 which I'd like to tell you. So the problem, we looked at percolation and some interpretation of what it means to have renormalization for it. The other thing which I like to do is to talk about the conformal field theory of it, which is slightly more involved. So imagine that I have on the complex plane some percolating cluster. Then I choose a domain on the plane and I ask a simple question. What is the probability that this cluster enters in an arc and lives in another arc? So if I name these points A, B, C, D, I can take this out of the percolating cluster. So I have a domain like that, an arc here and another arc here. What is the probability that this happens? This is called a crossing probability. In fact, for any random process you can define it, you are interested in the case of percolation. It is just because we are now focused on percolation, I like to ask what is the probability that this happens? In fact, one of the exercises I have given you is a similar problem in the exercise sheet that you do by simulation. So the way I organize this is control the boundary conditions. So if I let's suppose that percolation cluster is black. So if I make this arc all white and I make this arc all white and I make this black and I make this black. In fact, now the percolating cluster doesn't have a choice other than to start here and leave from there. In the exercise I simplified it to start at a point and leave at a point. But in general, you can make it a little wider that is it can start on any point here and leave through any point there. I can, in fact, because of the Riemann mapping theorem, change that into any shape I like. So I should be able to map this to any shape I like, any old shape with just four points. It shouldn't make a difference. What I therefore need is to have a method of controlling the boundary condition to get the answer. It's only the boundary which is fixing the answer. The shape is not. The geometrical shape, its volume is not. It's just the four points because you can, in fact, by a conformal mapping shape, map it into anything you like. So at this point I need to insert something which I claim that it is some operator phi of z i. And what phi of z a does is that at a, at z a, it changes the boundary condition. In fact, I have just two boundary conditions. It can either be white or black. So if I start with this segment being black, then acting with this operator phi of z will change it into white. Then again, I act with phi of z d. And it changes the boundary condition here, again at c, and finally at b. So it's essentially these four operators which when you put in there, if you can construct and recognize these four operators, it will, it will create a setting for this entity. But I need a number because it's a probability. So I claim that this expectation value is this crossing probability. And these have to be operators from some field theory. And as you can guess, these are operators from conformal field theory with charge equal to 0. Charge equal to 0 because we know from reasons I haven't told you that this is the percolation theory. So the crossing probability for a percolating cluster to come in and leave like this is given by this four point function. And we have a calculation for four point functions. So I can simply insert it from there. So this probability p will be, this is the cross-ratio of these particular four points. And this is the hypergeometric function. If you remember, we went through a symmetry calculation showing that this cannot have any other shape than this shape. I have, of course, forgotten some terms here. Yes. Particle, no. I have a percolating cluster. Percolating cluster is in the percolation problem. You turn the squares on and off. And it forms a cluster. No, in percolation problem. Percolation problem, you turn them on and off. And you look if a cluster goes from bottom to top. Even that happens, you are at the critical point, the percolating cluster. And then you say, OK, I take this circle and I throw it on the complex plane. What is the probability that the percolating cluster comes in and goes out of this circle? You make it a little bit harder. What is the probability that it comes in this arc and goes out through that arc? Sorry, I didn't hear that. This I cannot do. I have to go through a lot of conformal field theory to tell you exactly why this happens, that this is true. Just let's accept that it's given by this four-point function. We make some observations. This probability has to be conformally invariant, that it is conformally invariant is the case. Because if you make a conformal transformation, you can take this domain to any shape you like. It should not change. So it has to be a conformally invariant quantity. It must depend on these four points. So it does. So this is a quantity which is conformally invariant and just function of the points and not their geometrical relation to each other. Yes, this is the hypergeometric function, F21, which is a solution of hypergeometric equation. And eta is this cross-ratio. If you remember when I did this calculation for four-point function of any conformal field theory, I got that it has some parallel dependence and the function dependence. The function dependence at that time I did not specify. I just left this as a general undetermined function, F of eta. But now for this specific problem, because it is the standard classical percolation, I can specify it because I now know that it has to correspond to central charge equal to 0. So there are pieces of information which I cannot give one is, why is percolation related to c equal to 0? And how do you exactly prove this? However, we can do a neat trick. Take this and take it into a triangle, triangle of side one. So where are my four points? A, B, C, and D. So the problem now becomes percolating cluster coming up and leaving here. So these points are again where the boundary condition changing operator is acting. So these phi's are now acting on the four points A, B, C, and D on one of the sides. And I choose the sides at exactly distance 1. So it's an equilateral triangle with side 1. But this does is that it simplifies this relationship a lot because everywhere you put 1 and these things all disappear. So now it's obvious that the probability of exit is just this distance. So p has to be called x. Rotationally the same, A, B, C, D. Now, why is the probability exactly equal to x? Because if a percolating cluster comes in here and this side is black and this side is white, it has no choice other than to go out that side. But on this side, I've made this part equal to white. So it only has a choice of going out through this window. Hence the length of this window will determine the probability of it. If I make the window as big as the side D, it will certainly go out somewhere. So the probability becomes 1. It's a statement about percolation which has been tested numerically many times. And it is strange that it's true, but it is true. There is a comfort. Yes, will you get that by conform? No, there is no problem with sharp points on the boundary. It's, in fact, there is a general theorem which says that you can do that. And you actually, in complex analysis, can take a boundary with a smooth arcs to one with points, a polygon. You cannot break the boundary if you are deaf, that's what I mean. You cannot break the boundary, but you can make it sharp like this, make good points on it. Okay, the problem is that we might exit from the same side we have, and with fixing the boundary condition. You see, I can, it's true that I draw these shapes, but essentially the percolation comes in through the whole side, and the option of going out the same side is not available. It has to go out through this window which I give it. So you can also imagine it here like that. In fact, a percolating cluster coming in here, going out there. The boundary conditions, the probability, I see it is just one, right? I mean, if you fix the boundary condition, that can go out in this way. No, the idea is that you have a percolating cluster, you throw this triangle on it. So it is possible that you have this situation. You have this situation, don't you? So the probability of doing it hitting this side plus the probability of, sorry, I understand what you mean. I should not have said white here. Yeah. Okay, it's not obvious that it has to be X, but it is. There is the transformation which takes this into that, has the effect that takes this hypergeometric function into just X. To go into the conformal field here of percolation, you have to admit a lot of difficulties, but we saw that just like Ising, percolation has a renormalization group and it has a CFD representation. Next. My play by what? You should apply another parenthesis, without being, and in the denominator, the parenthesis, this is not the frustration, I mean. What is missing? Z of D, Z of D should be Z of A. Okay, the cross-ratio has some rules that, okay, it's not correct. You must assume that the probability does not change under conformal. The probability does not change under conformal. And it is an assumption, or is proven? No, we accept that percolation at the critical point is conformally invariant. How? Well, it is critical percolation, it's not under. So it is conformally invariant. So if you ask a question about it, the answer should be conformally invariant. And the answer in this, when I ask this sort of question, is where are the points? Where are the four points? The only thing that matters is the position of these four points with respect to each other. So intuitively, if the segment is a bit larger. Yes. The other one becomes smaller, sorry. Yes, yes, exactly. Or you can, it doesn't matter if you make this domain, if you change this domain by this shape. The four points are in the same place, but the arcs change. Still, you get the same answer. That is with different unit cells of different size. They're just a conformal transformation of. That point is true. As I said to you yesterday, I know this problem. It doesn't have an obvious answer. If it has, I don't know it. You are right. The details of the lattice should not matter, especially when I'm at a critical point. There is something missing there. I cannot quite get hold of it. So obviously, this sort of problem will not depend on the shape of the lattice, because what is the lattice is changing is just the critical percolation value. And that is not an invariant property. For instance, in critical temperature, you can change things with changes to critical temperature, but does not change the exponents. Yes. Much together, many circles, and in India, I'll form a big cluster. Similarly, I can do in case of triangles, and I might expect the same kind of threshold probability in both the cases. Then the PC is coming out to be same from this argument for different kinds of unit cells. Yes, I'm not sure. I don't know with that sort of argument if you can change the value of PC. And you may be able to, but I haven't thought about it. Yes? It's invariant under conformal transformation. Probability of crossing, yes. But in that case, you say P is equal to x. But with conformal transformation, I could move the point C in the segment BD. Yes. Not forbidden, so I could modify the probability. Yes. That's true. So it's not. Sorry, when you change x, if you change C and bring it here, you can take C and bring it here. But then that is not, you cannot change that by conformal transformation to itself. You cannot do this by conformal transformation. You have that, there you're changing something about the physics of it. In the same sense that here I can add this arc, but if I move, if I decide to move this point here, this is not a conformal transformation. It is changing the definition of the problem. I don't think so. It is, in a sense it's obvious it's not because if you change it, the cross-ratio will change. That's the cross-ratio is supposed to be invariant. Okay, so I'm going to stop here and start a new chapter in this course and that is to look at Schaum-Lufner evolution. So the next two lectures, that is this lecture and tomorrow, I will spend on SLE and then I will stop. On Thursday I will just, I will use Thursday as a day to solve problems. No new torture. So that as much as I can get this in to today and tomorrow is my luck. So this guy, Schaum-Lufner evolution, usually in the literature referred to as SLE-COPPA, is a very powerful for analyzing critical curves. When I say critical curves, I mean curves which are scale-free paths and this happened at the point of critical phenomena when two different phases meet. Also critical paths, as I said, in the beginning of my lectures, appear when you have random, random paths such as the self-avoiding walk. So in fact, I have by this method a way of producing many critical paths by just changing COPPA. So I change COPPA, COPPA when I have COPPA equal to two, I generate loop-raised random walk. When COPPA is eight-thirds, I generate a self-avoiding walk. For COPPA equal to three, I generate the critical ising interface. Another path which is called harmonic explorer comes out at COPPA equal to four, percolation comes out at COPPA equal to six and the uniform spanning three at COPPA equal to eight. And perhaps it's not, these are not all of them. There are other paths as well, some which I know fall out of this range and some maybe have not yet been discovered. So it offers me a very powerful tool to discuss. And therefore, a methodology on the complex plane is necessary to deal with it. And then the methodology is a very strange way, if you like, or innovative way to look at a path on the complex plane. So usually we are used to referring to a path on the complex plane by its tip. So gamma of t is a complex number and it starts at some point on the real axis, hence gamma of zero is equal to A and it goes up and from t onwards it can also continue. This is the usual way to think about it. However, what Lovner suggested was doing this. He said, let us find a mapping of the complex plane to itself such that this random path comes down and is absorbed in the real axis. What is meant by the hull here is any region of the upper complex plane that the motion of the path has restricted. Restricted in the sense that it is no longer available to the path. So I have such a map which absorbs the path into the real axis. This means that it has to change with time because with time the path changes. So for each value of t, I will have a different complex function which absorbs the path and the areas which it has made inaccessible to itself. Therefore, g of zero of z is z. Why? Because the path has not existed yet. So the only map you need here is the identity. Just maps it to itself. It is possible that the path gamma comes and touches the x axis as it has done here. Not a very good drawing. It looks like it has crossed it but it really just touched it. It is allowed to touch the real axis and then continue again. Again now you see a hull has been formed. This is a bit more complex than the previous hull but still it is something which is not accessible to gamma that is this path cannot go back in there. The hull copper is now something bigger and more complex and it's not just the path but some areas of the complex plane which have been excluded by the motion of the path. What Loveder succeeded in proving was that for even for these sort of situations there is a map which absorbs all of that into the real axis. So this map g z of t still exists even for this kind of a hull. We also put in some more conditions and g t of z and then it becomes unique. The other condition we put on g t of z is that at infinity complex plane is essentially the identity. So it has a Lorentz expansion at infinity is starting with z meaning that I absorb this copper this k into the real axis but if you go far away from real axis I don't do anything. The infinity is left alone. All these would not be much fun if finding g would be very difficult. However it's not t is time yes. Well it's actually I call it time but it's actually not time. It is a real number which counts along the path. t can be finite for z infinity. It can be so for that t has to be infinite. So we are always looking at this at finite t but very large t. So the point is that at finite t the infinity is left untouched. Okay if this differential equation didn't exist it would be a waste of time talking about g because finding g would be very difficult. The genius of Lovner is that all that is explained you can just solve this differential equation and the answer does all that you want. Very, very powerful result. It's an amazing result. So this equation says that I solve this differential equation with the boundary condition g zero of z is z and g at infinity has that expansion. So the answer comes out unique for any given a of t. A of t is called the driving function. You give me a, I calculate g for you by solving this differential equation. So I need to go back to this, to my standard understanding of what a path is. So my standard understanding of a path is this guy, gamma of t. What Lovner is saying is that in fact this gamma of t is by g of t is brought back into the real axis or there is another way you can look at it. You start at origin and you apply g of t a step by step and you build your path. So which is the other expression below. And in this way I have a method of describing paths on the complex plane. The last equation is a formal writing. So g of t is applied to zero for a small t. Then you repeat for the next year. Go epsilon ahead in t and then repeat g of t. So it's really, I should have, this is really should is something like this. G of delta t on origin and then g of two delta t on this and so on. Okay, let's do a very simple example to understand what is going on. So I take a point, I take a constant value for a and that gives me a differential equation which is relatively easy to solve. So I see I have a factor round there. Two times t minus p. No, no, sorry. I think that it has a two here and that's where that two is coming. So solve this to get that shape. However, I know that if I set t equals to zero, I should get the z and t equal to zero b determines the value of t at zero. So I have made a shift in time. So this is easily solved to get this formula. So the final answer is this guy. Gt of z is a very simple function of t and z and of course it's simple because the driving function was taken a constant. I can expand it near z equal to infinity. I get the right asymptotic behavior and at t equals to zero, I take the positive square root. A is canceled and Gt of zero z is just set. So this is a solution, an algebraic or a solution to the differential equation but what is it geometrically? Yes, in the what? Yes, yes, yes. This particular equation. In the exercises in the homework, as you say, I asked you to solve it for a driving function which you like. No, there is no partial differential equation to solve. It's just an exact differential equation to solve but for the constant it's easy for any other function you may have to do it numerically or you can maybe in a clever way find an analytic function other than the constant which you can solve by hand. So what is it geometrically? This solution, what is it do? It's a path which starts at A and just goes up, just goes up. So gamma of t is at any point at the position A plus IB, A is fixed, B is to root t. Just goes up and as I said, gamma of t is the action of inverse of G on A which has an obvious reason. It's because at this point, Lovner's equation becomes singular on the right hand side. So at the point where G equals to A, you have singularity. So that singularity is the tip of your path, gamma of t. Hence, the tip of the path is at G minus one of A. Where is the rest of the path? So when G acts on this complex plane, what it does is that it takes gamma of t to A on the real axis. It's obvious from here. G of gamma is A. What, where does it take A to, it takes it to some point further back. So it's actually taken, what it has done is that it has taken this path and absorbed it into the real axis. Relatively easy to believe, very difficult to believe when you have a complex stochastic process. But for this simple segment map, it is easy to believe what is happening. Hence, we can do a little as well and see that the origin is actually gone here. In other words, there is some point back here which is acting as the origin. Now, as correctly pointed out, this is the exercise. Two functions. One is the sine of t and the other is t sine t. You put it in there and see what sort of lovner paths you get. It is interesting paths come out. But they have the property that they don't touch. The path does not touch itself. It's a, in other words, it's a simple path. It does not, sorry, cross itself. I shouldn't say touch. It does not cross itself. It is a simple path. There is no point of crossing. And as you choose different functions, you will see that it has no point of crossing. Now that was lovner 1924. Then we time translate forward to 2000. Schram made an observation that if you put in, instead of the driving function, a Brownian motion, you get something interesting. When you put in a Brownian motion, you no longer get just the path. You get a collection of paths. You get a collection of paths, which are stochastically or generated. So he said put Brownian motion instead of a. And of course, this is Brownian motion with variance equal to one. So I put a square root of kappa to get any variance I like. And now the question is what do I get? I get random paths like that starting at a point on the real axis. Usually we take it to be the origin and it goes up. These paths are random paths which don't cross themselves, but it's not a self avoiding path. Self avoiding random walk is in this class of paths, but it's not all of them are that kind. It's a concept which at least for me was a little, took me a while to digest. It is a path, it is random, it doesn't cross itself, but it's not a self avoiding random walk. It can be for a special value of kappa, but in general it is not. It is statistically different. The only similarity is that it doesn't cross itself. So in this way I can generate a lot of paths which are not self crossing and they are random. Hence it's a very good candidate for critical curves or interfaces of interfaces, random interfaces. So here is, I could have given this as an exercise as well, but I thought it is harder. Just put in different values of kappa and see what you get. So for different values of kappa for equal to one, three and six, these are the paths you get. So we notice that there is an obvious qualitative difference between here and there. And I will argue that kappa is smaller than four which is here and kappa is bigger than four which is here are qualitatively different. We will get to that point. Okay, just a slide about Brownian motion in case there is someone in the class who doesn't know what is a Brownian motion. Brownian motion is a stochastic process which has following properties. It's continuous, its increments are independent random variables and in fact these increments have a Gaussian distribution. There are these BT and BT plus H are normally distributed. It has the property that this probability distribution function is a solution of the diffusion equation. Hence it is conformally invariant. In other words, if you take a Brownian path in a domain and you map it to another domain and hence a phi of gamma and phi is conformal. Again, phi of gamma will be Brownian. In physics literature we are used to describe its distribution by this path integral. Strictly speaking, it's a wrong thing to do because B is not differentiable, it's just continuous. But like many other things we do in physics, we do this. And maybe in this representation it's easier to see that now any mapping of B by a conformal map will give us another Brownian distribution. However, Brownian motion is allowed to repeatedly cross itself. It's the motion of a Brownian particle and if you've ever seen it under a microscope it continually crosses itself. SLE doesn't, when it goes inside the Lovner equation the produced path does not cross itself anymore. So B of t as a candidate for critical curves although it is conformally invariant is a bad candidate because we cannot have self-crossing in interfaces, critical interfaces. We are dealing with two dimensions and it's very important to note this property of Brownian motion that it is recurrent. It repeatedly comes back to the origin and that is going to play a very important role. That's this, they are. Why they are? Because they are a scaling invariant as they should be and they are not self-crossing as they should be. This is the property that interfaces have. Now the contribution made by Schramm, I should say that he actually called, in his paper called this stochastic Lovner revolution but it was this S and this S are in the right places so now we call it the Schramm-Lovner revolution. Yes, yes. And then we see that it really, the final line reproduces quite well the boundaries in critical domains. But this thing, in the sense that the complex construction that we built to obtain this curves are physically related to the problem or it's just by chance that finally we reconstructed such complex lines so that by chance they reproduce the boundaries or there is a deeper connection between the construction and the physical properties of them? In a, there is a deeper connection and you are jumping this presentation a little. There is a deeper connection but still you could say that even that deeper connection happened by chance. I don't know how much Schramm knew that this connection is going to be made. He just observed that these are conformally invariant distributions. The observation which was made later and I'm not sure who made it, maybe Karl did it was that the theory of these curves is related to conformal field theory. And since conformal field theory is related to critical phenomena in two dimensions there must be a relation between these guys and critical phenomena. However, this connection is not complete. I mean, not all critical phenomena can be described in this way, just some of them. And the connection to conformal field theory is obscure, that means not all quantities of conformal field theory can be explained in terms of these, just some. But one of the things which we are interested in which is the fractal dimension of these curves comes out to have very easy expression. They have in common the scaling events so this common feature connects them up. Okay, so Schramm claimed that they have two properties. One is the Markov property. The Markov property says that if you have a domain like this and you have an SLE path going from R1 to R2, you choose a point on the middle tour. Then you make a two-step calculation. You first do a SLE mapping, bring tour to the edge. Then it is true that if you take this gamma one and you first absorb gamma one into the edge, then tour to gamma two is again conformally invariant. So that whether it goes from tour to gamma two is independent of how it came to tour. So this is the Markov property that these paths have. And the other one is of course the conformal invariance of it, that is if you take a distribution over a domain of paths which connect R1 and R2 and you make a conformal mapping, it changes the domain to D prime, changes the path to phi of gamma and it changes the points to phi of R1, phi of R2 which are written as R1 prime, R2 prime. The probability does not change. So this is in shapes what I just said. Conformal invariance means that you make a mapping with these two end points and the path connecting them from D to D prime and you get new points and a new path and a new domain. The probability distribution over the new domain is the same as the old domain. So this is the hall which I already explained that as the path moves out of the real axis as it comes out here, it takes some parts of the complex plane which are then not available to it. And obviously what is available to it is what is left out. I didn't make this obvious. In this theory, because of the way we set up the differential equation and so on, we specifically work with the upper complex plane. And so of course, arbitrary. I just need to define a real axis and then everything works above it. Okay, now let's take the Loewner equation and make a transformation on it. And that is define a new function which we have called F here and put it in Loewner's equation. We get something simpler which is this guy. What I now have here is the time derivative of the Brownian motion. Therefore, this is a white noise. So I have now a Langevin equation with a drift force and a white force, white noise force. And we know how to solve these differential equations in physics. This particular one is called the Bessel process which is by the way very useful in financial mathematics. It has been used for a long time in financial mathematics. But for us, I mean people who don't do finance, what is important in it is that it has a drift force which is one over F. So actually a one over F force derives the particle away from the origin. If this was not here, I would just have a Brownian motion. F would be just the Brownian motion. And the Brownian motion has the property that it plays around the origin. So the Bessel process we know in physics is the balance of between these two forces, the deterministic drift pushing it away from the origin, the stochastic force bringing it back. So let's take these two steps separately. I can assume this to be zero. So I set K kappa equal to zero. Sorry, here I have done it the other way around. First, I ignore this guy. Yeah, I ignore this guy, that's true. So F squared d by dt one half is equal to two. Therefore, F squared is four dt. That kappa there is unnecessary. I mean, it's just a force of habit when I write a diffusion equation, I just automatically put a constant here. I don't need it. So what does this say? This is that you have a path which is going to infinity like a square root of t. And this is what I expected with this repulsive force. Just goes to infinity like a square root of t. Alternatively, I can forget about that. And so this and that will give me a force which comes back. So question is, when do these two forces equal each other? Of course you need a better proof, but the implication here is that there is a point for exactly kappa equal to four. Here I have a returning process and here I have a path going to infinity. And these two forces equalize exactly at kappa equal to four. So three regions can be identified. One is for kappa equal to four so that we have a path which starts at origin is a random path and a simple path and it goes up to infinity. Above four, and above four it's a path which can touch itself, it can touch the real axis so it will cut out a hull and it's a growing hull as gamma of t goes up to infinity, the hull grows. So still the path goes towards infinity but it gives you a growing hull like that. Now there is from somewhere else I know that kappa equal to eight, it becomes a space filling path. That means that it visits every point on the space as it goes up. So essentially I have three phases for SLE. Below four which is this sort of random path not touching itself, exactly at four which is a shape somewhat like that. And above four but smaller than eight which it becomes denser and denser here until it becomes totally dense. Another property of the path is that I can ask what happens if I scale time. Scaling in a space was accepted because we are at the critical point so you know that a scale invariance in a space is needed that is a scaling in the z direction. However you can ask what happens if I scale it in the t direction. t is the time variable, it goes along the path so if you scale it you are actually going up or down in the path. The flight noise has this scaling property that if you scale it by a squared, a comes out. You can take that and show that the novena process has to have this particular scaling. That g has to scale by one over a when you scale time by a squared. And therefore it follows that the hull will also scale with a like that. Yes, it doesn't matter, it's a function so you define a new function as g minus a and that goes in the denominator. It's a very normal operation on stochastic differential equations that it is in the denominator does not play any important role. So another question which we can ask is the locality of it. Is the path local or it knows of its entire existence elsewhere? So one question is that is it a random process which sort of takes this tip and then pushes it a little further ignoring the rest of things which are on the complex plane. Or suppose I have a domain in the complex plane a very far from my path and my path at some point has been in there and has come out now and is continuing here. Is it, is there, what is the condition that this gamma does not know about a? If it knows about a then it's a non-local entity. It knows about all its past as it moves up. You may want to say that for some physics processes I like to have one which is blind to its past. I will jump the calculation and just give you the answer that if kappa equals to six, it will be a path which is local. It doesn't see its past. So it can, the tip of it just follows a Brownian motion. Only one value of kappa has the property that it is local. So locality requires kappa equal to six. Yes, there was a question. Say again. Yes. Yes, we mean the path, right? Yes. I had some slides to make this calculation but I had decided not to give you the mathematics of it. So to ask if the path knows about a you look at the expected time to hit a and then you do that calculation on that and you see that if that expected time to hit a is a divergent then it will never hit it and it doesn't know about. The alternative, the sort of dual question to that is that if I require the path to not visit a fixed domain A in the upper half plane. A similar question, a slightly different from not knowing where it is and you would expect that this to be related. Again, some calculation and the answer comes out that restriction requires kappa equal to eight thirds. Eight thirds. So you might guess that these two numbers are related and in fact you are right because the SLE has a property called duality and I want to see not duality means. So take an SLE path, it is going up into infinity, it creates a hall and then I can make a cover for the hall which is this red path. In fact, this picture gives you the impression that my SLE path is the red line which is actually not, it's the blue line. So there are, when you construct an SLE path you have two dual paths. One is the covering of the other and this happens only for kappa equal bigger than four because smaller than four you don't have such a process. If you look at this now you see that this path which is the cover of this other path is an SLE which is not touching itself, it's a rather simple SLE and this is in fact the case that you have two pictures. One is an SLE path which is going to infinity, rather jagged and complex and the other is the path which just traces out its perimeter which is this path and they are both SLE, this is an SLE, this is also an SLE and their couples are related by this relationship, yeah. The restriction is that we want a path not to visit a part of the complex path to just not go in there. It's different from the other one that it has a section of itself that it sees or does not see that it moves independent of it but here I have an area and I want this path never to go in there. Now the duality says that they are both SLE paths with kappa kappa prime equal to 16. If you think of their phases of SLE the minimum was two, the maximum was eight so that minimum, maximum are related by this relationship two times eight being 16. The other way to look at it is that these two paths have fractal dimensions and if you calculate their fractal dimensions and multiply them by each other you get this relationship. What is amazing is that this relationship before SLE was known. So DH is the whole fractal dimension, DEP is the external perimeter fractal dimension and for a critical phenomenon this was known. However the explanation comes as these guys can be interpreted as SLEs. So this case six times eight over three is also 16. That means that these two paths, the locality and restriction are dual to each other. Okay I can think of SLE as a stochastic process and it is because I can write a Langevin equation for it. So I must also be able to write a Fokker-Planck equation for it. So if PSI is the probability distribution function of SLE it must satisfy this differential equation which is the Fokker-Planck equation for this guy here. Now you look at this and you say I've seen this differential equation somewhere else. And where have you seen it? It's the level two null vector which I gave you when I was telling you about null vectors except that this coefficient is now given in terms of kappa. Hence you suddenly observe that the PDF of SLE is in fact something which satisfies the level two null vectors and it's therefore a wave function or a distribution which you can get out of CFD provided you fix this constant kappa over four. Okay we fixed the constant kappa over four in front. Two amazing things happens. One is that you get the central charge of the CFD in terms of kappa and the other is that you get the conformal rate of the field associated, the H21 field associated with this C which is just six minus kappa over two kappa. This is in fact a very revealing thing. It is the question you asked a little sooner and getting here that there is really a deep connection because of this simple observation between CFD and SLE. So I can perhaps get a lot of calculations of SLE out of CFD by using this connection. Now this connection, these expressions are interesting. One of them is this that they see vanishes for kappa being six or being eight thirds which are these dual pictures. The other one is that the H21 field has conformal rate equal to zero because at kappa equal to six. So when you're dealing with kappa equal to six, whatever it is, you have a zero field and zero central charge. But we know that is because half an hour ago I was telling you that percolation has C equals to zero and H21 equal to zero of the fields which I introduced. There is another property here which I wonder if I have given. It's not obvious but this expression for the central charge is invariant under this transformation. So if you change kappa to 16 over kappa, C does not change. So I can now just make a table of the central charge and kappa and for each central charge, of course I have a statistical physics model which I can write next to it. And now I expect this to be related to this SLE and these are very famous models. The looper is random walk corresponds to kappa equal to two self avoiding walk or KPC level sets to eight thirds. A spin cluster boundaries in the Ising model or W03 level sets. This is tungsten oxide. Double diamond model, harmonic explorer, level sets of the Gaussian free field. They all correspond to central charge one on kappa equal to four. FK clusters, Ising model, but FK clusters corresponds to 16 over three. Percolation cluster boundaries corresponds to six and it so happens that 2D turbulence also corresponds to kappa equal to six. And finally the uniform spanning trees which fill out the space as you know are related to the sand pile model and shortest distance on the UST of sand pile model is the looper is random walk. So this is the other limit where kappa is eight. So these two theories are dual by this understand. This third column is still something I have to do. That is in terms of this knowledge that I have, I can construct the fractal dimension of these curves. Only one which is obvious is this UST at the very end, uniform spanning trees. And it's because they are a space filling, they have the same fractal dimension as the ambient space. This is a two dimensional curve. These ones I have to show you how I drive, but I want to stop here. Thank you. Tomorrow will be the last part.