behavior of the system changes drastically when P pushes beyond PC. What do I mean by that? If P is slightly smaller than PC and you look at a very large picture, then almost nothing is connected. The white regions are small islands or very small lakes inside a vast area of black, and the situation reverses when P is above PC. This is very easy. It's harder to determine what PC is, and for some percolation models, we will probably never know what PC is. For this particular model that I've described here, it has been known. Well, Harris proved one half of the theorem. He showed that at P equals one half, there is no infinite white component. There is no infinite cluster. This means that PC is at least one half. Twenty years later, Keston showed that PC actually is one half because above PC, above one half, you do have infinite cluster. The one half here is a value which is easy to guess. This is because at one half, you have a symmetry of white and black. They behave the same. If you believe my story about one completely dominating the other when you're off of PC, then you should believe that one half has to be the critical value. Putting it was a significant achievement, and based on an earlier important result of Uso Simo in Welsh. Well, Harris's theorem tells us that at one half, we don't have an infinite cluster, and that is the PC. What happens if you condition on this rare event that the cluster of the origin is going to be large? If you sample percolation, you know that with probability one, the cluster of the origin is going to be finite. Typically, you'd expect it to have maybe size five, six, zero hexagons, something like that. But you can run your simulations a few times, and you can wait until you get a simulation which has many hexagons in the cluster of the origin, and then this is roughly what the cluster looks like if you wait till you get a sample with a thousand hexagons or more. You can go on and you can ask yourself, well, how long will I have to wait till I see a cluster of a thousand hexagons? How unlikely does the probability of having a large cluster, how fast does it decay as the notion of large increases? Such questions are governed by what's known as critical exponents, and physicists such as Denise Nienhoes-Cardy, they've predicted the values of these exponents. In particular, the probability that the origin is in a cluster of diameter, at least r, behaves like r to the minus five over 48 with some low-order corrections as r tends to infinity. If you ask a similar question but require that the connection be inside the half plane, then the exponent you get is minus a third instead of minus five over 48. These were conjectures and the arguments typically assume asymptotic conformal invariance and many other unproven properties. The way the physicists like to assume conformal invariance, they look at, say, the probability that two hexagons are in the same cluster and they believe that there's going to be a limit to that as the measure finds after you rescale appropriately and then they say, well, this limit should have nice conformal invariance properties and make similar assumptions. Then once they have a few assumptions in place, they can work some machinery to calculate the values. Conformal invariance plays an important role. Now, there are also predictions which are more refined than just saying an exponent. See, an exponent just gives you one number which is very important because it tells you what these rare events, how unlikely they are and so forth. As useful as it is, there are more precise things to know about. For example, Cardi had the following prediction. If you look at the rectangle and you take the mesh very fine and you ask, what is the probability to have a connection in white hexagons from the left to the right? In the limit as the mesh refines, this quantity approaches this expression right here. This is the gamma function. F2,1 is the hypergeometric function and eta is the cross ratio of these four points after you map the rectangle conformally to the upper half plane. This is a reasonably complicated formula. Leonard Carlson found a nicer form for this formula. As far as conjectures go, you want to assume it naturally because otherwise you can't get even started. One's formula is the following, you take a triangle and you want to connect the bottom edge to a segment on the right-hand edge. If the edge length of this triangle is 1 and the length of this segment is x, then the probability to have this connection converges to x as the mesh goes to 0. This is somewhat simpler than Cardi's form. Just so that we don't get carried away with percolation, I want to mention a few other models because percolation, though it's a central theme, it's not the only topic I want to discuss. Here I can use this picture to describe three different models. What is this picture? You have a white grid here and on the white grid, you should think of the white grid as a graph. Well, it's all a graph, but you should think of an n by n grid, square grid, and the white is a spanning tree in that grid. If you notice, the white maze here has no cycles and it spans every vertex in the appropriate grid. If you have that grid, n by n grid, you can think about choosing a spanning tree from that grid and you can say, well, I want to be unbiased, I'm going to choose the spanning tree uniformly among all spanning trees in that grid. Well, at first I was kind of skeptical about this model. Why would the uniform measure be a natural thing to study? Well, it turns out this uniform spanning tree has many nice connections to different types of questions such as electricity and random walks and so forth. So it is very natural, in fact, to take the uniform measure. So our white is the uniform spanning tree. The pink is also a uniform spanning tree of the dual of the original graph. The pink is just essentially the complement of the white. And the black curve between them is what's called the piano path associated with the uniform spanning tree and it's also known as the Hamiltonian path on the Manhattan lattice. Now those, so here we have a two-dimensional system. We have one curve associated with it and in fact there's another nice curve associated to that system. If you fix two vertices, say white vertices and you fix them before you see the tree, then you sample the tree and there's just one path in the maze connecting the two vertices because it's a tree. That path is what's known as the loop-race random walk. The loop-race random walk is a model invented by Greg Lawler and it is very intimately connected to the uniform spanning tree. Well so there was a belief that the uniform spanning tree in the loop-race random walk ought to be conformally invariant. And there has been some partial progress in that direction. Rick Kenyon proved that some properties of the uniform spanning tree in loop-race random walk are conformally invariant in the scaling limit. And here's an example for a property which he proved. If you take here a domain and you fix three points in the boundary and you take the uniform spanning tree of the grid restricted to this domain, then near these three points you can take vertices in your grid and join them up by the paths that join them in the tree and you will have a kind of a Y-shaped topological Y-tree spanning these three points and there will be a special meeting point, this red meeting point. And if you can do this thing in another domain and you can, I'm assuming the domains are simply connected and you can map one domain conformally to the other domain and what Rick Kenyon has proved is that the distribution of this red meeting point is conformally invariant, namely the image of the measure here of the law of the red point is going to be the law of the red point here provided the three base points are mapped to the three base points. That is a good bit of progress. It doesn't quite prove conformal invariance of this whole system, for example. You don't yet know that these paths are actually have the same distribution in both pictures. Okay, so to study percolation it turns out that we want to look at an interface. Let me, I have a demo here showing this interface. So you start out by declaring that you want the boundary hexagons to be, to change color across your origin here. This is what's called as the devotion boundary conditions. And then you can start an interface between the blue and the yellow. And you want to know how this interface continues so you look at what this hexagon is going to do. And this hexagon, you flip the coin, you see it's yellow so you turn right and then you look at the next hexagon, it turned out blue, you turn left and the next hexagon and the next hexagon. Now sometimes you flip one hexagon and at this point we flipped one hexagon and we could, that's made three steps of the interface because the interface when it hits hexagons that has, that have been colored before, it doesn't need to sample a new one. And you continue and so on and so on, so on. And you continue like that and you build this interface which has no choice but to go out to infinity. You should think of it as an interface connecting the two boundary switching points which is zero and infinity. So now we're going to take, well so far I haven't done really any math, I just told you about math. Now we're going to, for four slides roughly, we're going to actually do some math here. And I'm going to try to walk you through the intuition for why you want to consider Levener's equation with Brownian motion as the driving term, though you don't need to know any of these words. We're going to see, and this is roughly the way this actually, the original motivation, though the original motivation was in the context of Lupe or Israel and more, but the picture is similar. So we take our curve and suppose that we only look at part of it and we stop when it reaches size epsilon. And I'm not going to be very clear at this point at what size means, we'll see later what size means. And then we believe in conformal invariance for now, the task of justifying conformal invariance we'll discuss later. We believe conformal invariance, we have sampled the part of the walk, of this interface and we'd like to know what the rest looks like. Well, the trouble is that this piece here is very complicated, it's kind of fractal shaped and we want to simplify it, but we believe in conformal invariance and see the boundary of this domain is, after you remove the path is colored by two colors, the white and the black here and they change in exactly one point which is the tip of the curve. And therefore, we can simplify the domain, we can map over by the conformal map, we use Riemann's mapping theorem telling us that the complement of this region that we've examined already is simply connected, therefore we can map it over to the upper half plane again. And when we map it over to the upper half plane, there will be a special point where the image of the black curve meets the image of the white and the red here is indicated to show what part of the boundary corresponds to things that are adjacent to the curve. And this map that we're going to use has some power series expansion near infinity and we normalize it so it fixes infinity and the power series expansion looks like that. This is all very easy consequence of the Riemann mapping theorem. And if we think about the continuation of the path here, that's going to look in distribution about the same as what the original path would have looked if we started it at the tip at the boundary point right here because we believe conformal invariance. And now we can continue the procedure, right? We can, you know, after we can start here, we can build the interface until it has size epsilon and then map over by another map, F2. And then F2 has essentially the same distribution as F1 except that it's translated to start at the point here where black changes to white is not necessarily the same point as black change to white here. And so F2 has given that we know F1, F2 has the distribution of F1 except translated therefore conjugated by this translation. And this here denotes equivalence in law. And we can continue inductively. If WJ denotes the place where the image of the jth map takes, the jth composition takes the tip to some place, then we have this composition GN which is the composition of all these Fs and each Fj has essentially the same law of F1 except conjugated by translation depending on this WJ. And this GN is just the map from the complement of the curve we've examined so far to the upper half plane because it's, you know, each time we've added a piece and we've taken the map that makes it disappear. And now each Fj is going to be very close to the identity because our epsilon is going to be very small. And therefore we can attempt to think of this composition as a flow rather than a composition of discrete steps because if you have a map that's close to the identity you can think of it as approximately a vector field. And to understand this vector field we're going to essentially find out what it is. We can look again at this map F1 which has this power series representation near infinity. And now to simplify things we want to choose cleverly how we measure the size of the initial segment. It turns out that A1 is monotone in the path as you extend the path A1 increases and therefore to make life easier we choose A1 to be essentially to epsilon. Therefore we stop the curve once A1 reaches size 2 epsilon. And then easy calculation with power series and the scaling argument implies that F1 is well z plus 2 epsilon z inverse plus the other terms which are bounded by epsilon to the three halves. So we're not going to do this calculation for you now. And then the consequence is that Fj plus 1 is going to be the same thing except conjugated by the translation to Wj and therefore has this expression. Fj plus 1 is the same thing except you have minus Wj. And therefore so this is essentially a vector field where this is the vector at the point gt of z. And therefore instead of thinking of the composition of these new identity maps we can rather flow according to this vector field where this Wt is some continuous version of the Wj's. And well what is this continuous version of the Wj's? Well the Wj's have stationary independent increments. As you go from Wj to Wj plus 1 the distribution of that translation is going to be the same as from W1 to W2 from W2 to W3. So all the same because we've remarked that the law of these maps are all the same except for the translation. Once you take away the translation the law is the same and therefore the Wj's are have the same, it's like a simple random walk except the steps not necessarily have a fixed length but have a random length but they have the same random length each time. And so this Wt which is a limit of these Wj's is going to be a process that has stationary independent increments. It's symmetric because our problem is symmetric and it's continuous that you have to argue by some other means but that's not too hard. And it follows easily once you know a little bit about Brownian motion that it's a multiple of Brownian motion. And this leads us to the definition of what SLE is. SLE is stochastic levonner evolution. Well we fix the parameter kappa and we define Wt as Brownian motion at time kappa t and this Brownian motion is one dimensional. And then you solve this differential equation starting at g0 of z equals z. This is essentially what is supposed to be the conformal map from the complement of the interface to the upper half plane at time t. And this equation is known as Levener's equation. He's used it to study the Biberbach conjecture. And well he studied a different, mostly used a different, slightly different form of the equation so this is why it's called Cordo. And once you solve this equation you can recover the path itself by taking gt inverse of Wt. So what is the basic idea here? The idea here is that we want to invoke differential equations to study our percolation. Now the percolation interface is a fractal object and it's hard to study fractals with differential equations. Differential equations are better suitable to smooth objects. Well we have a smooth object at our disposal, the complement of the interface. That's a domain and therefore a smooth object. So instead of studying the evolution of the interface we study the evolution of the complement and that's amenable to differential equations and therefore we can not only find out what we guess it to be but also then we can do calculations to calculate properties of the complement and therefore of the interface itself. Well this whole discussion was conditional on having conformal invariance but well we didn't have conformal invariance so what do we do? Well it seems just to be a nice physics like prediction, not so satisfying for a mathematician. At this point Vendeline Verne came up with a brilliant idea that well we have a fractal at our disposal which is known to be conformal invariant but still has some nice questions that haven't been answered yet so why don't we apply this method of understanding conformal invariant fractal to Brownian motion. If you look at Brownian motion itself in some ways it's very simple. However if you look at the outer boundary of Brownian motion that's not very amenable to analysis directly from the definition of Brownian motion. So what you see in this picture is Brownian motion started at the origin and stopped when it hits the boundary of the unit disc and the red is all the places visited by the motion and the black is the outer boundary of the Brownian motion. The set of points so that you can get to from the exterior without hitting any other points before that. There was a conjecture of Benoit Mandelbrot that the outer boundary of Brownian motion has dimension four thirds and well Vendeline was right. SLE was the right tool to attack this problem and several related problems about the Brownian motion. I think Vendeline will tell you more about this later this afternoon but let me just give one of the punch lines. Greg Lawler Vendeline and I proved that indeed the Mandelbrot conjecture is correct. The outer boundary has dimension four thirds and you can extract other information about the curve. For example the set of cut points has house of dimension three fourths. The cut points are those points where if you remove them you disconnect the Brownian motion. And this is based on first, well this is not how it was written but a posteriori we now know that in fact the outer boundary of this Brownian motion is essentially the same as the outer boundary of SLE six and that's what permits the connection between why studying SLE can tell us something about the Brownian motion. At first we didn't know that they're actually the same but we knew that the dimensions should be the same. Well okay so that was one use but we didn't have to wait very long and Stas Mianoff actually proved Cardi's formula in Carlson's form and in fact the way the simple form of thinking about triangles and the hexagonal lattice turned out to be an important observation leading to the proof. The proof is quite brilliant but and amazingly simple and short. So Stas proved that in fact Cardi's formula is correct and from this it's not hard to conclude that the percolation interface scaling limit the same curve that I showed on the opening slide is SLE six. So six here is the parameter cap that you choose remember SLE has a free parameter. And this theorem allowed the determination of many of the exponents associated with percolation in particular we showed that the exponent that I discussed in the beginning the probability to connect to distance r indeed decays like r to the minus five over 48 with some lower order corrections and many other exponents and properties were proved by Smeonoff and Verna based on earlier work by Keston relating all these exponents to one another that SLE point of view only helps to determine things at PC at the critical probability and there are some exponents relating to off critical but Keston linked these with the critical exponents. Subsequently the looper is random walk was proved to converge to SLE two in the uniform spanning tree. Piano path scaling limit was proved to be SLE eight. This was with Loller and Verna and here the situation is kind of the opposite in the sense that first convergence to SLE was proved and the consequence of that was conformal invariance rather than the other way around and here you see a picture of what the looper is random walk looks like. So you see that with different values of the parameter cap are give entirely different processes and this may be a little bit surprising at first that you can change things like the house of dimension and properties just by changing the speed of the Brownian motion. It is like you can think of the Brownian motion as a driver in a taxi driver so if the taxi driver has one beer maybe can still drive so and reasonably well but after a few more drinks becomes erratic and the consequences are easy to see. So those are two phase transitions in the behavior of SLE as depending on Kappa this was proved with Stefan Rode. If Kappa is at most four it is a simple path if it is between four and eight then it is not a simple path anymore it kind of swallows parts of the domain as it goes along that is roughly what the SLE six pictures look like if you remember we will see them shortly again and when Kappa is eight or larger the curve is actually space filling it visits every point. Now the SLE path it cannot cross itself but it can touch itself therefore in this case once you circle some part of the domain you can't go in there and fill that up that is one difference from Brownian motion. Brownian motion can make a loop and then go in the loop and out again that is why Brownian motion is more difficult to study from this point of view than SLE. Now notice the parameter eight is precisely what the uniform spanning tree piano path converges to and it is easy to see from the definition that it has to converge to a space filling curve and that is exactly the transition where the Kappa where the transition to space filling occurs. There is some information about the dimension of the SLE in this paper with Stefan we showed that the expected box count dimension well expected box count dimension I am not going to define this concept because you can probably guess what it means by the from its name the expected box count dimension of the path is one plus Kappa over eight when Kappa is between zero and eight when Kappa is larger than eight it is of course two because it is space filling and in the range where Kappa is larger than four the outer boundary of the path is not the same as the path itself because it is not a simple path and the expected box count dimension of the outer boundary is given by this formula which was earlier conjectured by Rick Kenyon and Vincent Befar in a quite clever and technical work managed to show that this expected box count dimension of the path is in fact the actual house dwarf dimension. The expected box count dimension is much simpler to determine because you just need to know you need to calculate expectations and for house dwarf dimensions essentially you need to calculate second moments and that is more difficult. I think that the corresponding statement for this is still open but probably won't be for very long. I want to tell you about yet another model which whose relation to SLE has been determined this is the Gaussian free field. What you should think let's discuss the discrete Gaussian free field there is also a continuous version which is of the same flavor only harder to define. You should think first of a graph and then at each vertex you have some value H of V and you want to penalize discrepancies. You can think of it as a model for a surface which has penalties for discrepancy from being smoothness. You penalize by H of V minus H of U squared for vertices that are close by so the total penalty is the exponential of minus the sum over all neighbors of this square difference and this is a nice formula because the distribution of H under this probability density is going to be a Gaussian. The Kenyon showed that this Gaussian free field is a scaling limit of the domino tiling height function which is another discrete model which I'm not going to describe to you but this was partly a motivation to study the Gaussian free field. Here's what you do with the Gaussian free field if you want to do you put the Brouche and boundary values again you put boundary values plus A here and minus A here and then you can look at the set of edges. You can color the vertices one color if they're positive and another color if they're negative and then you can take the interface and this is a graphical depiction of this interface. Scott Schaaffield and I proved that this interface of the discrete Gaussian free field converges to SLE4. Now 4 is the special value where the interface above 4, the SLE above 4 stops being a simple curve and you can see that this is an actual simulation. You can see that this curve is almost not a simple curve anymore and the shades of blue and red they depict the heights of the field. Now in order for this statement to hold true you need to pick this value of A, a very specific value. If you take a different value then you get convergence to something very closely related which we also know how to describe. And there's an analogous statement about the continuum Gaussian free field but that's harder to formulate. There's a conjecture and Stas Miernoff reported on much progress on this conjecture that the critical easing model is converging to SLE3. This is this picture and Stas also shows that with other parameters things associated with the easing model also converge to SLE but with a different value of kappa and this figure is drawn by David Wilson. And so here the white is plus spins and the dark is minus spins. If I assume you're familiar with the easing model if not then think of this as just another pretty picture, hopefully pretty. Yes. And if you take the uniform measure on lattice paths that are self-avoiding and stay in the upper half plane the belief is that that converges to SLE8. I can't say I can't report any progress on this conjecture but it's supported experimentally by simulations by Tom Kennedy and this figure is being drawn by Tom Kennedy. Contrary to all the other figures in this talk this figure is one which you really need to work hard to produce. All the others have very simple algorithms. Well, time is running short and I'm going to give a quick summary. Here a list of things we know SLE describes the scaling limit of. Critical site percolation on the triangular grid and the value of kappa is 6. Lupo-Aisland and walk and the value is 2. The piano curve the value of 8. Gaussian free field and the value is kappa. Like explore is a simpler model that is kind of built, defined to be easy for this statement. And I'm not going to discuss it. And here are some conjectures regarding other models converging to SLE. Well there are other models of critical percolation which should converge to the boundaries should converge to SLE 6 but that's not been proven. The easing model with different parameters converges to 3 and 6 and we hope to see a proof by Stas pretty soon. And the FK cluster boundaries well I haven't defined them but I mentioned them here because they actually there's another parameter they're Q and therefore you have a whole range of kappa between 4 and 8 and similarly with the ON models. The self avoiding walk is of course very challenging. The double domino has relations to the Gaussian free field we've mentioned this already. And there are some things that SLE does not give. It does not give DLA. There's been a Lerven type analysis of DLA by Carlson and Makarov and by Hastings-Levitov that actually proceeds this SLE definition. It appears that the paths of the minimal spanning tree are not described by SLE and are probably not conformally invariant. This is based on simulations by Willand and Wilson and SLE doesn't apply to many interesting questions in dimension higher than 2. Let me mention that percolation in high dimensions is almost completely understood due to the work of Haar and Slade and others with a method called the lace expansion which is very different than what I described today. And the existence of the looper-ace random walk scaling limit in R3 has been proven recently by Gadi-Cosma but Gadi hasn't determined what the limit is just shown that it exists but that's quite a remarkable achievement. Well, so we have seen things that you can do with SLE, things that hopefully will be done and things that you need to use other methods. Thank you very much for attending my talk.
 So it's my privilege to introduce this final plenary speaker who is older, Sram from Microsoft  in the United States.  So Sram has a background in complex analysis.  He was a student of Thurston and he worked early on on packing problems and the proof  of the Riemann mapping theorem that way.  And then he was interested in graphs and in probability aspects of that.  But the real jackpot he hit by having the idea that one should study the Lerner equation  not with only smooth coefficients as was Lerner's original idea but also in the general case,  in more general cases.  And that the choice of the Brownian motion as the leading function in the differential  equation was going to produce interesting results.  And he was indeed completely right.  It was later supplemented by his work together with Lawler and Werner who was one of the  recipients of the Fields Medal this year.  And as you know, they were able to prove, for example, the Mandelbeut conjecture about  the dimension of the outer boundary of the Brownian motion, that the outer dimension  is four-third.  And if you had mentioned to me ten years ago that somebody was going to prove this conjecture,  I would have said that you were out of your mind that there is no chance that this can  ever be proved in a strictly mathematical sense.  So I think we have witnessed here one of the most interesting development and it would  be a great opportunity for you, if you don't know it already, to learn about this in the  lecture that Odechram is going to deliver.  I cannot help making a personal comment about the method.  So it's all based on the deformation of a domain in the complex plane according to the  learner scheme.  And in this way it is a flow where time plays an essential role.  And it's really remarkable to think about that.  The proof of the, it started with the Biebebach conjecture which is based precisely on a deformation  also of a domain into what was supposed to be the extrema case.  And in this case you could get monotonicity.  And of course one can think about Perlman's proof of the Frank-Array conjecture as based  on completely the same idea that you have a difference equation according to which you  deform a domain and then you, there the great difficulties are to understand the singularities.  So I think this method of deformation of domains in the complex is one of the greatest  achievements of our time in analysis.  And I think there are prospects for very interesting developments in the future also.  Personally I've been involved in trying to do what is called the DLA.  And there is a healer show flow which one doesn't really understand which is similar  to that.  So I think you will see even a continuation of this method.  But right now we look forward to order some lecture on conformal invariance scaling limits.  Thank you.  Thank you very much, Lena, to a great honor of being here.  Okay, well most speakers had some slides showing a plan of the talk and this is my plan.  We're going to look at random systems such as the one you see on the slide.  This one has a very simple description.  All the hexagons are independently white or black with probability one-half.  And we're going to study such systems.  And the way we're going to learn information about that is by studying random curves that  appear in these systems such as the red curve here.  Percolation is going to be the primary example but not the only one.  So this is called percolation.  There are two closely related talks, actually very closely related talks.  I'd like to mention the talk by Stanislav Smirnov yesterday and Wendelin Werner's talk  later today.  And some things which I will omit for lack of time, you can probably learn from Wendelin  Werner's talk.  Let's start with something rather simple.  Let's discuss one-dimensional Brownian motion.  One way to think of one-dimensional Brownian motion is a limit of simple random walk.  Here you see a simple random walk where the x-axis is the time axis and the y-axis is  the spatial coordinate and at specific time units on an intergeal lattice, the walk with  probability one-half goes up or down regardless of what has happened in the past.  When you get such a graph, the shape, these rectangular steps are not drawn with the height  difference equal to the time difference because we're going to take a limit as the time and  space scales and get a Brownian motion.  And you have to scale time and space differently if you scale space by delta, you have to scale  time by delta squared.  So this is some definition and a refresher on Brownian motion.  Here are some properties Brownian motion has.  It is a continuous path with probability one.  It satisfies the Markov property which means that if you want to know, if you're a time  t and you want to predict something about the future, not with certainty but with some  reliability, well, the only information that's relevant for you that's in the past is your  current position.  And Brownian motion satisfies Brownian scaling.  Delta times the Brownian motion is the same in law, has the same distribution as the Brownian  motion with time scaled by delta squared.  Let's go on to two dimensions.  The rest of the talk will be entirely two-dimensional.  When you take two-dimensional Brownian motion, you have two natural ways to define it.  Since we define Brownian motion as the limit of simple random walk, you can also define  two-dimensional Brownian motion as a limit of simple random walk on the square grid.  Alternatively, you can take two independent Brownian motion, one in the x-coordinate,  one in the y-coordinate, and therefore get a two-dimensional curve.  And these two definitions are equivalent.  Both definitions suggest that Brownian motion will actually, well, both definitions use  the coordinates, the x and y coordinates, but a nice property of two-dimensional Brownian  motion and higher-dimensional Brownian motion is that Brownian motion, in fact, doesn't depend  on the coordinate system, only depends on the inner product, namely, its rotationally  invariant.  In two dimensions, it has a higher form of symmetry.  It's conformally invariant.  That means the following.  You have here a domain, and you have a base point, and you start Brownian motion from  the base point, you stop when you hit the boundary, and you consider an analytic map  or a conformal map to another domain.  You start in the other domain a Brownian motion from the image of the base point.  Well, let me put it this way.  The image of this Brownian motion in this domain is the same as a Brownian motion starting  at the image of the base point, except that the time parameterization changes.  You should keep this in mind because later on we will discuss various conformal invariants  of different models.  I will be rather loose, and I will not say what conformal invariance means, but you should  think of something in this flavor.  Maybe I should say a simple effect is just that the hitting point on the boundary is  conformally invariant, but this conformal invariance means much more.  That means the whole distribution of the curve is conformally invariant.  Let's move on to a more difficult topic, which is percolation.  Here you have a mesh of hexagons.  This is the most successful two-dimensional percolation model right now.  If you're doing a Bernoulli-P percolation, each hexagon is white or open with probability  P independently.  Then, well, so far it's just a system of independent bits, but once you start studying  connected components of white or red or black regions, then you're doing percolation.  There are various alternative models.  There's a large variety of alternative models of percolation.  One alternative model is you work on a grid and you flip a coin for each edge to decide  if you keep it or throw it away.  Now that's percolation.  Now we will discuss critical percolation.  It turns out, and that's pretty easy to see, though I will not prove it to you, that there  is some number between 0 and 1.  See, we have this P as our parameter, which we can decide.  P is the probability for each hexagon to be white.  We can play with this parameter.  There is some specific value that is crucial, and it's called P sub C, the critical value  of P, and it can be defined in various different ways, has been defined in various different  ways, but the simplest definition mathematically is by saying that if P is larger than that  value, you will have an infinite white component.  If P is smaller than this value, then the probability for the existence of an infinite  white component is going to be 0.  This is a good model for a phase transition because the large-scale behavior of the system  changes drastically when P pushes beyond PC.  What do I mean by that?  If P is slightly smaller than PC and you look at a very large picture, then almost nothing  is connected.  The white regions are small islands or very small lakes inside a vast area of black, and  the situation reverses when P is above PC.  This is very easy.  It's harder to determine what PC is, and for some percolation models, we will probably  never know what PC is.  For this particular model that I've described here, it has been known.  Well, Harris proved one half of the theorem.  He showed that at P equals one half, there is no infinite white component.  There is no infinite cluster.  This means that PC is at least one half.  Twenty years later, Keston showed that PC actually is one half because above PC, above  one half, you do have infinite cluster.  The one half here is a value which is easy to guess.  This is because at one half, you have a symmetry of white and black.  They behave the same.  If you believe my story about one completely dominating the other when you're off of PC,  then you should believe that one half has to be the critical value.  Putting it was a significant achievement, and based on an earlier important result of  Uso Simo in Welsh.  Well, Harris's theorem tells us that at one half, we don't have an infinite cluster, and  that is the PC.  What happens if you condition on this rare event that the cluster of the origin is going  to be large?  If you sample percolation, you know that with probability one, the cluster of the origin  is going to be finite.  Typically, you'd expect it to have maybe size five, six, zero hexagons, something like that.  But you can run your simulations a few times, and you can wait until you get a simulation  which has many hexagons in the cluster of the origin, and then this is roughly what  the cluster looks like if you wait till you get a sample with a thousand hexagons or more.  You can go on and you can ask yourself, well, how long will I have to wait till I see a  cluster of a thousand hexagons?  How unlikely does the probability of having a large cluster, how fast does it decay as  the notion of large increases?  Such questions are governed by what's known as critical exponents, and physicists such  as Denise Nienhoes-Cardy, they've predicted the values of these exponents.  In particular, the probability that the origin is in a cluster of diameter, at least r, behaves  like r to the minus five over 48 with some low-order corrections as r tends to infinity.  If you ask a similar question but require that the connection be inside the half plane,  then the exponent you get is minus a third instead of minus five over 48.  These were conjectures and the arguments typically assume asymptotic conformal invariance and  many other unproven properties.  The way the physicists like to assume conformal invariance, they look at, say, the probability  that two hexagons are in the same cluster and they believe that there's going to be  a limit to that as the measure finds after you rescale appropriately and then they say,  well, this limit should have nice conformal invariance properties and make similar assumptions.  Then once they have a few assumptions in place, they can work some machinery to calculate  the values.  Conformal invariance plays an important role.  Now, there are also predictions which are more refined than just saying an exponent.  See, an exponent just gives you one number which is very important because it tells you  what these rare events, how unlikely they are and so forth.  As useful as it is, there are more precise things to know about.  For example, Cardi had the following prediction.  If you look at the rectangle and you take the mesh very fine and you ask, what is the  probability to have a connection in white hexagons from the left to the right?  In the limit as the mesh refines, this quantity approaches this expression right here.  This is the gamma function.  F2,1 is the hypergeometric function and eta is the cross ratio of these four points after  you map the rectangle conformally to the upper half plane.  This is a reasonably complicated formula.  Leonard Carlson found a nicer form for this formula.  As far as conjectures go, you want to assume it naturally because otherwise you can't get  even started.  One's formula is the following, you take a triangle and you want to connect the bottom  edge to a segment on the right-hand edge.  If the edge length of this triangle is 1 and the length of this segment is x, then the  probability to have this connection converges to x as the mesh goes to 0.  This is somewhat simpler than Cardi's form.  Just so that we don't get carried away with percolation, I want to mention a few other  models because percolation, though it's a central theme, it's not the only topic I want  to discuss.  Here I can use this picture to describe three different models.  What is this picture?  You have a white grid here and on the white grid, you should think of the white grid as  a graph.  Well, it's all a graph, but you should think of an n by n grid, square grid, and the white  is a spanning tree in that grid.  If you notice, the white maze here has no cycles and it spans every vertex in the appropriate  grid.  If you have that grid, n by n grid, you can think about choosing a spanning tree from  that grid and you can say, well, I want to be unbiased, I'm going to choose the spanning  tree uniformly among all spanning trees in that grid.  Well, at first I was kind of skeptical about this model.  Why would the uniform measure be a natural thing to study?  Well, it turns out this uniform spanning tree has many nice connections to different types  of questions such as electricity and random walks and so forth.  So it is very natural, in fact, to take the uniform measure.  So our white is the uniform spanning tree.  The pink is also a uniform spanning tree of the dual of the original graph.  The pink is just essentially the complement of the white.  And the black curve between them is what's called the piano path associated with the  uniform spanning tree and it's also known as the Hamiltonian path on the Manhattan lattice.  Now those, so here we have a two-dimensional system.  We have one curve associated with it and in fact there's another nice curve associated  to that system.  If you fix two vertices, say white vertices and you fix them before you see the tree,  then you sample the tree and there's just one path in the maze connecting the two vertices  because it's a tree.  That path is what's known as the loop-race random walk.  The loop-race random walk is a model invented by Greg Lawler and it is very intimately connected  to the uniform spanning tree.  Well so there was a belief that the uniform spanning tree in the loop-race random walk  ought to be conformally invariant.  And there has been some partial progress in that direction.  Rick Kenyon proved that some properties of the uniform spanning tree in loop-race random  walk are conformally invariant in the scaling limit.  And here's an example for a property which he proved.  If you take here a domain and you fix three points in the boundary and you take the uniform  spanning tree of the grid restricted to this domain, then near these three points you can  take vertices in your grid and join them up by the paths that join them in the tree and  you will have a kind of a Y-shaped topological Y-tree spanning these three points and there  will be a special meeting point, this red meeting point.  And if you can do this thing in another domain and you can, I'm assuming the domains are  simply connected and you can map one domain conformally to the other domain and what Rick  Kenyon has proved is that the distribution of this red meeting point is conformally invariant,  namely the image of the measure here of the law of the red point is going to be the law  of the red point here provided the three base points are mapped to the three base points.  That is a good bit of progress.  It doesn't quite prove conformal invariance of this whole system, for example.  You don't yet know that these paths are actually have the same distribution in both pictures.  Okay, so to study percolation it turns out that we want to look at an interface.  Let me, I have a demo here showing this interface.  So you start out by declaring that you want the boundary hexagons to be, to change color  across your origin here.  This is what's called as the devotion boundary conditions.  And then you can start an interface between the blue and the yellow.  And you want to know how this interface continues so you look at what this hexagon is going  to do.  And this hexagon, you flip the coin, you see it's yellow so you turn right and then you  look at the next hexagon, it turned out blue, you turn left and the next hexagon and the  next hexagon.  Now sometimes you flip one hexagon and at this point we flipped one hexagon and we could,  that's made three steps of the interface because the interface when it hits hexagons that has,  that have been colored before, it doesn't need to sample a new one.  And you continue and so on and so on, so on.  And you continue like that and you build this interface which has no choice but to go out  to infinity.  You should think of it as an interface connecting the two boundary switching points which is  zero and infinity.  So now we're going to take, well so far I haven't done really any math, I just told  you about math.  Now we're going to, for four slides roughly, we're going to actually do some math here.  And I'm going to try to walk you through the intuition for why you want to consider Levener's  equation with Brownian motion as the driving term, though you don't need to know any of  these words.  We're going to see, and this is roughly the way this actually, the original motivation,  though the original motivation was in the context of Lupe or Israel and more, but the  picture is similar.  So we take our curve and suppose that we only look at part of it and we stop when it reaches  size epsilon.  And I'm not going to be very clear at this point at what size means, we'll see later  what size means.  And then we believe in conformal invariance for now, the task of justifying conformal  invariance we'll discuss later.  We believe conformal invariance, we have sampled the part of the walk, of this interface and  we'd like to know what the rest looks like.  Well, the trouble is that this piece here is very complicated, it's kind of fractal  shaped and we want to simplify it, but we believe in conformal invariance and see the  boundary of this domain is, after you remove the path is colored by two colors, the white  and the black here and they change in exactly one point which is the tip of the curve.  And therefore, we can simplify the domain, we can map over by the conformal map, we use  Riemann's mapping theorem telling us that the complement of this region that we've  examined already is simply connected, therefore we can map it over to the upper half plane  again.  And when we map it over to the upper half plane, there will be a special point where  the image of the black curve meets the image of the white and the red here is indicated  to show what part of the boundary corresponds to things that are adjacent to the curve.  And this map that we're going to use has some power series expansion near infinity and we  normalize it so it fixes infinity and the power series expansion looks like that.  This is all very easy consequence of the Riemann mapping theorem.  And if we think about the continuation of the path here, that's going to look in distribution  about the same as what the original path would have looked if we started it at the tip at  the boundary point right here because we believe conformal invariance.  And now we can continue the procedure, right?  We can, you know, after we can start here, we can build the interface until it has size  epsilon and then map over by another map, F2.  And then F2 has essentially the same distribution as F1 except that it's translated to start  at the point here where black changes to white is not necessarily the same point as black  change to white here.  And so F2 has given that we know F1, F2 has the distribution of F1 except translated  therefore conjugated by this translation.  And this here denotes equivalence in law.  And we can continue inductively.  If WJ denotes the place where the image of the jth map takes, the jth composition takes  the tip to some place, then we have this composition GN which is the composition of all these Fs  and each Fj has essentially the same law of F1 except conjugated by translation depending  on this WJ.  And this GN is just the map from the complement of the curve we've examined so far to the  upper half plane because it's, you know, each time we've added a piece and we've taken  the map that makes it disappear.  And now each Fj is going to be very close to the identity because our epsilon is going  to be very small.  And therefore we can attempt to think of this composition as a flow rather than a composition  of discrete steps because if you have a map that's close to the identity you can think  of it as approximately a vector field.  And to understand this vector field we're going to essentially find out what it is.  We can look again at this map F1 which has this power series representation near infinity.  And now to simplify things we want to choose cleverly how we measure the size of the initial  segment.  It turns out that A1 is monotone in the path as you extend the path A1 increases and therefore  to make life easier we choose A1 to be essentially to epsilon.  Therefore we stop the curve once A1 reaches size 2 epsilon.  And then easy calculation with power series and the scaling argument implies that F1 is  well z plus 2 epsilon z inverse plus the other terms which are bounded by epsilon to the  three halves.  So we're not going to do this calculation for you now.  And then the consequence is that Fj plus 1 is going to be the same thing except conjugated  by the translation to Wj and therefore has this expression.  Fj plus 1 is the same thing except you have minus Wj.  And therefore so this is essentially a vector field where this is the vector at the point  gt of z.  And therefore instead of thinking of the composition of these new identity maps we can rather flow  according to this vector field where this Wt is some continuous version of the Wj's.  And well what is this continuous version of the Wj's?  Well the Wj's have stationary independent increments.  As you go from Wj to Wj plus 1 the distribution of that translation is going to be the same  as from W1 to W2 from W2 to W3.  So all the same because we've remarked that the law of these maps are all the same except  for the translation.  Once you take away the translation the law is the same and therefore the Wj's are have  the same, it's like a simple random walk except the steps not necessarily have a fixed length  but have a random length but they have the same random length each time.  And so this Wt which is a limit of these Wj's is going to be a process that has stationary  independent increments.  It's symmetric because our problem is symmetric and it's continuous that you have to argue  by some other means but that's not too hard.  And it follows easily once you know a little bit about Brownian motion that it's a multiple  of Brownian motion.  And this leads us to the definition of what SLE is.  SLE is stochastic levonner evolution.  Well we fix the parameter kappa and we define Wt as Brownian motion at time kappa t and  this Brownian motion is one dimensional.  And then you solve this differential equation starting at g0 of z equals z.  This is essentially what is supposed to be the conformal map from the complement of the  interface to the upper half plane at time t.  And this equation is known as Levener's equation.  He's used it to study the Biberbach conjecture.  And well he studied a different, mostly used a different, slightly different form of the  equation so this is why it's called Cordo.  And once you solve this equation you can recover the path itself by taking gt inverse  of Wt.  So what is the basic idea here?  The idea here is that we want to invoke differential equations to study our percolation.  Now the percolation interface is a fractal object and it's hard to study fractals with  differential equations.  Differential equations are better suitable to smooth objects.  Well we have a smooth object at our disposal, the complement of the interface.  That's a domain and therefore a smooth object.  So instead of studying the evolution of the interface we study the evolution of the complement  and that's amenable to differential equations and therefore we can not only find out what  we guess it to be but also then we can do calculations to calculate properties of the  complement and therefore of the interface itself.  Well this whole discussion was conditional on having conformal invariance but well we  didn't have conformal invariance so what do we do?  Well it seems just to be a nice physics like prediction, not so satisfying for a mathematician.  At this point Vendeline Verne came up with a brilliant idea that well we have a fractal  at our disposal which is known to be conformal invariant but still has some nice questions  that haven't been answered yet so why don't we apply this method of understanding conformal  invariant fractal to Brownian motion.  If you look at Brownian motion itself in some ways it's very simple.  However if you look at the outer boundary of Brownian motion that's not very amenable  to analysis directly from the definition of Brownian motion.  So what you see in this picture is Brownian motion started at the origin and stopped  when it hits the boundary of the unit disc and the red is all the places visited by the  motion and the black is the outer boundary of the Brownian motion.  The set of points so that you can get to from the exterior without hitting any other points  before that.  There was a conjecture of Benoit Mandelbrot that the outer boundary of Brownian motion  has dimension four thirds and well Vendeline was right.  SLE was the right tool to attack this problem and several related problems about the Brownian  motion.  I think Vendeline will tell you more about this later this afternoon but let me just  give one of the punch lines.  Greg Lawler Vendeline and I proved that indeed the Mandelbrot conjecture is correct.  The outer boundary has dimension four thirds and you can extract other information about  the curve.  For example the set of cut points has house of dimension three fourths.  The cut points are those points where if you remove them you disconnect the Brownian motion.  And this is based on first, well this is not how it was written but a posteriori we now  know that in fact the outer boundary of this Brownian motion is essentially the same as  the outer boundary of SLE six and that's what permits the connection between why studying  SLE can tell us something about the Brownian motion.  At first we didn't know that they're actually the same but we knew that the dimensions should  be the same.  Well okay so that was one use but we didn't have to wait very long and Stas Mianoff actually  proved Cardi's formula in Carlson's form and in fact the way the simple form of thinking  about triangles and the hexagonal lattice turned out to be an important observation  leading to the proof.  The proof is quite brilliant but and amazingly simple and short.  So Stas proved that in fact Cardi's formula is correct and from this it's not hard to  conclude that the percolation interface scaling limit the same curve that I showed on the  opening slide is SLE six.  So six here is the parameter cap that you choose remember SLE has a free parameter.  And this theorem allowed the determination of many of the exponents associated with percolation  in particular we showed that the exponent that I discussed in the beginning the probability  to connect to distance r indeed decays like r to the minus five over 48 with some lower  order corrections and many other exponents and properties were proved by Smeonoff and  Verna based on earlier work by Keston relating all these exponents to one another that SLE  point of view only helps to determine things at PC at the critical probability and there  are some exponents relating to off critical but Keston linked these with the critical  exponents.  Subsequently the looper is random walk was proved to converge to SLE two in the uniform  spanning tree.  Piano path scaling limit was proved to be SLE eight.  This was with Loller and Verna and here the situation is kind of the opposite in the sense  that first convergence to SLE was proved and the consequence of that was conformal invariance  rather than the other way around and here you see a picture of what the looper is random  walk looks like.  So you see that with different values of the parameter cap are give entirely different  processes and this may be a little bit surprising at first that you can change things like  the house of dimension and properties just by changing the speed of the Brownian motion.  It is like you can think of the Brownian motion as a driver in a taxi driver so if the taxi  driver has one beer maybe can still drive so and reasonably well but after a few more  drinks becomes erratic and the consequences are easy to see.  So those are two phase transitions in the behavior of SLE as depending on Kappa this  was proved with Stefan Rode.  If Kappa is at most four it is a simple path if it is between four and eight then it is  not a simple path anymore it kind of swallows parts of the domain as it goes along that  is roughly what the SLE six pictures look like if you remember we will see them shortly  again and when Kappa is eight or larger the curve is actually space filling it visits  every point.  Now the SLE path it cannot cross itself but it can touch itself therefore in this case  once you circle some part of the domain you can't go in there and fill that up that is  one difference from Brownian motion.  Brownian motion can make a loop and then go in the loop and out again that is why Brownian  motion is more difficult to study from this point of view than SLE.  Now notice the parameter eight is precisely what the uniform spanning tree piano path  converges to and it is easy to see from the definition that it has to converge to a space  filling curve and that is exactly the transition where the Kappa where the transition to space  filling occurs.  There is some information about the dimension of the SLE in this paper with Stefan we showed  that the expected box count dimension well expected box count dimension I am not going  to define this concept because you can probably guess what it means by the from its name the  expected box count dimension of the path is one plus Kappa over eight when Kappa is between  zero and eight when Kappa is larger than eight it is of course two because it is space filling  and in the range where Kappa is larger than four the outer boundary of the path is not  the same as the path itself because it is not a simple path and the expected box count  dimension of the outer boundary is given by this formula which was earlier conjectured  by Rick Kenyon and Vincent Befar in a quite clever and technical work managed to show  that this expected box count dimension of the path is in fact the actual house dwarf dimension.  The expected box count dimension is much simpler to determine because you just need to know  you need to calculate expectations and for house dwarf dimensions essentially you need  to calculate second moments and that is more difficult.  I think that the corresponding statement for this is still open but probably won't be  for very long.  I want to tell you about yet another model which whose relation to SLE has been determined  this is the Gaussian free field.  What you should think let's discuss the discrete Gaussian free field there is also a continuous  version which is of the same flavor only harder to define.  You should think first of a graph and then at each vertex you have some value H of V  and you want to penalize discrepancies.  You can think of it as a model for a surface which has penalties for discrepancy from being  smoothness.  You penalize by H of V minus H of U squared for vertices that are close by so the total  penalty is the exponential of minus the sum over all neighbors of this square difference  and this is a nice formula because the distribution of H under this probability density is going  to be a Gaussian.  The Kenyon showed that this Gaussian free field is a scaling limit of the domino tiling  height function which is another discrete model which I'm not going to describe to you  but this was partly a motivation to study the Gaussian free field.  Here's what you do with the Gaussian free field if you want to do you put the Brouche  and boundary values again you put boundary values plus A here and minus A here and then  you can look at the set of edges.  You can color the vertices one color if they're positive and another color if they're negative  and then you can take the interface and this is a graphical depiction of this interface.  Scott Schaaffield and I proved that this interface of the discrete Gaussian free field converges  to SLE4.  Now 4 is the special value where the interface above 4, the SLE above 4 stops being a simple  curve and you can see that this is an actual simulation.  You can see that this curve is almost not a simple curve anymore and the shades of blue  and red they depict the heights of the field.  Now in order for this statement to hold true you need to pick this value of A, a very specific  value.  If you take a different value then you get convergence to something very closely related  which we also know how to describe.  And there's an analogous statement about the continuum Gaussian free field but that's  harder to formulate.  There's a conjecture and Stas Miernoff reported on much progress on this conjecture that the  critical easing model is converging to SLE3.  This is this picture and Stas also shows that with other parameters things associated  with the easing model also converge to SLE but with a different value of kappa and this  figure is drawn by David Wilson.  And so here the white is plus spins and the dark is minus spins.  If I assume you're familiar with the easing model if not then think of this as just another  pretty picture, hopefully pretty.  Yes.  And if you take the uniform measure on lattice paths that are self-avoiding and stay in the  upper half plane the belief is that that converges to SLE8.  I can't say I can't report any progress on this conjecture but it's supported experimentally  by simulations by Tom Kennedy and this figure is being drawn by Tom Kennedy.  Contrary to all the other figures in this talk this figure is one which you really need  to work hard to produce.  All the others have very simple algorithms.  Well, time is running short and I'm going to give a quick summary.  Here a list of things we know SLE describes the scaling limit of.  Critical site percolation on the triangular grid and the value of kappa is 6.  Lupo-Aisland and walk and the value is 2.  The piano curve the value of 8.  Gaussian free field and the value is kappa.  Like explore is a simpler model that is kind of built, defined to be easy for this statement.  And I'm not going to discuss it.  And here are some conjectures regarding other models converging to SLE.  Well there are other models of critical percolation which should converge to the boundaries should  converge to SLE 6 but that's not been proven.  The easing model with different parameters converges to 3 and 6 and we hope to see a  proof by Stas pretty soon.  And the FK cluster boundaries well I haven't defined them but I mentioned them here because  they actually there's another parameter they're Q and therefore you have a whole range of  kappa between 4 and 8 and similarly with the ON models.  The self avoiding walk is of course very challenging.  The double domino has relations to the Gaussian free field we've mentioned this already.  And there are some things that SLE does not give.  It does not give DLA.  There's been a Lerven type analysis of DLA by Carlson and Makarov and by Hastings-Levitov  that actually proceeds this SLE definition.  It appears that the paths of the minimal spanning tree are not described by SLE and are probably  not conformally invariant.  This is based on simulations by Willand and Wilson and SLE doesn't apply to many interesting  questions in dimension higher than 2.  Let me mention that percolation in high dimensions is almost completely understood due to the  work of Haar and Slade and others with a method called the lace expansion which is very different  than what I described today.  And the existence of the looper-ace random walk scaling limit in R3 has been proven recently  by Gadi-Cosma but Gadi hasn't determined what the limit is just shown that it exists  but that's quite a remarkable achievement.  Well, so we have seen things that you can do with SLE, things that hopefully will be  done and things that you need to use other methods.  Thank you very much for attending my talk.
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So it's my privilege to introduce this final plenary speaker who is older, Sram from Microsoft in the United States. So Sram has a background in complex analysis. He was a student of Thurston and he worked early on on packing problems and the proof of the Riemann mapping theorem that way. And then he was interested in graphs and in probability aspects of that. But the real jackpot he hit by having the idea that one should study the Lerner equation not with only smooth coefficients as was Lerner's original idea but also in the general case, in more general cases. And that the choice of the Brownian motion as the leading function in the differential equation was going to produce interesting results. And he was indeed completely right. It was later supplemented by his work together with Lawler and Werner who was one of the recipients of the Fields Medal this year. And as you know, they were able to prove, for example, the Mandelbeut conjecture about the dimension of the outer boundary of the Brownian motion, that the outer dimension is four-third. And if you had mentioned to me ten years ago that somebody was going to prove this conjecture, I would have said that you were out of your mind that there is no chance that this can ever be proved in a strictly mathematical sense. So I think we have witnessed here one of the most interesting development and it would be a great opportunity for you, if you don't know it already, to learn about this in the lecture that Odechram is going to deliver. I cannot help making a personal comment about the method. So it's all based on the deformation of a domain in the complex plane according to the learner scheme. And in this way it is a flow where time plays an essential role. And it's really remarkable to think about that. The proof of the, it started with the Biebebach conjecture which is based precisely on a deformation also of a domain into what was supposed to be the extrema case. And in this case you could get monotonicity. And of course one can think about Perlman's proof of the Frank-Array conjecture as based on completely the same idea that you have a difference equation according to which you deform a domain and then you, there the great difficulties are to understand the singularities. So I think this method of deformation of domains in the complex is one of the greatest achievements of our time in analysis. And I think there are prospects for very interesting developments in the future also. Personally I've been involved in trying to do what is called the DLA. And there is a healer show flow which one doesn't really understand which is similar to that. So I think you will see even a continuation of this method. But right now we look forward to order some lecture on conformal invariance scaling limits. Thank you. Thank you very much, Lena, to a great honor of being here. Okay, well most speakers had some slides showing a plan of the talk and this is my plan. We're going to look at random systems such as the one you see on the slide. This one has a very simple description. All the hexagons are independently white or black with probability one-half. And we're going to study such systems. And the way we're going to learn information about that is by studying random curves that appear in these systems such as the red curve here. Percolation is going to be the primary example but not the only one. So this is called percolation. There are two closely related talks, actually very closely related talks. I'd like to mention the talk by Stanislav Smirnov yesterday and Wendelin Werner's talk later today. And some things which I will omit for lack of time, you can probably learn from Wendelin Werner's talk. Let's start with something rather simple. Let's discuss one-dimensional Brownian motion. One way to think of one-dimensional Brownian motion is a limit of simple random walk. Here you see a simple random walk where the x-axis is the time axis and the y-axis is the spatial coordinate and at specific time units on an intergeal lattice, the walk with probability one-half goes up or down regardless of what has happened in the past. When you get such a graph, the shape, these rectangular steps are not drawn with the height difference equal to the time difference because we're going to take a limit as the time and space scales and get a Brownian motion. And you have to scale time and space differently if you scale space by delta, you have to scale time by delta squared. So this is some definition and a refresher on Brownian motion. Here are some properties Brownian motion has. It is a continuous path with probability one. It satisfies the Markov property which means that if you want to know, if you're a time t and you want to predict something about the future, not with certainty but with some reliability, well, the only information that's relevant for you that's in the past is your current position. And Brownian motion satisfies Brownian scaling. Delta times the Brownian motion is the same in law, has the same distribution as the Brownian motion with time scaled by delta squared. Let's go on to two dimensions. The rest of the talk will be entirely two-dimensional. When you take two-dimensional Brownian motion, you have two natural ways to define it. Since we define Brownian motion as the limit of simple random walk, you can also define two-dimensional Brownian motion as a limit of simple random walk on the square grid. Alternatively, you can take two independent Brownian motion, one in the x-coordinate, one in the y-coordinate, and therefore get a two-dimensional curve. And these two definitions are equivalent. Both definitions suggest that Brownian motion will actually, well, both definitions use the coordinates, the x and y coordinates, but a nice property of two-dimensional Brownian motion and higher-dimensional Brownian motion is that Brownian motion, in fact, doesn't depend on the coordinate system, only depends on the inner product, namely, its rotationally invariant. In two dimensions, it has a higher form of symmetry. It's conformally invariant. That means the following. You have here a domain, and you have a base point, and you start Brownian motion from the base point, you stop when you hit the boundary, and you consider an analytic map or a conformal map to another domain. You start in the other domain a Brownian motion from the image of the base point. Well, let me put it this way. The image of this Brownian motion in this domain is the same as a Brownian motion starting at the image of the base point, except that the time parameterization changes. You should keep this in mind because later on we will discuss various conformal invariants of different models. I will be rather loose, and I will not say what conformal invariance means, but you should think of something in this flavor. Maybe I should say a simple effect is just that the hitting point on the boundary is conformally invariant, but this conformal invariance means much more. That means the whole distribution of the curve is conformally invariant. Let's move on to a more difficult topic, which is percolation. Here you have a mesh of hexagons. This is the most successful two-dimensional percolation model right now. If you're doing a Bernoulli-P percolation, each hexagon is white or open with probability P independently. Then, well, so far it's just a system of independent bits, but once you start studying connected components of white or red or black regions, then you're doing percolation. There are various alternative models. There's a large variety of alternative models of percolation. One alternative model is you work on a grid and you flip a coin for each edge to decide if you keep it or throw it away. Now that's percolation. Now we will discuss critical percolation. It turns out, and that's pretty easy to see, though I will not prove it to you, that there is some number between 0 and 1. See, we have this P as our parameter, which we can decide. P is the probability for each hexagon to be white. We can play with this parameter. There is some specific value that is crucial, and it's called P sub C, the critical value of P, and it can be defined in various different ways, has been defined in various different ways, but the simplest definition mathematically is by saying that if P is larger than that value, you will have an infinite white component. If P is smaller than this value, then the probability for the existence of an infinite white component is going to be 0. This is a good model for a phase transition because the large-scale behavior of the system changes drastically when P pushes beyond PC. What do I mean by that? If P is slightly smaller than PC and you look at a very large picture, then almost nothing is connected. The white regions are small islands or very small lakes inside a vast area of black, and the situation reverses when P is above PC. This is very easy. It's harder to determine what PC is, and for some percolation models, we will probably never know what PC is. For this particular model that I've described here, it has been known. Well, Harris proved one half of the theorem. He showed that at P equals one half, there is no infinite white component. There is no infinite cluster. This means that PC is at least one half. Twenty years later, Keston showed that PC actually is one half because above PC, above one half, you do have infinite cluster. The one half here is a value which is easy to guess. This is because at one half, you have a symmetry of white and black. They behave the same. If you believe my story about one completely dominating the other when you're off of PC, then you should believe that one half has to be the critical value. Putting it was a significant achievement, and based on an earlier important result of Uso Simo in Welsh. Well, Harris's theorem tells us that at one half, we don't have an infinite cluster, and that is the PC. What happens if you condition on this rare event that the cluster of the origin is going to be large? If you sample percolation, you know that with probability one, the cluster of the origin is going to be finite. Typically, you'd expect it to have maybe size five, six, zero hexagons, something like that. But you can run your simulations a few times, and you can wait until you get a simulation which has many hexagons in the cluster of the origin, and then this is roughly what the cluster looks like if you wait till you get a sample with a thousand hexagons or more. You can go on and you can ask yourself, well, how long will I have to wait till I see a cluster of a thousand hexagons? How unlikely does the probability of having a large cluster, how fast does it decay as the notion of large increases? Such questions are governed by what's known as critical exponents, and physicists such as Denise Nienhoes-Cardy, they've predicted the values of these exponents. In particular, the probability that the origin is in a cluster of diameter, at least r, behaves like r to the minus five over 48 with some low-order corrections as r tends to infinity. If you ask a similar question but require that the connection be inside the half plane, then the exponent you get is minus a third instead of minus five over 48. These were conjectures and the arguments typically assume asymptotic conformal invariance and many other unproven properties. The way the physicists like to assume conformal invariance, they look at, say, the probability that two hexagons are in the same cluster and they believe that there's going to be a limit to that as the measure finds after you rescale appropriately and then they say, well, this limit should have nice conformal invariance properties and make similar assumptions. Then once they have a few assumptions in place, they can work some machinery to calculate the values. Conformal invariance plays an important role. Now, there are also predictions which are more refined than just saying an exponent. See, an exponent just gives you one number which is very important because it tells you what these rare events, how unlikely they are and so forth. As useful as it is, there are more precise things to know about. For example, Cardi had the following prediction. If you look at the rectangle and you take the mesh very fine and you ask, what is the probability to have a connection in white hexagons from the left to the right? In the limit as the mesh refines, this quantity approaches this expression right here. This is the gamma function. F2,1 is the hypergeometric function and eta is the cross ratio of these four points after you map the rectangle conformally to the upper half plane. This is a reasonably complicated formula. Leonard Carlson found a nicer form for this formula. As far as conjectures go, you want to assume it naturally because otherwise you can't get even started. One's formula is the following, you take a triangle and you want to connect the bottom edge to a segment on the right-hand edge. If the edge length of this triangle is 1 and the length of this segment is x, then the probability to have this connection converges to x as the mesh goes to 0. This is somewhat simpler than Cardi's form. Just so that we don't get carried away with percolation, I want to mention a few other models because percolation, though it's a central theme, it's not the only topic I want to discuss. Here I can use this picture to describe three different models. What is this picture? You have a white grid here and on the white grid, you should think of the white grid as a graph. Well, it's all a graph, but you should think of an n by n grid, square grid, and the white is a spanning tree in that grid. If you notice, the white maze here has no cycles and it spans every vertex in the appropriate grid. If you have that grid, n by n grid, you can think about choosing a spanning tree from that grid and you can say, well, I want to be unbiased, I'm going to choose the spanning tree uniformly among all spanning trees in that grid. Well, at first I was kind of skeptical about this model. Why would the uniform measure be a natural thing to study? Well, it turns out this uniform spanning tree has many nice connections to different types of questions such as electricity and random walks and so forth. So it is very natural, in fact, to take the uniform measure. So our white is the uniform spanning tree. The pink is also a uniform spanning tree of the dual of the original graph. The pink is just essentially the complement of the white. And the black curve between them is what's called the piano path associated with the uniform spanning tree and it's also known as the Hamiltonian path on the Manhattan lattice. Now those, so here we have a two-dimensional system. We have one curve associated with it and in fact there's another nice curve associated to that system. If you fix two vertices, say white vertices and you fix them before you see the tree, then you sample the tree and there's just one path in the maze connecting the two vertices because it's a tree. That path is what's known as the loop-race random walk. The loop-race random walk is a model invented by Greg Lawler and it is very intimately connected to the uniform spanning tree. Well so there was a belief that the uniform spanning tree in the loop-race random walk ought to be conformally invariant. And there has been some partial progress in that direction. Rick Kenyon proved that some properties of the uniform spanning tree in loop-race random walk are conformally invariant in the scaling limit. And here's an example for a property which he proved. If you take here a domain and you fix three points in the boundary and you take the uniform spanning tree of the grid restricted to this domain, then near these three points you can take vertices in your grid and join them up by the paths that join them in the tree and you will have a kind of a Y-shaped topological Y-tree spanning these three points and there will be a special meeting point, this red meeting point. And if you can do this thing in another domain and you can, I'm assuming the domains are simply connected and you can map one domain conformally to the other domain and what Rick Kenyon has proved is that the distribution of this red meeting point is conformally invariant, namely the image of the measure here of the law of the red point is going to be the law of the red point here provided the three base points are mapped to the three base points. That is a good bit of progress. It doesn't quite prove conformal invariance of this whole system, for example. You don't yet know that these paths are actually have the same distribution in both pictures. Okay, so to study percolation it turns out that we want to look at an interface. Let me, I have a demo here showing this interface. So you start out by declaring that you want the boundary hexagons to be, to change color across your origin here. This is what's called as the devotion boundary conditions. And then you can start an interface between the blue and the yellow. And you want to know how this interface continues so you look at what this hexagon is going to do. And this hexagon, you flip the coin, you see it's yellow so you turn right and then you look at the next hexagon, it turned out blue, you turn left and the next hexagon and the next hexagon. Now sometimes you flip one hexagon and at this point we flipped one hexagon and we could, that's made three steps of the interface because the interface when it hits hexagons that has, that have been colored before, it doesn't need to sample a new one. And you continue and so on and so on, so on. And you continue like that and you build this interface which has no choice but to go out to infinity. You should think of it as an interface connecting the two boundary switching points which is zero and infinity. So now we're going to take, well so far I haven't done really any math, I just told you about math. Now we're going to, for four slides roughly, we're going to actually do some math here. And I'm going to try to walk you through the intuition for why you want to consider Levener's equation with Brownian motion as the driving term, though you don't need to know any of these words. We're going to see, and this is roughly the way this actually, the original motivation, though the original motivation was in the context of Lupe or Israel and more, but the picture is similar. So we take our curve and suppose that we only look at part of it and we stop when it reaches size epsilon. And I'm not going to be very clear at this point at what size means, we'll see later what size means. And then we believe in conformal invariance for now, the task of justifying conformal invariance we'll discuss later. We believe conformal invariance, we have sampled the part of the walk, of this interface and we'd like to know what the rest looks like. Well, the trouble is that this piece here is very complicated, it's kind of fractal shaped and we want to simplify it, but we believe in conformal invariance and see the boundary of this domain is, after you remove the path is colored by two colors, the white and the black here and they change in exactly one point which is the tip of the curve. And therefore, we can simplify the domain, we can map over by the conformal map, we use Riemann's mapping theorem telling us that the complement of this region that we've examined already is simply connected, therefore we can map it over to the upper half plane again. And when we map it over to the upper half plane, there will be a special point where the image of the black curve meets the image of the white and the red here is indicated to show what part of the boundary corresponds to things that are adjacent to the curve. And this map that we're going to use has some power series expansion near infinity and we normalize it so it fixes infinity and the power series expansion looks like that. This is all very easy consequence of the Riemann mapping theorem. And if we think about the continuation of the path here, that's going to look in distribution about the same as what the original path would have looked if we started it at the tip at the boundary point right here because we believe conformal invariance. And now we can continue the procedure, right? We can, you know, after we can start here, we can build the interface until it has size epsilon and then map over by another map, F2. And then F2 has essentially the same distribution as F1 except that it's translated to start at the point here where black changes to white is not necessarily the same point as black change to white here. And so F2 has given that we know F1, F2 has the distribution of F1 except translated therefore conjugated by this translation. And this here denotes equivalence in law. And we can continue inductively. If WJ denotes the place where the image of the jth map takes, the jth composition takes the tip to some place, then we have this composition GN which is the composition of all these Fs and each Fj has essentially the same law of F1 except conjugated by translation depending on this WJ. And this GN is just the map from the complement of the curve we've examined so far to the upper half plane because it's, you know, each time we've added a piece and we've taken the map that makes it disappear. And now each Fj is going to be very close to the identity because our epsilon is going to be very small. And therefore we can attempt to think of this composition as a flow rather than a composition of discrete steps because if you have a map that's close to the identity you can think of it as approximately a vector field. And to understand this vector field we're going to essentially find out what it is. We can look again at this map F1 which has this power series representation near infinity. And now to simplify things we want to choose cleverly how we measure the size of the initial segment. It turns out that A1 is monotone in the path as you extend the path A1 increases and therefore to make life easier we choose A1 to be essentially to epsilon. Therefore we stop the curve once A1 reaches size 2 epsilon. And then easy calculation with power series and the scaling argument implies that F1 is well z plus 2 epsilon z inverse plus the other terms which are bounded by epsilon to the three halves. So we're not going to do this calculation for you now. And then the consequence is that Fj plus 1 is going to be the same thing except conjugated by the translation to Wj and therefore has this expression. Fj plus 1 is the same thing except you have minus Wj. And therefore so this is essentially a vector field where this is the vector at the point gt of z. And therefore instead of thinking of the composition of these new identity maps we can rather flow according to this vector field where this Wt is some continuous version of the Wj's. And well what is this continuous version of the Wj's? Well the Wj's have stationary independent increments. As you go from Wj to Wj plus 1 the distribution of that translation is going to be the same as from W1 to W2 from W2 to W3. So all the same because we've remarked that the law of these maps are all the same except for the translation. Once you take away the translation the law is the same and therefore the Wj's are have the same, it's like a simple random walk except the steps not necessarily have a fixed length but have a random length but they have the same random length each time. And so this Wt which is a limit of these Wj's is going to be a process that has stationary independent increments. It's symmetric because our problem is symmetric and it's continuous that you have to argue by some other means but that's not too hard. And it follows easily once you know a little bit about Brownian motion that it's a multiple of Brownian motion. And this leads us to the definition of what SLE is. SLE is stochastic levonner evolution. Well we fix the parameter kappa and we define Wt as Brownian motion at time kappa t and this Brownian motion is one dimensional. And then you solve this differential equation starting at g0 of z equals z. This is essentially what is supposed to be the conformal map from the complement of the interface to the upper half plane at time t. And this equation is known as Levener's equation. He's used it to study the Biberbach conjecture. And well he studied a different, mostly used a different, slightly different form of the equation so this is why it's called Cordo. And once you solve this equation you can recover the path itself by taking gt inverse of Wt. So what is the basic idea here? The idea here is that we want to invoke differential equations to study our percolation. Now the percolation interface is a fractal object and it's hard to study fractals with differential equations. Differential equations are better suitable to smooth objects. Well we have a smooth object at our disposal, the complement of the interface. That's a domain and therefore a smooth object. So instead of studying the evolution of the interface we study the evolution of the complement and that's amenable to differential equations and therefore we can not only find out what we guess it to be but also then we can do calculations to calculate properties of the complement and therefore of the interface itself. Well this whole discussion was conditional on having conformal invariance but well we didn't have conformal invariance so what do we do? Well it seems just to be a nice physics like prediction, not so satisfying for a mathematician. At this point Vendeline Verne came up with a brilliant idea that well we have a fractal at our disposal which is known to be conformal invariant but still has some nice questions that haven't been answered yet so why don't we apply this method of understanding conformal invariant fractal to Brownian motion. If you look at Brownian motion itself in some ways it's very simple. However if you look at the outer boundary of Brownian motion that's not very amenable to analysis directly from the definition of Brownian motion. So what you see in this picture is Brownian motion started at the origin and stopped when it hits the boundary of the unit disc and the red is all the places visited by the motion and the black is the outer boundary of the Brownian motion. The set of points so that you can get to from the exterior without hitting any other points before that. There was a conjecture of Benoit Mandelbrot that the outer boundary of Brownian motion has dimension four thirds and well Vendeline was right. SLE was the right tool to attack this problem and several related problems about the Brownian motion. I think Vendeline will tell you more about this later this afternoon but let me just give one of the punch lines. Greg Lawler Vendeline and I proved that indeed the Mandelbrot conjecture is correct. The outer boundary has dimension four thirds and you can extract other information about the curve. For example the set of cut points has house of dimension three fourths. The cut points are those points where if you remove them you disconnect the Brownian motion. And this is based on first, well this is not how it was written but a posteriori we now know that in fact the outer boundary of this Brownian motion is essentially the same as the outer boundary of SLE six and that's what permits the connection between why studying SLE can tell us something about the Brownian motion. At first we didn't know that they're actually the same but we knew that the dimensions should be the same. Well okay so that was one use but we didn't have to wait very long and Stas Mianoff actually proved Cardi's formula in Carlson's form and in fact the way the simple form of thinking about triangles and the hexagonal lattice turned out to be an important observation leading to the proof. The proof is quite brilliant but and amazingly simple and short. So Stas proved that in fact Cardi's formula is correct and from this it's not hard to conclude that the percolation interface scaling limit the same curve that I showed on the opening slide is SLE six. So six here is the parameter cap that you choose remember SLE has a free parameter. And this theorem allowed the determination of many of the exponents associated with percolation in particular we showed that the exponent that I discussed in the beginning the probability to connect to distance r indeed decays like r to the minus five over 48 with some lower order corrections and many other exponents and properties were proved by Smeonoff and Verna based on earlier work by Keston relating all these exponents to one another that SLE point of view only helps to determine things at PC at the critical probability and there are some exponents relating to off critical but Keston linked these with the critical exponents. Subsequently the looper is random walk was proved to converge to SLE two in the uniform spanning tree. Piano path scaling limit was proved to be SLE eight. This was with Loller and Verna and here the situation is kind of the opposite in the sense that first convergence to SLE was proved and the consequence of that was conformal invariance rather than the other way around and here you see a picture of what the looper is random walk looks like. So you see that with different values of the parameter cap are give entirely different processes and this may be a little bit surprising at first that you can change things like the house of dimension and properties just by changing the speed of the Brownian motion. It is like you can think of the Brownian motion as a driver in a taxi driver so if the taxi driver has one beer maybe can still drive so and reasonably well but after a few more drinks becomes erratic and the consequences are easy to see. So those are two phase transitions in the behavior of SLE as depending on Kappa this was proved with Stefan Rode. If Kappa is at most four it is a simple path if it is between four and eight then it is not a simple path anymore it kind of swallows parts of the domain as it goes along that is roughly what the SLE six pictures look like if you remember we will see them shortly again and when Kappa is eight or larger the curve is actually space filling it visits every point. Now the SLE path it cannot cross itself but it can touch itself therefore in this case once you circle some part of the domain you can't go in there and fill that up that is one difference from Brownian motion. Brownian motion can make a loop and then go in the loop and out again that is why Brownian motion is more difficult to study from this point of view than SLE. Now notice the parameter eight is precisely what the uniform spanning tree piano path converges to and it is easy to see from the definition that it has to converge to a space filling curve and that is exactly the transition where the Kappa where the transition to space filling occurs. There is some information about the dimension of the SLE in this paper with Stefan we showed that the expected box count dimension well expected box count dimension I am not going to define this concept because you can probably guess what it means by the from its name the expected box count dimension of the path is one plus Kappa over eight when Kappa is between zero and eight when Kappa is larger than eight it is of course two because it is space filling and in the range where Kappa is larger than four the outer boundary of the path is not the same as the path itself because it is not a simple path and the expected box count dimension of the outer boundary is given by this formula which was earlier conjectured by Rick Kenyon and Vincent Befar in a quite clever and technical work managed to show that this expected box count dimension of the path is in fact the actual house dwarf dimension. The expected box count dimension is much simpler to determine because you just need to know you need to calculate expectations and for house dwarf dimensions essentially you need to calculate second moments and that is more difficult. I think that the corresponding statement for this is still open but probably won't be for very long. I want to tell you about yet another model which whose relation to SLE has been determined this is the Gaussian free field. What you should think let's discuss the discrete Gaussian free field there is also a continuous version which is of the same flavor only harder to define. You should think first of a graph and then at each vertex you have some value H of V and you want to penalize discrepancies. You can think of it as a model for a surface which has penalties for discrepancy from being smoothness. You penalize by H of V minus H of U squared for vertices that are close by so the total penalty is the exponential of minus the sum over all neighbors of this square difference and this is a nice formula because the distribution of H under this probability density is going to be a Gaussian. The Kenyon showed that this Gaussian free field is a scaling limit of the domino tiling height function which is another discrete model which I'm not going to describe to you but this was partly a motivation to study the Gaussian free field. Here's what you do with the Gaussian free field if you want to do you put the Brouche and boundary values again you put boundary values plus A here and minus A here and then you can look at the set of edges. You can color the vertices one color if they're positive and another color if they're negative and then you can take the interface and this is a graphical depiction of this interface. Scott Schaaffield and I proved that this interface of the discrete Gaussian free field converges to SLE4. Now 4 is the special value where the interface above 4, the SLE above 4 stops being a simple curve and you can see that this is an actual simulation. You can see that this curve is almost not a simple curve anymore and the shades of blue and red they depict the heights of the field. Now in order for this statement to hold true you need to pick this value of A, a very specific value. If you take a different value then you get convergence to something very closely related which we also know how to describe. And there's an analogous statement about the continuum Gaussian free field but that's harder to formulate. There's a conjecture and Stas Miernoff reported on much progress on this conjecture that the critical easing model is converging to SLE3. This is this picture and Stas also shows that with other parameters things associated with the easing model also converge to SLE but with a different value of kappa and this figure is drawn by David Wilson. And so here the white is plus spins and the dark is minus spins. If I assume you're familiar with the easing model if not then think of this as just another pretty picture, hopefully pretty. Yes. And if you take the uniform measure on lattice paths that are self-avoiding and stay in the upper half plane the belief is that that converges to SLE8. I can't say I can't report any progress on this conjecture but it's supported experimentally by simulations by Tom Kennedy and this figure is being drawn by Tom Kennedy. Contrary to all the other figures in this talk this figure is one which you really need to work hard to produce. All the others have very simple algorithms. Well, time is running short and I'm going to give a quick summary. Here a list of things we know SLE describes the scaling limit of. Critical site percolation on the triangular grid and the value of kappa is 6. Lupo-Aisland and walk and the value is 2. The piano curve the value of 8. Gaussian free field and the value is kappa. Like explore is a simpler model that is kind of built, defined to be easy for this statement. And I'm not going to discuss it. And here are some conjectures regarding other models converging to SLE. Well there are other models of critical percolation which should converge to the boundaries should converge to SLE 6 but that's not been proven. The easing model with different parameters converges to 3 and 6 and we hope to see a proof by Stas pretty soon. And the FK cluster boundaries well I haven't defined them but I mentioned them here because they actually there's another parameter they're Q and therefore you have a whole range of kappa between 4 and 8 and similarly with the ON models. The self avoiding walk is of course very challenging. The double domino has relations to the Gaussian free field we've mentioned this already. And there are some things that SLE does not give. It does not give DLA. There's been a Lerven type analysis of DLA by Carlson and Makarov and by Hastings-Levitov that actually proceeds this SLE definition. It appears that the paths of the minimal spanning tree are not described by SLE and are probably not conformally invariant. This is based on simulations by Willand and Wilson and SLE doesn't apply to many interesting questions in dimension higher than 2. Let me mention that percolation in high dimensions is almost completely understood due to the work of Haar and Slade and others with a method called the lace expansion which is very different than what I described today. And the existence of the looper-ace random walk scaling limit in R3 has been proven recently by Gadi-Cosma but Gadi hasn't determined what the limit is just shown that it exists but that's quite a remarkable achievement. Well, so we have seen things that you can do with SLE, things that hopefully will be done and things that you need to use other methods. Thank you very much for attending my talk.
probability aspects of that. But the real jackpot random curves that appear in these systems such is some definition and a refresher on Brownian go on to two dimensions. The rest of the talk have a mesh of hexagons. This is the most percolation, each hexagon is white or open with models. There's a large variety of alternative models of percolation. One alternative model is been defined in various different ways, but the and for some percolation models, we will probably what's known as critical exponents, and physicists like to assume conformal invariance, origin here. This is what's called as the which is a limit of these Wj's is going to be a the definition of what SLE is. SLE is stochastic questions that haven't been answered yet so why problems about the Brownian motion. I think in the behavior of SLE as depending on Kappa this and are probably not conformally invariant. This