 All right, so let me get started now. So initially, what I was planning on doing today was I was going to give the construction of the metric for leave of quantum gravity. But what I decided to do is to postpone that to the next lecture. And the reason is that I want to kind of redo something I did at the very end last time a bit more carefully, because this is actually very important for the construction of the metric. And so it's very important to explain that more clearly. So let me begin by reminding you of the statement of the quantum Ziprah theorem. And again, this is due to Sheffield. And the statement is that you have a free boundary Gaussian free field H. And this is on the left. On the upper half plane, then you have an SLE Kappa process, eta, which is independent of your Gaussian free field. And I'm going to let G sub t be its Lovner revolution. So with Lovner revolution, G sub t. And I also let F sub t be equal to G sub t minus W sub t. And this is convenient to look at because then F sub t of the tip of the path is always equal to 0. Then the statement of the theorem is that for each fixed t greater than or equal to 0, we have that H composed with Ft inverse plus 2 over the square root of Kappa times the log of Ft inverse plus q times the log of Ft inverse prime is equal in distribution to H plus 2 over the square root of Kappa times the log function. So that's the abstract statement of the quantum Ziprah theorem. And what does it actually mean? Oh, by the way, so here q is equal to 2 over gamma plus gamma over 2, and gamma is always equal to the minimum of the square root of Kappa and 4 over the square root of Kappa. And so again, what's going on is that you have your free boundary Gaussian free field plus 2 over the square root of Kappa times the log function. And then you cut it along an SLE curve eta. And then the statement is that if you then map back, then over here what you get after mapping back has the same law as what you started with. So the field H composed with Ft inverse plus 2 over the square root of Kappa times the log of Ft inverse plus q times the log of Ft inverse. And when you start off with your Gaussian free field and you cut along your SLE curve, then this is the unzipping operation. And you can always invert this procedure and go the other direction. And when you go the other direction, this is the zipping up operation. And both unzipping and zipping up preserve the law of your Gaussian free field. OK, so that was the basic statement of the quantum zipper theorem. And now what I'm going to do is I'm going to re-explain what I did at the very end of last time, which is I'm going to derive the structure of the bubbles that you get when you put an SLE curve when capital is between 4 and 8 on top of a levial quantum gravity surface. OK, so let's start to do that. So I'm going to derive more carefully the structure of the bubbles, which are cut out by an SLE curve, which has capital between 4 and 8, which is the regime where SLE actually makes bubbles. And this is when it's drawn on an independent levial quantum gravity surface. And in particular, you'll hopefully see more clearly the connection between SLE, levial quantum gravity, and levy processes. OK, so throughout, I'm going to assume that I have fixed kappa between 4 and 8. And gamma is always 4 over the square root of kappa. And q is 2 over gamma plus gamma over 2. OK, and then the idea is that using the quantum zipper theorem, we can construct a time stationary pair. And this is what I'm going to call h tilde superscript t eta tilde superscript t. And this is defined for all real numbers t. And for each t eta tilde superscript t, this is going to be a free boundary Gaussian free field plus 2 over the square root of kappa times the log function. And eta tilde of t, this is going to be an SLE kappa curve. And moreover, h tilde of t and eta tilde of t are going to be independent of each other. OK, and in what sense is this a stationary pair? So what we have is that for each S positive, what you can do is you can start off with h tilde of t and eta tilde of t. And you can unzip for S units. And what you get is h tilde of t plus s comma eta tilde of t plus s. So this is the forward direction. So this is just performing the operation where you start off with your Gaussian free field and then you cut along it SLE. And you can also do the reverse, the inverse of this operation, which is where you start off with h tilde of t, eta tilde of t. And then you zip up for S units. And that corresponds to going to h tilde of t minus s eta tilde of t minus s. OK, so you have this stationary field decorated by an SLE where you can cut and you can unzip and you can zip. Now, one thing which is really very important here is that this field is just a free boundary Gaussian free field plus a function. And free boundary Gaussian free fields are only defined modulo additive constant. So all of these fields here, they don't have well-defined values. They're only defined modulo and additive constant. And so the first thing that we're going to do is we're going to fix the additive constant for h tilde of 0. And we're going to do this by setting its average on the boundary of the unit ball, which is inside of the upper half plane to be equal to 0. So if I have a field which is defined modulo additive constant and then I fix its average on any set, then I get something which actually has well-defined values. So after I do this, then h tilde of 0 integrated against phi, this makes sense, or this is actually defined for all smooth test functions. All right, so now I've fixed the additive constant for the field at time 0. But this in turn fixes the additive constant for all times. And the reason for this is that the way that I transform from one field to another is just by performing the zipping or the unzipping operation. And I can determine what the additive constant is for any time t just by either zipping or unzipping to get to time 0. So in particular, h tilde of t integrated against any smooth test function is defined. OK, so that was the basic setup. And again, the picture that you should have in mind, when you do the quantum zipper, when kappa is between 4 and 8 is the following. So you have your upper half plane. And then what you can do is you can imagine, well, let me draw the SLE. This is the SLE up to some time. And then up to a further time, let me make it in red like this. And let's say this is eta at time t. Then what you can do is you can unzip up to time t. And if you unzip up to time t, it just corresponds to applying the conformal transformation which uniformizes the complement of the white curve, so everything that's in the unbounded component of the white curve. And what you're going to get is the image of the red curve over here. But of course, there's also some extra information because you have the field inside of the holes which was cut out by the white curve. And you can think of this somehow as being some bubbles which are hanging below the red curve. And when you perform the zipping up operation, which is you go this direction, then in some sense what you're doing is you're taking these bubbles down here and you're pushing them into the field to get the picture on the left-hand side. And what we want to do is we want to determine the structure of the field inside of these bubbles and relate them to the quantum disks. OK, so that's the basic picture. Now, so let me draw it again over here. So let's suppose that I've got my SLE like this. And now, if I look at a given bubble like this bubble here, then I can divide, sorry, this bubble here, I can divide the boundary of the bubble into two parts. So there is sort of the top of the bubble, which I'm making red. And then the other part of the boundary is the bottom. And let me make that orange. OK, so each bubble boundary has two parts. So you have the top. And this is just the part of the boundary which is in the upper half plane. And then you have the bottom. And this is the part of the boundary in the real line, OK? And OK, now the first kind of very important point is that if you've fixed a bubble like this and you look at the conditional law of the Gaussian free field, so let's say h tilde of t, given its values on the top of the bubble, the law of h tilde of t in the bubble is very simple to describe. It's a Gaussian free field with free boundary conditions on the bottom, and it has fixed boundary conditions on the top, so Dirichlet boundary conditions, OK? And moreover, the field in the bubble, so if you look at what you get when you restrict the field to this bubble here, let's say, then what you're going to get is conditionally independent of the field outside, OK? And so this last fact is going to be important when we try to show that these bubbles actually have some kind of a Poissonian structure, OK? Now, one final observation is that the Liouville quantum gravity boundary length, this is a well-defined quantity, and it's well-defined for both the top and the bottom, OK? So the reason that the Liouville quantum gravity boundary length makes sense for the bottom is that on the bottom of the bubble, you just have a free boundary Gaussian free field, so that makes perfect sense. And on the top, it also makes sense because if you start to unzip, so you cut along the boundary here, then the top of the bubble is going to get mapped into the upper half plane, and so it also just looks like a free boundary Gaussian free field on the boundary of the upper half plane. So you can compute boundary lengths both on the top and the bottom, OK? So now, what we want to do is we want to understand the law of the quantum surfaces that you see inside of these bubbles. And so to do this, what I'm going to do is I'm going to fix two parameters, epsilon and delta, very small. And eventually, I will first take a limit as epsilon goes to 0, and then eventually as delta goes to 0. OK, and so what do you do? So you imagine you've performed the zipping up procedure, and you zip up until the first time, you find a bubble with boundary length on the top equal to epsilon, and quantum mass in the interior greater than or equal to delta. So the picture is that you have your SLE like this, and then sort of hanging below your SLE, you have these bubbles that you're pushing in. And by the way, the way that these bubbles get pushed in is they don't sort of instantaneously appear. What happens is that part of the bubble gets sort of pushed into the interior, and then it takes some time for the entire bubble to get pushed into the interior. So if you take any bubble, there's always going to be a time where the boundary length of its top is equal to some fixed value. So these stopping times actually do make sense for this particular procedure. OK, all right, so you can imagine that you've performed this procedure until this time, and you have a bubble. So maybe the bubble that you're looking at is this one here. So this is its top, which has boundary length epsilon, and here's the bottom. And then what I know is that the amount of the mass and the inside is not too small. It's at least delta. And what I want to do is I'm going to try to determine the law of this bubble when I take a limit as epsilon goes to 0. So the first thing to do is just to apply the thing that I said just a moment ago to get that the conditional law of the field in the bubble is going to be a Gaussian free field with Dirichlet boundary conditions on top and free boundary conditions on the bottom. And what I want to do is I want to understand what happens when I condition on less and less information, in particular when I condition on less and less information that corresponds to the values of the field along the top of the bubble. So that's what we're doing. All right, so let me draw the picture over here one more time. So I have my SLE. And let's say I'm working on this particular bubble here. So again, here's its top and here's the bottom. And by the way, I only want to understand what's going on in the field here as a quantum surface, which means I only care about the structure of the field modulo conformal transformation. And so what I really want to do is I want to uniformize this domain to something which is more convenient. And the choice that I'm going to make is that I want to send it to the strip. So this is the strip. And for me, the strip is just the product of the real line with the interval from 0 to pi. OK. And what I want to do now is I want to conformally map this bubble over to the strip. But the annoying thing is that there are at this point only two marked points. And so in order to determine my conformal map, I need to find a third marked point. And how do I generate a third marked point? I can generate a third marked point by taking this law. And I can weight it by, so I can weight the law of the bubble, by the quantum length of the bottom. So I'm just taking the law of this bubble here. And I'm generating a new law where the Rad-Anikadim derivative is given by the length of this orange part here. And this is exactly the same thing as taking my field, let's say, h hat to be given by h plus gamma over 2 times the green function, where h has the unweighted law, g is the green's function, and z is going to be a uniform point in the bottom of the bubble. So whenever you take a little quantum gravity surface and you weight it by something like boundary length or area, you're just adding a third marked point plus with a log singularity at that point. And this is the same thing as h minus gamma times the log of z plus a harmonic function. All right, so now I have my third marked point. And this is going to be some point which is sitting down here. And what I'm going to do is I'm going to describe the law of this bubble after performing this weighting. And then at the end, I'm going to undo the weighting. So I'm going to unweight. OK, and so how is the conformal map chosen? So here I'm going to choose my conformal map so that z goes to infinity, z goes to plus infinity. And I'm also going to map the top of the bubble to the boundary of the strip. So this is the top of the bubble. OK, so now my conformal map is well-defined. And I can look at the field that I get after performing this change of coordinates. So this is h hat composed with phi inverse plus q times the log of phi inverse prime. And I'm going to try to understand what happens to this field when I take a limit as epsilon goes to 0 over on this side. OK, so let me just quickly tell you what the law of the field that you get over here is. So let me call that h circle. So the law of h circle, given its values on the left side of this strip, so the interval from 0 to pi i, this is a Gaussian free field with fixed boundary conditions on the interval from 0 to pi i and free boundary conditions on the boundary of the strip, which is not in this interval. So that's just describing this term here, well, part of this term, actually. But we also have the part which comes from the coordinate change term. And we have this extra log singularity. And what those translate into is that you have, in addition, the function gamma minus q times the real part of z. OK, and the reason that the extra function that you get looks like that is that near this point z here, this map just looks like the log function. And there's one other thing. And that's that, remember, I conditioned on this field having mass at least delta. And this is actually a very unlikely event. So this event that I'm conditioning on is very unlikely. And the reason is that epsilon is small. And epsilon, epsilon, remember, is the boundary length of the top, which then corresponds to the boundary length of this part here. And if it's small, it means that the field values here are very, very small. So they're very negative numbers. And so it's very hard to have positive mass because at the same time, the drift that I've added is negative. So this is a very unlikely event since epsilon is small and the drift gamma minus q is negative. Right. And so now what we want to do is we just want to analyze what happens when epsilon goes to 0. So we're going to determine the law as epsilon goes to 0. And then we're going to unweight after unweighting by the boundary length. OK, so the way that you should imagine this picture is that you have your strip. Let me make this strip very long, like that. And the way that we've set things up are so that the boundary length here, this is very small. So this has boundary length epsilon. And somehow, the overall mass that you see inside of the field in this strip is at least delta. And if epsilon is very small, that means that the field values are here are very, very small. And it's going to take some time before they get big again, before they get big enough to create mass delta. So in reality, the action in the field, so where you see the mass, is going to be very, very far away from this boundary point here. So the part of the strip where the mass is located, this is going to be far from the interval from 0 to pi i. Just one. I don't get it. Why are the boundary values so small as well? Oh, it's because roughly speaking, the length is going to be e to the gamma over 2 times the average of the field here. And if this is equal to epsilon, then the average of the field here has to be equal to, so a is going to be equal to 2 over gamma times the log of epsilon. If epsilon is very small, you're going to get a very negative number. But it means that you're going to disregard many of the bubbles before that. Well, so here, when I send a limit as epsilon goes to 0, I'm going to end up just focusing on the bubbles which have mass greater than or equal to delta. I'm going to look at some subset of the bubbles, just the ones with mass at least a certain size. I'll determine their structure. I'll see that they're independent of each other given their boundary length. And then I'll send delta to 0. And I'll get the whole plus on the instructor. Yeah, so the reason that you need to condition on the event that you have a certain amount of mass when you send epsilon to 0 is that the natural drift here is negative. So there's no way that you're going to, it's very unlikely that you're going to get positive mass. And so if you look at a typical bubble with boundary length epsilon, it's going to have mass, well, which goes to 0 as epsilon goes to 0. Right. And the reason that you have to go quite far away from the boundary before you start to see the mass is that if you look at the process A sub u, which is equal to the average of the field on the line with real part equal to u, then this process has a particular form. It's just going to be a Brownian motion. Run at twice the speed plus a drift gamma minus q times u. So here b is a standard Brownian motion. The reason that you have the speed is double. It just has to do with the fact that the greens function near the boundary looks like 2 times log and not log. And so if you watch this process A sub u, what it looks like is that if you were to draw its graph, it's going to start off, what it wants to do is it starts off at whatever its average is here, and it's just drifting downward like this. And in order to get a large amount of mass, somehow this average at some point has to get large. And when you condition a Brownian motion with downward drift to get large, that's the same thing as reflecting its drift about 0 until it eventually does get large, and then it will just come back down again. I think that. And so if you start at a very small value, it's always going to take a certain amount of time until you get to any fixed value. Right, OK. So let me first talk about A sub u for a second before I talk about the rest of the field. So in all of this stuff, it's very convenient to make the connection between Bessel processes. And I'm going to set x sub u to be the exponential of gamma over 4 times A sub u. And the reason for making this particular choice is that you can calculate the quadratic variation of x sub u. And this is its entire quadratic variation from 0 to infinity. And if you just calculate this, it's going to be a constant times the integral from 0 to infinity of the exponential of gamma over 2 times A sub u du. And it turns out that this is the same thing as the same constant, which I don't want to specify, times the expectation of the boundary length of the bubble given the whole process A. And this is approximately the same thing as the same constant times the boundary length of the bubble itself, OK? So up to a factor which is tight. This is a good proxy for the boundary length of the bubble. OK. And then there's a general fact, which is very handy, which everybody should learn when they learn stochastic calculus. And that's that if B is a Brownian motion, then and you have a constant A and R, then the process e to the bt plus at is what you get when you exponentiate the Brownian motion. This is a Bessel process. And it's a Bessel process with dimension 2 plus 2A, at least up to time change. And so the right time change to perform is you just want to make its quadratic variation equal to dt. So if you reparameterize it so the quadratic variation is at constant speed, then you get a Bessel process. OK. And so what do you get in this particular case? So in our case, if you take x and you time change it appropriately, so in other words, you define u of s so that what you get when f to u time change has constant quadratic variation, then if you just apply this fact about Bessel processes, you see that x of u of s, this is going to be a Bessel process of dimension 4 minus 8 over gamma squared. OK. So this is just a little algebra calculation that one has to do. OK. And of course, because the drift of my Brownian motion is negative, the dimension of this Bessel process is going to be smaller than 2. So 4 minus 8 over gamma squared is always less than 2 when gamma is less than 2. OK. So that's a good sign. Right. OK. And the time interval that the time changed process, this Bessel process, x of u of s is defined on. This is the quadratic variation of x before the time change. And remember, this is just given by a constant times the conditional expectation of the boundary length of the bubble given A, which is approximately the same thing as the boundary length of the bubble. So up to something which is some nice random variable, the length of this Bessel excursion is just equal to the boundary length of this bubble. OK. All right. So now what happens when you tend to limit as epsilon goes to 0? So when epsilon goes to 0, the boundary length that you see here is getting smaller and smaller. And it takes a longer and longer time before you get to the interesting part of the field where you actually see the mass. And so in fact, what happens is that this picture is sort of mixing in the sense that when you go far enough along this strip so that you actually see the mass over here, you are just going to completely forget about what your boundary values were here. So there's some argument that it takes to justify this. But somehow what you see here does not degenerate in such a way when epsilon goes to 0, that these boundary values are so bad that they cause a long-term problem where you actually start to see the mass over on this side. And essentially the reason for this is that when epsilon goes to 0, if you're working on a given bubble, so let's say you're working on this bubble here, and this is its top, so making epsilon small just corresponds to cutting more of the top away. So you're just sort of cutting away along this part of this picture. And if you make epsilon say extremely small, the top will just become to this last little bit of the top when epsilon was bigger. And the point is that if you take an SLE, let's say an SLE kappa, or kappa is between 4 and 8, and you look at the local picture where it hits the boundary, so you sort of zoom in here when epsilon is very small, then you can actually describe the scaling limit of this picture. So it's not very hard to construct the scaling limit. You can describe it using also the Gaussian free field, so imaginary geometry. And because this has a scaling limit, it's sort of enough to deduce that nothing really bad is going to happen here, because this is just going to be described by what you see when you zoom in at this point there. OK, and so the point is that you can give an explicit recipe, and this is actually one way of seeing that the definition of a quantum disk is correct in deriving it. You can describe an explicit recipe for sampling the limiting object. So when you take a limit as epsilon goes to 0, it's natural not to work on the half strip, so the strip which, let's say, starts at 0. You want to work on the whole strip, so the field that you get as epsilon goes to 0 can be sampled from in the following way. So number one, you pick your Bessel excursion. So you pick z from the Bessel excursion measure for the Bessel measure with dimension 4 minus 8 over gamma squared, and you're going to take your process, a sub u, to be equal to 4 over gamma times the log of z parameterized by quadratic variation. And the correct time parameterization should have constant quadratic variation two times speed 1, so speed 2. And the reason is that the average field is supposed to be a Brownian motion at twice the speed. And this determines the average of your field on the vertical line, so the line with real part equal to u to be a sub u. So this determines one part of the field law. And for the other part, you take the rest of it, which is just the orthogonal projection onto the complementary space. So I've described the field values on averages. So you have your strip. I've just described what happens when you average it on the lines. You just get this process a sub u. And so we just have to describe what you get in the complementary space. And the projection onto the complementary space, this is just given by corresponding projection of a Gaussian free field. Now what I've just described here is actually an infinite measure, because the Bessel excursion measure is an infinite measure. But I have conditioned it on a positive and finite probability event. Because remember, I always assumed that the mass of my bubble was at least delta. So this is always conditioned on the event that you have mass greater than or equal to delta. Yep? How do you express the mass in terms of the Bessel process? Oh, how is the mass expressed in terms of the Bessel process? Well, so the length of the Bessel excursion gives you the conditional expectation of the mass. And the way that you get the actual mass, I mean, the way that it works is that this Bessel process is determining the Bessel process determines the orthogonal projection of the field onto one subspace. But there's still the other subspace. And so what you have to do when you actually calculate it is you have to include the other subspace. And that's sort of like an extra random factor. But there are also different ways of describing this. I mean, another way of describing it is that you can actually just take the length of the Bessel excursion to be the mass. But then that changes some of these conditional laws. So the different ways of doing it that give the same answer. OK, right. OK, and this is now an explicit description of a field on the strip. And this turns out to be exactly the same thing. What I've just described is a quantum disk. It's not exactly a quantum disk, because remember, I had to add my third mark point. So it's weighted by boundary length. And it's conditioned on my event, which is that it has mass greater than or equal to delta. And one can check that this construction of a quantum disk is exactly the same thing. It gives you exactly the same answer as what you would have gotten if you had taken a quantum wedge. Like I tried to describe earlier. And then you condition on the formation of a bottleneck. So you condition on something like that happening. And if you sort of send the bottleneck size to 0, you get exactly the same thing as what I just described here. Well, except for, of course, you won't have this extra weighting. OK, so that's OK. So now I've just described the law of these bubbles, but with this extra weighting. So one thing I don't get is that you could have the bubbles which have much bigger than its size and with some mass. Yes, of course. So remember, I'm just talking about the top of the bubble. So what happens when a, I mean, what does the life of a bubble look like? So you have your bubble down here. And all of this stuff, I'm imagining that I'm zipping things in. So I'm pushing stuff in. And what happens as this bubble comes in is that only a very tiny, when it comes in, only a very tiny little bit of the top is going to appear. And so that means the boundary length of this part is going to be very, very small. And the rest of the boundary length is still here. So sort of like at the instant of time where this bubble comes in, the boundary length has a jump because it's like you've just attached it at essentially one point like this. And then as you move time in, the bubble sort of gets pushed further and further in, which corresponds to changing the picture so that it looks more and more like this. So this is now the boundary. And then you have a little bit of top right here. And as you kind of push it further and further in, the picture looks more and more like this. And then eventually, when you push it further and further in, it will become one of the bubbles of the SOV, which is not touching the boundary. So this top boundary length is actually changing continuously in time as you push it in. And it doesn't reflect the overall boundary length of the bubble. This is just a convenient way of kind of regularizing this procedure so that you only focus on a finite collection of grid bubbles. Yep. If I understand correctly that the quantum zipper is purely SLE statement, no gamma in it, the gamma you need to put in the exponent shop to get the quantum surface. And there you set the gamma to be the special value for the square root kappa. Yep. Where does it enter into the proof that you need this crucial value to compute the length in the area? Well, I mean, that comes in because you need the quantum zipper theorem to be true. Because if the quantum zipper thing doesn't hold, then you can't perform the reverse zipping operation. And the matching between gamma and kappa and q just comes into the proof of the quantum zipper theorem. So when you do the stochastic calculus part of the proof, things will not, you will not get a martingale if gamma, q, and kappa are not matched. This I understand, but in the quantum zipper, q has no gamma in it. It's the minimum of four over square kappa and square kappa over four, right? So that's what gamma is. And q is two over gamma plus gamma over two. Yeah, but it's later that you set gamma, if I understand correctly, it's later that you set gamma is equal to this value in order to be able to compute the length and the area. Oh, so gamma, okay, gamma is universally fixed. Gamma is always equal to the square root of kappa in the minimum, sorry, the minimum of the square root of kappa and four over the square root of kappa. Now here, and q is always equal to two over gamma plus gamma over two. When I started talking about lengths, I don't know, it's possible that you're referring to this part with the Bessel process. And here I just chose this particular exponential just so that when I computed a quadratic variation, I would see gamma over two, and that's the one that corresponds to quantum length. But yeah, I mean all of the parameters are just universally fixed from the beginning. The only sort of choice that's made in some sense is that you can describe quantum disks and bubbles using many different Bessel process dimensions. It's just that only one of them has the right interpretation, or two of them actually. So if that didn't answer your question, we can always discuss it. Okay, so now, right. So now let me just go back to this for a second. So here I want to just now explain why the structure of the bubbles is related to a Levy process. So remember, the quantum disk measure is based on the infinite measure associated with a Bessel process of this dimension. And remember that measure for the length, so the length of a Bessel process of this type is a constant times t to the delta over two minus two dt, where t is the length of the Bessel excursion. And if I plug in my value of delta, then I get t to the minus four over gamma squared dt. Okay, so that's the measure that I'm using to pick lengths from. But t here, remember t is approximately the same thing as the boundary length of the bubble. And we waited by the boundary length of the bubble in order to get this Bessel process measure. And so we have to unweight. And when you unweight, what you get is t to the minus four over gamma squared minus one dt. And this is exactly the Levy measure for a kappa over four, which is equal to four, a four over gamma squared, which is the same thing as kappa over four, stable Levy process. Okay, and so what this tells you is that the bubbles, which are cut off by the SLE kappa, these are in correspondence with the jumps of a kappa over four stable Levy process. Okay, now there's a small thing that I'm glossing over here and that's that you can take, when you set all of this up, you can make a decision as to what the Bessel process is actually going to correspond to. One possibility is what I did up here, which is where the length of the Bessel excursion corresponds only to a conditional expectation of the length, but not the length itself. Another way of setting it up is that you can make it so that the length of the Bessel excursion is the boundary length of the bubble itself. Okay, and when you use the second perspective, unweighting in the way that I just said does actually just correspond to changing the exponent by one, but if you use the other perspective, you're not exactly unweighting in this particular way. So there's a small thing here that has to do with that fact, but when you set it up correctly or in the second way, this is the right formula. Okay, so now what I'm going to do, perhaps after a short break, is explain how one can start to think of these types of theorems as giving away to represent processes like SLEs as gluings of pairs of trees. And then I'll try to explain further how one can extract the corresponding types of results when one works not on a quantum wedge, but rather on a sphere, because the sphere is really what we're going to be interested in when we construct the actual metric next time. Excuse me, I'm confused. Your stable process is a denominator? No, cap was between four and eight. Yeah, so what do you mean that it has some... Yeah, so what's going on is... That's only positive jumps? Well, it depends which direction you go in. So if you go in, there are two directions here. When you do the quantum zipper. So when you go in the forward direction, it's going to have downward jumps. And the downward jumps come... Let me process this representing the boundary length. And you have a downward jump right when you cut this off, because you've lost this boundary length as viewed from infinity. So downward jumps are going to correspond to boundary lengths of bubbles. And when you, yeah, as viewed from infinity. If you go in the reverse direction, you have upward jumps and the upward jumps correspond to the bubbles appearing when you zip them in. Right, and yeah, and of course it's napped. If you think of this from the discrete perspective, you see that from the discrete side that the process is not just going down, because when you do say a percolation exploration, there are two types of triangles that you can see on a random triangulation. The process sort of goes up when you just discover a triangle that looks like that. And so boundary length has increased and boundary length goes down when you do something like that. And so these types of triangles are kind of what are providing the compensation for the Levy process so that you get something non-trivial when you subtract off those downward jumps. Okay, so the Levy process, does it measure exactly because of the intersection? Oh, what is it going to measure exactly? So the nicest way to set it up is actually to have two Levy processes. And what the two Levy processes correspond to is the following thing. So you have your SLE up to a given time. And what you can do is you can measure, let's say it starts from zero, then of course the length of this ray and of this ray are both going to be infinite. And the length of this ray here is going to be infinite and so is this one. But you can ask, what is this minus that that makes sense? And the Levy process is going to represent how that difference evolves. Yeah, exactly. And you have two of them because what happens on the left and the right side is they're going to be independent from each other. And all of this is very natural by the way from the discrete models because when you look at the discrete model you have formulas for everything and you can see that this is going to be the right answer. Okay, so maybe we'll just take a short break before doing the next part. All right, so maybe I'll start again. Okay, so what I was just trying to explain leads to the following fact. So if you take a quantum wedge and you draw on top of it an SLE kappa, so this is SLE kappa on a quantum wedge, then the left and right boundary lengths, these evolve as independent kappa over 4 stable Levy processes. And if you go in the forward direction, these only have downward jumps. And what is a boundary length? When I talk about say the left boundary length, what I'm talking about is the difference between this length here and this length. So this minus this evolves as a kappa over 4 stable Levy process. The same thing is true for the right side and these two processes are independent of each other. And so what you have, I'm going to try to draw this now more pictorially, is you have a correspondence between SLE kappa on Leville quantum gravity and pairs of kappa over 4 stable Levy processes. And here's how we like to think of them. Here's the picture. So if I let x is going to be equal to the left boundary length and y is going to be equal to the right boundary length, then I can draw a picture of what that actually looks like so what does the left boundary length look like? It's a Levy process which has only downward jumps so it for example can, we'll go for a little while and then there's a downward jump like this. And these correspond to whenever the left side of the curve is hitting either the boundary or itself and let me actually make these downward jumps red. So this is maybe what that looks like and that's going to be x. And then I also want to draw y next to this but I want to reflect y. So I'm just going to draw C minus y where C here is just a very large constant and now it looks like this. Oops, where the downward jumps became upward jumps because I reflected it and then we make those also red. Okay and this is the left and right boundary lengths of my SLE and now I can start to glue this picture together and the way that I do that is I'm just going to draw lines, the lines which lie entirely below this graph like this and I'm just going to declare points to be equivalent if they can be connected by one of these purple lines which is entirely below let's say the graph of x. And then what I'm going to get is one of these loop trees where each loop in this picture corresponds to one of the downward jumps in x. Okay so if I glue this picture together I get this loop tree and I have another loop tree over here which corresponds to y. So maybe this one looks like this. So you have this sort of tree of bubbles and then I can take this picture and I can glue it together. Sorry let me first draw in the lines for y and I can glue the two trees together and the way that I do that is I'm just going to draw orange vertical lines going this way and I'm going to declare points to be equivalent if they can be connected by one of these orange vertical lines in addition to the previous equivalents. And this induces a gluing over here of these two loop trees. So somehow these guys are going to be glued together and what I've just described for you is an abstract topological space because I just constructed an equivalence relation let's say on this rectangle and that induces a gluing of these two loop trees and that's just some abstract topological space and what comes out of this is that what you get when you glue two loop trees together like this is you get something which has exactly the same topology as SLE kappa on Leaville quantum gravity. Okay so somehow the topology of the holes in an SLE kappa here you have all these disks which are sort of cut up by the curve and they correspond to holes. The way that these things are glued together is encoded by how these two stable loop trees are glued together here. So these things have exactly the same topology. And if you don't care about quantum gravity but you care about SLE this tells you that something which is actually very powerful and that's that you can convert any question about the topology of SLE into an equivalent question about the topology of a gluing of loop trees stable loop trees. Okay so any question that you can ask about how the bubbles are hooked up up there is corresponds to an equivalent question about how this picture here is glued together. And it actually turns out that the correspondence is much stronger than that. And I'm not going to say anything about this because there's not enough time but it turns out that the following is true. It turns out that one can recover the SLE and the quantum wedge in a measurable way. So there's some embedding function which is measurable. If you just observe the levy processes that's not quite enough information but you also need to observe the quantum discs that correspond to the holes. Okay so if I just show you this picture which was generated from an SLE in the way that I described earlier and I tell you what to put into the holes then you can recover in a measurable way what the SLE must have been to sort of off with. Okay so that means that you can convert any question about SLE itself into a question about gluings of stable loop trees plus the extra information which is inside of these discs. And it turns out that this is very powerful and we just give you one quick example of this before I move on to something else. And that's that it turns out that you can calculate you can convert any SLE dimension problem. So let's say you want to calculate a house or dimension associated with SLE like the double point dimension or boundary intersection dimension or whatever into a dimension question for levy processes which somebody has probably studied and then there's a formula that tells you how to convert the levy process answer into the SLE answer. Okay so for example, let me just give you one simple example of this. Let's say you have an SLE or a campus between four and eight and you draw it on top of the quantum wedge and then you ask for example what times correspond to when this SLE is in this boundary ray. And remember the right boundary length let's say is just this length here minus that length. And so if you're in the boundary it means that you've just cut a bubble off let's say and you're actually at the running in a femum. So boundary intersection times this is equal to the set of times when either the left or the right levy process is that it's running in a femum and we know the dimension of the time set where a levy process is at it's running in a femum and then there's a formula that you can apply which then gives you the corresponding dimension for the spatial set when SLE is intersecting the boundary. And this is just one example of a type of theorem that one can prove by converting questions about the geometry or structure of SLE into questions about the gluings of these types of trees that you get from this type of theorem. Right and then this particular result this is part of a paper that I wrote with two students at MIT. So they're you and Gwen and Nina Holden and then also myself. Okay, so that's just a side remark that says that there's sort of a world of questions that one can try to address about SLE using these levial quantum gravity representations but let me not talk more about that and focus on the stuff that we need in order to finish constructing or in order to construct the metric for levial quantum gravity. So as I just said, if you have a quantum wedge and you decorate it with SLE then this is the same thing as what you get when you have a pair of Kappa over four stable levy processes. Now when we do the metric construction we're going to be interested in what happens not when you work on a quantum wedge but rather when you work on a quantum sphere. So let me first tell you what the corresponding result is on a quantum cone. So on a quantum cone it turns out that if you take a quantum cone and you decorate it with a whole plane SLE so this is a process which starts from the origin and goes to infinity and it can wrap around, et cetera. So if you decorate a quantum cone with let's say an SLE six process and you ask how does the boundary length of the outer boundary evolve? So how does this boundary of the undoubted component evolve? Then it turns out that this is going to be a three halves stable levy process which is conditioned to be greater than or equal to zero. Okay, so that's how this boundary length is going to evolve. And I don't want to go through the process of deriving this but this is proved in essentially the same way as the quantum wedge case is proved. It's just that the quantum wedge one is sort of the nicest one to do. But you can believe that if you work on a cone and look at the outer boundary length and this is sort of the natural thing which is going to come up. Okay. And what I want to do now is I want to explain the sphere version of this. So the quantum cone statement is proved essentially the same way that the quantum wedge one is proved. The quantum sphere one is a bit trickier. It's actually quite a bit trickier to prove. And let me explain how that's going to work. So for a quantum sphere, it's going to turn out that a quantum sphere plus an SLE six is going to correspond to a three halves stable levy excursion, okay? And right, okay. So what I want to do now is I just want to explain kind of roughly speaking at a high level how one extracts what's going on with the sphere from the quantum cone case. Okay. So for a quantum cone, so what is a quantum cone again? So if you parameterize a quantum cone by the cylinder, then it looks like this. And what you have is a Gaussian free field on the cylinder plus the function gamma minus Q times the real part of Z, okay? And so what this means is that if you look at the average of your field along lines with, you know, vertical lines on the cylinder, then the way that this is going to evolve is you have this Brownian motion with downward drift and it's a Brownian motion with downward drift Q minus gamma or gamma minus Q times U, okay? Okay, and then the way that we constructed a sphere, the, what is a quantum sphere? A quantum sphere, this is what you get when you, in some way you pinch a bubble of mass off of a cone, quantum cone. And the point is that there are different ways of doing that. So if this is your quantum cone, let's say, so this is a quantum cone, then you can again look at this process which is given by the, this is the average of the field on vertical lines. And this is the Brownian motion with a given drift. So it has a downward drift as you go off to plus infinity. That's one process you can look at. The other one that you can look at is you can draw your SLD6 on top of it. So this is plus infinity and I start to explore my SLD6 this way. Sometimes it wraps around, et cetera. And you can watch the boundary length. Let me make it a little bit simpler. So let me, you can watch the boundary length of the component which contains minus infinity, okay? And that's going to evolve when you go in this direction as a stable levy process with downward jumps conditioned to be positive. So it looks something like this. Okay, so this is the boundary length of the component which contains infinity, containing minus infinity. Okay, and there are sort of two ways, natural ways to pinch a bubble of mass off. So one thing that you can do is you can condition on this process getting very, very small before it gets big and then it comes back down again. And the other option is that you can condition on the event that the SLD has a time where the unbounded component has a very, very small boundary length. Because it has a very, very small boundary length, it's also going to create some kind of a bubble. So you have two ways of doing it. You have two options and basically what you have to do is you have to show that these are equivalent. So to create the sphere, you can, you have two options. So option one, you condition the process which is the Brownian motion with negative drift to take on a large value after a very small value after a very small value. Okay, so that's where the picture kind of looks like this. Your process is going down and then you condition it to get big again after getting small. And this gives you the Bessel, sort of the Bessel description of a sphere. And option two, you can condition on the boundary length of the complimentary component of the SLE to get very small after getting big. So here you have your levy process which is keeping track of the boundary length, the orange region up there. And it's always conditioned to be positive. Let's say this is the zero line. And you can condition this process to do something very unlikely which is to get very, very close to zero before it sort of continues on. And if you send this height that you want it to reach after getting big to actually tend to zero, what you need to show is that this sort of conditioning also corresponds to this type of conditioning. Okay, so you have two different ways to condition on an unlikely event and you have to show that somehow they're equivalent. And once you do that, you see that if you explore a quantum sphere with an SLE six, then the way it's sort of encoded is in terms of one of these stable levy excursions. Okay, but this is something which is very natural. You would certainly guess this to be true, but it's actually quite a pain, significant pain to show that these two forms of conditioning on this unlikely event actually works. So I don't want to go into details of how one actually justifies that. Okay, so I, okay, and so what you sort of get is that there are different ways of representing quantum spheres. You can represent them in terms of Bessel excursion measures. You can represent them in terms of levy excursion measures and there's a third representation which I won't say anything about. Okay, so I just want to explain one last thing. About what happens when you do SLE six on a sphere, a quantum sphere. So you have your picture that looks like this. This is my cylinder and I have here a quantum sphere. Then I can draw my SLE six on it and sometimes it wraps around, et cetera. And what do I know? I know that the boundary length of the unbounded component, so the one which contains minus infinity, this is evolving as a three half stable levy excursion. And then the other thing that I know is that the holes that I'm cutting out, these are just going to be conditionally independent quantum disks. And the reason that these things are going to be disks is that when you perform this type of conditioning here, where you condition on the boundary length to get very, very small after getting big, you're only affecting the boundary length process, you're not affecting what's inside of the holes and we know what's inside of the holes are actually quantum disks. And what I want to do, and this is the last thing I need to explain, before I can do the metric construction next time, is what is the law of the unbounded component as a surface? So let me just quickly explain what that is. So the last question I want to look at is, what is the law of the component which contains minus infinity? So let me quickly explain how you derive what it is. So the first thing I want to remind you is that when we constructed a quantum cone, the origin of the quantum cone was a special point. So the origin was a quantum typical point, something just chosen from the Liouville quantum gravity measure. Right, okay. And it turns out that in the quantum cone, when you parameterize it by the cylinder, the origin, this corresponds to plus infinity in the cylinder parameterization, okay. And it turns out that when you construct a quantum sphere, the origin, which is plus infinity, is also a special point, okay? So in a quantum sphere, because of this fact of how we built the sphere from the cone and how we built the cone by zooming in near a quantum typical point, it turns out that the point plus infinity is a quantum typical point. So this is a highly not obvious fact when you see the definition of the sphere. But what it says is that if you take a quantum sphere, it's invariant under the operation of picking a point at random and swapping it with plus infinity, let's say. Okay, so somehow what you can do, this is very not obvious from the definition of the sphere, but it turns out that you can just pick a point at random here from the quantum measure, let's say Z, and then you can apply the conformal map, which swaps these two points and leaves this one fixed in what you're going to get has the same law as what you started with, okay. And the same thing is also true for minus infinity. So minus infinity is also a quantum typical in the sense that you can resample it from the quantum measure and the law is preserved. And so what this means is that you can do the following thing. You have your SLE on your sphere and you can draw it up to a given time like this. And what we know is that the holes here, these are all quantum disks. And so if you take the point minus infinity and you resample it, there's some chance that it's going to be inside of one of these holes. So if you resample, if you sample a point Z from the quantum measure, there's going to be a chance. It lands in one of the holes, which is cut out by the SLE. And the probability that it lands in a particular hole is just going to be proportional to its mass. So the probability Z is in a given hole is going to be proportional to the mass of the hole cut out by the SLE, okay. Now you can always take an SLE six and you can retarget it at a different point, okay. So basically all that you do is you just take minus infinity, you resample it, and then you pretend that that's minus infinity and you draw your SLE towards that. And this will leave the whole picture invariant and therefore the unexplored region, so that the region which contains minus infinity, its law is very simple to describe. It's a quantum disk, which is weighted by its area, okay. So the unexplored region isn't exactly a quantum disk, it's weighted by its area and it's very natural to expect that to be the right answer because this component also has a marked point corresponding to minus infinity. Okay. Right, and so now I have all of the facts that I need about SLE six on Lievill quantum gravity surfaces in particular on the sphere, so that next time I can construct, I'll use SLE six and I will use these facts to build the metric on Lievill quantum gravity for gamma equal to the square root of eight thirds. Okay, so I think I'll stop there.