 So today, what I'm going to do is I'm going to construct the metric on square root 8 thirds, leave of quantum gravity. And the way that we're going to build it is we're going to construct a variant of the Eden model in the continuum using SLE6. So SLE6 on square root 8 thirds, leave of quantum gravity. OK, so before I do that, let me just quickly review the discrete story and try to motivate you why you should guess that any of this is going to work. So let me begin by supposing that we have a triangulation of the plane. So this is the uniform infinite planar triangulation. And I'm going to let t be the root triangle of the UIPT. Then what you can imagine doing is you can imagine performing an Eden exploration starting from this marked triangle. OK, and this is going to be in the dual graph. So how does that work? So you imagine that you have your initial triangle, t. Let's say that's t. And then each stage, what you do is you look at the cluster that you've made. You pick an edge at random on the boundary, and then you just explore the opposing triangle. So maybe the first triangle you see is this one, because you picked this edge. And then maybe in the next stage, you pick, let's say, this edge. Then maybe you pick this one. And then every once in a while, you might see something like this. So if I pick this edge, I can have a triangle come all the way around and cut off a big bubble, et cetera. And because I'm starting off with a uniform infinite planar triangulation, the holes that I cut out here, these are going to be conditionally independent triangulations of the disk given their boundary length. And it's also going to be true that the boundary length of the infinite component, this is going to be a Markov chain. This is a Markov chain. And you can write down what the transition probabilities for this Markov chain are. OK, so this is one very natural model. And another very natural model that you can do on the UIPT is site percolation. So another natural model is you can do site percolation. And so here, you start off with a triangle. Let's say your root triangle. And I can imagine that I've prescribed some boundary conditions. Maybe this is black, this is black, and this is white. And then I can do a percolation exploration. So starting from here, I reveal this triangle. And maybe I see something that looks like this, in which case I'm going to go this direction. And then I reveal this triangle, et cetera. And then just like before, sometimes I see triangles that do something more complicated. So I could see something that, see a triangle that does something like this, et cetera. And the point is that the holes which are cut out by the percolation exploration, these have the same law as the holes cut out by the Eden exploration. And not only that, but it's also true that the boundary length of the infinite component, this evolves in the same way. So this evolves in the same way as in the Eden model. And so the way that you want to think of this is that somehow percolation and the Eden model, they're exactly the same, except the only difference is that when you do a percolation exploration, you just have to keep track of where you're going to explore from next. So in some sense, the Eden model, this is the same thing as the percolation exploration, except for you sort of forget where you're going from at each stage. So the Eden model is the same thing as percolation, except at each stage, you forget from where you were growing from. You resample it uniformly at random, and then you reveal another triangle. OK, so that's sort of a very simple observation. And it's also very natural to expect that the Eden growth, this at large scales, this looks like a metric ball, looks like a metric ball growth. And this was, in fact, proved in a work, relatively recent work, of Nicola Currian and Jean-François Lagal. So this was proved by Currian and Lagal. OK, so what are we going to do? So we're going to use this intuition, but in the continuum. And what we're going to do is we are going to construct the metric for Li-Ville quantum gravity using the continuum analog of this discrete picture. And so the continuum analog of percolation is just going to be in SL-E6. And the planar map is going to be played, the role of that is going to be played by the Li-Ville quantum gravity surface. So we're going to construct this metric by reshuffling an SL-E6 curve. OK, so that's the general idea. So before I jump into the exact construction and the proof of the metric property, let me just do a very quick review of some of the things that we talked about last time. So let me remind you of some of the objects. So when we talked about Li-Ville quantum gravity spheres, we worked on the infinite cylinder, which is just the product of the real line and the interval from 0 to 2 pi with the top and bottom of the cylinder glued together. And the law of the distribution, which describes the Li-Ville quantum gravity sphere, is very simple to sample from. And how do you do that? So you do it in two steps. So first, you take its projection. So you take the projection of H, which is going to be the random field, onto the space of functions, which are constant on vertical lines, to be given by 2 over gamma times the log of z, where here z is a Bessel excursion of dimension 4 minus 8 over gamma squared. And once you form this process, you then have to reparameterize time appropriately. And the way that you do that is that you take its quadratic variation to be just given by dt. So it has constant speed. So to have a quadratic variation with constant speed. And this describes one part of the distribution. And the other part of it is what you get when you project onto the orthogonal complement and the projection onto the orthogonal complement of the space of functions which are constant on vertical lines, which happens to be the space of functions which have zero mean on vertical lines. You just take that to be the same as a Gaussian free field on the cylinder. And that's the construction of the Li-Ville quantum gravity sphere that we're going to work with. And let me just emphasize a couple of points about this construction, which are going to be very important for what we're going to do in a moment. And that's that, first of all, the points which lie at plus infinity and minus infinity. These are special points. They're marked points. And they turn out to be quantum typical. And what do I mean by that? This just means that, in other words, if you pick two new points, so if we sample x and y from the quantum measure, mu sub h, so this is the area measure associated with our sphere independently, and then we perform a change of coordinates, which swaps these two points with plus and minus infinity. So then we perform a change of coordinates, which swaps plus infinity and minus infinity with x and y. Then the result is going to have the same laws that we started with. So then the resulting field is left invariant. And this is true, actually, modulo one small thing. So this is true modulo horizontal translation and a rotation about the line from minus infinity to plus infinity. And the reason that this is the case is that, or one way of seeing it, it's not at all obvious from the definition that these points, I mean, they're obviously special from the definition, but it's not obvious why they're special. And the reason that this is the case is that the way that we constructed this sphere to begin with was by somehow pinching a bubble off a quantum cone. And the special point in a quantum cone is somehow you should think of it as a quantum typical point. So really, these are these spheres. They're doubly marked because they have these two special points at plus infinity and minus infinity. So the way that we like to think of this object, which it's a sphere, and it's parameterized by the infinite cylinder, and it's described by the distribution h, and it's marked by the points plus infinity and minus infinity. This is an embedding of a doubly marked quantum sphere. And if we don't want to specify how this is the particular choice of parameterization or field, sometimes we just write s for the quantum surface and then the two points x and y that we are marking it with. OK, and then the other thing we talked about is what happens when gamma is equal to the square root of 8 thirds. And we explore our sphere with an SLE6. So I'm going to let eta be a whole plane SLE6 from plus infinity to minus infinity. So the picture is that you have your cylinder that looks like this. Here is, sorry, I'm going to make it go from minus infinity to plus infinity. So here's the cylinder. And you can then draw your independent SLE6 on top of it. So you have this curve. Sometimes it wraps around, et cetera. And so I tried to explain last time that the holes that this SLE6 is cutting out, these have a very special structure. These are quantum disks. And moreover, they are conditionally independent, given their boundary length. So just like in the discrete story when we explored the UIPT with percolation, OK. And moreover, the point which contains plus infinity, that's special. And so if you run your SLE up to a given time, the component containing plus infinity, it's not exactly a quantum disk. It's sort of a special quantum disk. And the reason is that it somehow has to contain this extra marked point. And the way that that shifts its law is that it's just going to be a quantum disk, but it's weighted by its area. So it's size biased by its area. So we have this exact description of what's going on here. One other thing is that the process which describes the boundary length of the component containing plus infinity, this is going to evolve as the time reversal of a 3 half stable levy excursion, which has upward jumps, so with only upward jumps. So what do I mean by that? So the levy excursion itself has only upward jumps. But after you time reverse it, you get something which has downward jumps. And these jumps just correspond to the downward jumps of this time reversal. These just correspond to the boundary lengths of the holes that you cut out by the sl e 6. OK, so that's the general picture. And so what you get is the following thing. If you take a quantum sphere when gamma is the square root of 8 thirds and you explore it with an sl e 6, an independent whole plane sl e 6, then this produces for you a 3 half stable levy excursion plus the collection of quantum disks that correspond to the downward jumps. OK, so there's some way of going from this picture to this picture. And it's not at all obvious from what I've said so far, but it turns out that you can actually go in the other direction. So it's also possible to go this way. And what this means is that if you just observe this levy excursion and the quantum disks, as well as how they're oriented, then you can measurably recover the other picture. So these two things are actually encoding exactly the same amount of information. OK, so in other words, if you're given either structure, you can measurably recover the other. And this last point is something which I didn't say anything about, and unfortunately due to time constraints, I won't be able to say anything about how one constructs that arrow there. OK, so it turns out that these two objects are just exactly the same. And so what this means is that we have two ways of producing quantum spheres. We can either use our levy excursion or we can use a vessel excursion. There's actually a third way of producing a quantum sphere. Somehow using correlated Brownian motion, there's another excursion measure. But also due to time constraints, I'm not able to discuss that third construction. OK, so let me just mention that we have sort of two natural infinite measures that produce for us quantum spheres that we've talked about. So method one, this is where we describe the distribution, which corresponds to the field representation of the sphere. And this is based on a vessel excursion of dimension 4 minus 8 over gamma squared. And when gamma is equal to the square root of 8 thirds, this is just an excursion of a one dimensional vessel process or just Brownian motion reflected at 0. And the second method is you can just use a 3 half stable levy excursion. And what the 3 half stable levy excursion is encoding is it's encoding the quantum sphere plus the SLE6 on top of it. And this third construction, which I'm not going to talk about, somehow encodes the sphere. And then in addition to that, you have a space filling version of SLE6 rather than just SLE6 itself. OK, great. OK, so that's kind of the background, just reviewing what we did before. Now let me give you kind of the very high level idea for constructing this metric from these tools that I've just described for you. And I just want to emphasize a few things before jumping into the details so that there are a few things about it that hopefully you'll appreciate. So what we're going to do is we're going to build a growth process. And this is our QLE 8 third 0. And this is going to be built out of SLE6 on a square root 8 thirds level quantum gravity sphere. And this growth process, what it's going to be is it's going to be our candidate for describing metric ball growth. But it's not obvious from a mathematical perspective that this really is describing a metric. And then what we have to do is we have to check that the growth processes that we've defined, these actually do correspond to the growth of metric balls for some metric space. OK, so that's the general strategy. Let me make a few more kind of clarifying remarks about what we're going to do in just a moment before we actually do it. Number one, the metric that we're going to construct in just a moment, this is only going to be only defined, at least to start off with, on a countable dense set of points in our sphere. And these points, they're actually going to be, they're just going to be conditionally independent points. So IID points, x of i, which are all going to be chosen from the measure, so the area measure associated with this surface. And then it takes quite a bit of extra work beyond just checking the metric property. To show that, number two, the space that we construct is going to be, to show that the metric continuously extends to a metric on the whole sphere, so to a metric which is defined on the whole Lieveville quantum gravity sphere, which is homeomorphic to the Euclidean sphere. And it also requires quite a bit of extra work and another set of ideas to show that this metric space is equal to the Brownian map. So in the construction, if you're an expert in the Brownian map, you'll see some of the properties of the Brownian map sort of emerge, but proving that this is actually exactly the same as the Brownian map requires one to understand some additional properties which won't be obvious from this construction. And then finally, in the construction, we're going to show that this metric space is a measurable function of the field used to construct the Lieveville quantum gravity surface. But it's not going to be obvious that the Lieveville quantum gravity surface is a measurable function of the Brownian map structure. So there's also an extra argument to show that the metric space structure by itself determines the Lieveville quantum gravity surface that we started with. OK, and so these are all sort of separate issues. And what we're going to do today is we're going to focus on this one here, just constructing and making sense of the metric. And then if anybody's interested, I can explain later some of the additional ideas that go into establishing these points. Excuse me, Jason. Can you recall here, please, when you have the SLA6? How do you measure the boundary length? Oh, so the way that the boundary length is defined is that you can always apply a conformal transformation, which takes you, let me draw a new picture here. So you have your SLA6. You can always apply a conformal transformation, which takes you from the complement of this disk to, let's say, the upper half plane. And the field that you're going to get when you apply this conformal transformation is going to look like a free boundary Gaussian field on the boundary. So it has a well-defined notion of boundary length. Yeah, so in fact, I mean, what's very important is that the unexplored region is always a quantum disk, a levial quantum gravity disk weighted by its area. And these types of surfaces always come with an intrinsic boundary length measure. So you can always talk about the length of the boundary. That always makes sense. Right. But where it comes from is just the fact that these distributions arise from the free boundary Gaussian free field. And so you have the bicoformal transformation, then you have some multiplicative factor in the control zone. Yeah, that's right. There's always the correction term. That's right. So whenever you apply a change of coordinates, you always have the correction term. And that's just the right correction term so that areas and boundary lengths are preserved in the right way. Right. That's right. OK. Right. So let me write. OK. So before I do the metric construction, I have to emphasize one other point, which is very important to keep in mind when one works with these objects. And that's that there are, at this point, sort of two natural, well, let me just say, two parameterizations, which makes sense to talk about for the SLE6 on the square root 8 thirds Liouville quantum gravity sphere. So you have your SLE6. It's going from minus infinity to plus infinity, like this. And one kind of very important question is, what is the right time parameterization to talk about for this SLE6? And there are sort of two of them that make sense. So you can talk about, number one, the capacity parameterization as seen from plus infinity. That's one possibility. And the other one, which is going to be the one which we actually care about, is the time parameterization, which is associated with the 3 half stable Levy excursion. OK. And the reason that this is the one which is sort of natural to think about, the second one, is that this is just the one which corresponds to the continuum analog of what you would do if you were exploring a percolation exploration in the discrete one triangle at a time. So it's the continuum analog of the parameterization which corresponds to revealing one triangle at a time. OK. So when we talk about parameterizations for SLE6, we're not using the capacity parameterization that came from Schramm's original introduction of it. We're rather talking about the time parameterization which comes from the boundary length process. And the reason is that this is sort of the natural one to think of this somehow as being percolation in the continuum on Levy quantum gravity. And in a moment, there's going to be a third time parameterization, and that will be the one that we actually use to describe the metric. So now what I'm going to do is I'm going to start to construct the metric. And in the construction, it's very important to keep in mind the basic properties that one has for SLE6 on Levy quantum gravity, because the metric is going to have essentially the same, well, the right analogs of these properties. So how do you do that? I'm going to fix a positive parameter delta that I will eventually send to 0. And I'm going to define my approximations in the following way. OK, so here's the picture you should have in mind. So we have our sphere, which is described by the strip. And our process, I'm going to define my QLE process, which is going to go from minus infinity to plus infinity. And the way you should think about this is this is like a metric ball growth, where I'm only going to keep track of the component containing plus infinity, and the process itself is growing from minus infinity. OK, and so what do we do? So we are going to, number one, we're going to draw delta units of Levy process time. Sorry, let me say it this way, delta units of SLE6 with Levy process time, so the intrinsic notion of time, starting from minus infinity. And so I'm going to get my chunk of SLE6. And then what do I know? I know that the unexplored region is going to be a quantum disk weighted by area. And then I also know that the tip of my SLE6, this point turns out to be uniformly at random on the boundary of this component containing plus infinity. So what I can do is I can then resample the tip, so the location of the tip from the boundary measure. And then I can repeat. Just keep repeating this process. So in the next stage, I might, after resampling the tip, it might go to, say, this point here. And then I'm going to draw a bit more SLE6 from here. So maybe it does something like that. And then the tip of this SLE6, let's say at this point. And then I'm going to resample it from the boundary measure. Maybe I get something here. And then I draw more SLE6 again, like this. Here's its tip. And then you resample the tip, et cetera. And the point is that this is just like first passage percolation. The only difference is that rather than revealing the surface one triangle at a time, I'm revealing it by these chunks of percolation exploration. So this is just like the Eden model. This is an Eden model growth, at least in spirit, except my chunks that I'm adding, these are somehow pieces of SLE6, rather than somehow corresponding to a single triangle. And the point is that we don't know how to draw single triangles at a time in the continuum, but we can do these SLE6 explorations. And the reason that you would expect this to sort of work is that somehow it shouldn't matter what types of chunks you're adding at each stage, still the final answer should approximate the metric ball growth. Now, the reason for doing this construction in this particular way is that these approximations have a number of very special properties. And this is why the whole thing works. And that's that we know that, number one, the holes which are cut out by the exploration and the law of the unbounded component, including its boundary length, these things have, they're the same as what you get when you do SLE6. So they have the same law as in an SLE6 exploration of a levo quantum gravity surface. So we know kind of exactly what's going on, at least in terms of the holes, the boundary length process and the unbounded component of this process, regardless of how we chose delta. And so how are we going to define QLE? Well, at least initially, our candidate for metric ball growth is just going to be what you get when you take a sub-sequential limit as delta goes to 0 of this construction. So what's happening when delta goes to 0 is that you're just re-randomizing your tip more and more and more frequently. And because of how it was constructed, these two properties are also going to hold for the limit as delta goes to 0. So we're going to know that the boundary length, at least at the moment, evolves as a 3 half stable levy excursion up to time reversal. We also know that the holes are going to be quantum disks and the unbounded component is going to be a quantum disk weighted by area. So properties 1 and 2, these are going to hold for the sub-sequential limit. OK, and then let me also say that there are also some things that you could be worried about here. So number one, we took a sub-sequential limit and not a true limit. And you could ask whether or not the metric that we're going to define is going to depend on the choice of sub-sequential limit. It turns out that it's not going to. And sort of after the fact, much later, it's possible to show that this really is a true limit. And then the second thing you could worry about is that the construction involved additional randomness beyond the Leaville quantum gravity surface. So we had our SLE6s we were drawing. And we also had these points that we were re-sampling on the boundary of our growth. And you could worry that somehow this extra randomness is going to persist in the limit and not give you something which is measurable with respect to the Gaussian free field. But this also turns out not to be the case. And that will all come from the proof. So it really is a well-defined metric that is measurable with respect to the Gaussian free field and does not depend on the choice of subsequence. And as I said, sort of after the fact, it's possible to show that this really is a true limit and not a sub-sequential limit. OK, right. OK, so the notion of time that comes with this limit, the sort of a priori notion of time, is the wrong one because it's the one that comes from the SLE6. So this comes from the Leaville excursion. This is the wrong time parameterization for the metric. So this is not the correct time parameterization for the metric. Time equals distance to the root. In the metric, time equals distance to the root, right? That's right. That's right. So the notion of time that our growth process is currently equipped with is the one that you would get if you were doing the following thing. So let's say this is your root triangle and your UIPT. And let's say you did the Eden model where in each unit of time, you revealed one triangle. So that was one unit of time. Then the next unit of time, you did this. Maybe the next unit of time, you did that, et cetera. So that's sort of the wrong notion of time because when the boundary length of your cluster is getting longer and longer, you should be revealing more and more triangles because each unit of time, you're supposed to be revealing one distance one. So all of the triangles which are adjacent to your cluster. And so you have to make a time change. So again, in the Eden model, with the correct time parameterization, you add triangles to your cluster at a rate which is proportional to the boundary length. And so we want to make the corresponding time change now for our QLE. And so if x sub t is equal to the boundary of the complementary component, which is containing plus infinity when we're doing our QLE growth, then from the construction I've given so far, this is just given by the 3 half stable levy excursion up to time reversal. What we want to do is we want to make a time change so that we are in fact growing at a rate which is proportional to its boundary length. So we're going to change time by, we want our time to be, let's say, s of u, which is the first time t that the integral of 0 to t of 1 over xs ds is bigger than u. So that's the time parameterization which says that we're now growing at a rate which is proportional to boundary length. Is that the natural thing to do? And so what happens when you do that? So before the time change, x sub t was a 3 half stable levy excursion. And after the time change, so x sub s of u, for people who know about these things, it doesn't actually matter what the definition is. But this is going to be a continuous state branching process with branching mechanism u to the 3 halves, up to time reversal. So somehow when you perform this time reversal, your boundary length process goes from being a levy process to being a CSPP. In so particular, the distance from minus infinity to plus infinity, in terms of levy process time, it's just going to be equal to the integral from 0 up to, let me say, xc of 1 over xs ds, where xc is the first time t that xt is equal to 0, the first positive time that it's equal to 0. OK. All right, so now I think I'll maybe we'll just take a short break. And then in the second half, we'll come back and prove that this defines a metric space. OK, so maybe I'll start again. OK, so I just defined the metric ball growth that goes from minus infinity to plus infinity. How are we actually going to extend this? So we're going to assume that we have points xi. And these are going to be iid from the quantum measure, so from the area measure on your sphere. And what we're going to then do is given xi and h, what we're going to do is we're going to sample conditionally independent QLEs from each xi, so from each of the xi. And actually from each xi to each xj. And then what we're going to do is we're going to set the distance between xi and xj. This is just going to be the amount of time that takes the QLE from xi to absorb. OK, and now I've defined this function, which is sitting on top of the level quantum gravity surface marked by these points. But as I said before, there's in principle a lot of extra randomness here, because I have, well, in the definition of QLE itself, there's a lot of extra randomness. And beyond that, I have sampled a bunch of conditionally independent QLEs given my Gaussian free field and the collection of marked points. So that's something to keep in mind. OK, so what we're going to do now is we're going to show that this d is a metric on the points xi. And so to do this, there are three things to check. Well, actually, two things. We're going to show that d is symmetric. So this is something which is not at all obvious if you think about it, because let's say in the picture where I have minus infinity and plus infinity, what I have done is I've sampled two conditionally independent QLEs, one which is going in this direction. And I've defined the distance from here to here to the amount of time it takes this guy to make it to infinity. And the distance in the reverse direction is defined with a conditionally independent QLE going in the other way. And from the way things are defined, it's not so obvious actually that this is going to be something which is symmetric. And then number two, we need to show that it satisfies the triangle inequality. And actually, the proof of part two is going to somehow come from one. And the proof of one will also imply three, which is that the metric d, this is determined by h and the locations of the xi and does not depend on the subsequence. So it's not going to matter which sub-sequential limit we chose, and that will come from actually just the proof of one. So really, the heart of the whole argument is going to be this proof of symmetry, because once we have that, that will sort of give everything. OK. Excuse me, why would they all be conditionally independent? I've just chosen them to be. Right, but what's the motivation? Because that's the most natural way to couple things together when you don't understand their joint law at all. If you're going to put them in the same space and you don't understand how they're supposed to be related, just take them to be conditionally independent. It's also crucial for the proof, too. It's very important. OK, so now I'm going to explain the argument for symmetry. And so how does that work? So I'm going to imagine that I have, in general, my quantum sphere, Sxy, and it's marked by the points x and y, which are quantum typical. OK, and I'm just going to focus on the QLE, which goes from x to y, and the one which goes from y to x. So gamma, this is going to be the QLE, which goes from x to y. And gamma bar, this is going to be equal to the QLE, which goes from y to x. And here, just to emphasize it again, gamma and gamma bar, these are going to be conditionally independent, given the underlying surface. And what we want to show is we want to show that the amount of time it takes this QLE to go from x to y is the same amount of time it takes its time reversal, not its time reversal. It takes gamma bar, the other QLE, to go from y to x. And so how is that going to work? I'm going to define a law theta. And this is the law of S, x, y, gamma, gamma bar, and some additional randomness, u, where here u is going to be uniform between 0 and 1, independent of everything else. So that's what we're going to do. And the way that we're going to show that these two distances are the same is that we are going to define two new measures. So I'm going to define d theta. So theta x to y is going to be the measure whose radonichodium derivative with respect to theta is going to be the distance from x to y. And I'm going to define my other measure, d theta from y to x. This is going to be the measure whose radonichodium derivative is dyx. And because we don't know symmetry yet, it's not clear that these two measures are the same. And what's our goal going to be? Our goal is going to be to show that theta of x to y is equal to theta from y to x. And once we show that, then this has to be equal to that by the uniqueness of radonichodium derivatives. So this is what we want to do, because this then implies that the distance from x to y is equal to the distance from y to x. And as you're going to see, this is going to somehow boil down to the reversibility of whole plane SLE6. And I should mention that parts of this argument are actually based on another work, which is due to, so this is based on or related to a strategy developed in another work by Scott together with Sam Watson and Hao Wu, where they construct a metric on CLE4, somehow using an argument which is related to what I'm going to show you now. OK, so this is what we want to do. And there are sort of two steps to proving this symmetry statement. Here's the first one. So I'm going to define a stopping time tau. And tau is going to be a uniform between the distance of x and y. So it's just u times the distance from x. The amount of time it takes for the CLE to go from x to y. And I'm going to let tau bar be the first time t that gamma bar of t intersects gamma tau. So here what I'm doing is I'm drawing one of the QLEs up until a typical time between its distance from x to y. And I'm going to draw the other one until it hits it. And similarly, I'm going to let sigma bar, the u times, the time it takes to go from y to x. And I'm going to let sigma be the first time t that gamma of t intersect gamma bar sigma bar is not equal to the empty set. OK, and what is the first step? The first step is to show that, and this is really the heart of the argument, it's to show that the theta x to y law of s comma x comma y, gamma up to time tau, and gamma bar up until it hits it. This is equal to the theta y to x law of the same thing, except for its gamma up to time sigma, where remember sigma is the first time that gamma hits gamma bar up to time sigma bar. So we're first going to check that. And then there's a second step, which one also has to check. And that's that you need to show that the theta x to y law of gamma and gamma bar, given everything that we've seen so far, so s, x, y, gamma up to time tau, and gamma bar up to time tau bar, this is equal to the theta y to x law of gamma and gamma bar, given everything but with sigmas in places of tau. So s, x, y, gamma up to time sigma, and gamma bar up to time sigma bar. Because we checked those two things, then the two laws have to be the same, and therefore the radianicademe derivative is the same, and therefore the distance measured from the left is equal to the distance measured from the right. OK, so due to time constraints, I'm just going to focus on the first step, because the second step is actually very believable if you think about it, because this is just something about the sort of Markovian nature of this construction. Because we took everything conditionally independently, these are all sort of stopping times. It's natural to believe that these conditional laws should behave well. And it's sort of a technical exercise to check that this is all the case. So somehow all of the meat is in the first step of the argument. And so as I said before, the way that this is going to go is we're going to reduce step one somehow to the reversibility of whole plane SLE6. OK, so that's the idea. And now let me remind you that we have three, well, I've described one so far, but there are sort of three natural measures that we're going to consider on quantum spheres, on square root 8-thirds,劉ville quantum gravity spheres. And somehow by understanding these measures, this symmetry statement is just going to pop out. So the first one is the one which is induced by N, where N is the excursion measure for a 3-half stable levy excursion. OK, so if you have a 3-half stable levy excursion, then you can produce a quantum sphere with an SLE6 just by associating with each jump a conditionally-independent quantum disc. So there's this measure. The next natural one is you can take the one which is induced by taking the measure N. And then what you do is you add to it another variable T. OK, so there's a second variable here, small t. We're here dT is Lebesgue measure. T is equal to the length of the levy excursion, capital T. OK, so this second measure, all that I'm doing is I'm taking my levy excursion and I am biasing it by its length and then picking a typical time between 0 and the length of the excursion. So in this second construction, the marginal of the levy excursion, so of x, this is just given by just taking N and waiting it by its length. So it's just a size bias levy excursion. OK, and then the third possibility, the third measure is the one which is induced by the measure 1 over xT times the indicator of the event from 0 up to the length of the levy excursion. And then dT again is Lebesgue measure and this is the levy excursion measure. Right, OK, and so here in this third construction, the marginal on x is just N weighted by the integral from 0 to T of 1 over xS dS. OK, and so the way that we think of these things is that here you're just waiting by somehow the length of the levy excursion in construction 2. And in construction 3, if you think of this levy excursion as somehow being related to the QLE, then here you're somehow waiting by the distance. It's somehow a distance weighted construction. So this waiting is like waiting by the distance between two typical points on the sphere. OK, and let me give these measures names. So this one, this measure sometimes we just refer to as M2 SPH because this is describing a sphere with two marked points. This measure is a sphere with two marked points where we've weighted it by the amount of time it takes the SLE6 to go from one point to the other one. And then the last one is like it's a measure in a sphere with two marked points where somehow you're waiting it by the distance as measured by the SLE6 to go from one point to the other point. And what do we want to show? We want to show the following. So let's say this is my strip. Then if I produce a sample from M2 SPHD, so this consists of a sphere together with an SLE6. And then what I'm going to do is I'm going to draw my SLE6 up until the special time t. So eta here, this is the SLE6. It's going from minus infinity to plus infinity. And I've drawn it up to the special time t that's associated with this measure. And then I'm going to let eta bar be the time reversal of eta. And I'm going to let t bar be the first time that eta bar hits eta up to time t. So I have this picture where I have drawn one of the SLE6s, sorry, the SLE6 up until a typical time. And then I draw the other, I draw its time reversal, eta bar, up until it first hits eta. So this right here is eta of t. And this point down here is eta bar of t bar. And so what do we want to do now? So if I let x be the surface, which is separated from plus infinity by eta up to time t, x bar is going to be the surface separated from minus infinity by eta bar up to time t bar. And then b is going to be what's left. So it's the remaining surface. So in this picture, the stuff cut up by the white curve corresponds to x. The stuff cut off by the purple curve corresponds to x bar. And the stuff which remains is just b. And what we want to do is we want to show that, so our goal is to show that the law under this particular measure of x, comma, x bar, comma b, this is equal to the law of x bar, comma, x, comma b, OK? So we want to show that if you look at the surface cut up by the first SLE6 drawn up to the typical time, the second one until it hits it, then you can't tell which one is which. They look exactly the same. And the reason that that suffices to finish the proof, the reason that this suffices is that you can take this picture. So let me copy it down here. Save the same picture I had a moment ago. So here is eta up to time t. Here is eta bar up until it hits it. So this is eta bar of t. If you take this picture with the two meeting SLE6s, and then you reshuffle it, so you perform the tippery randomization procedure, then what you're going to do, what you're going to end up with are not two meeting SLE6s. You are going to end up with a pair of meeting t-releves. You'll have something like a QLE growth gamma up to time tau. And you'll end up with an eta bar will turn into gamma bar up to time tau bar. And if this picture is symmetric, then what you get when you perform the reshuffling operation down here is also going to be symmetric. And then this will then prove exactly what we need to show what we needed for the first step. So really the heart of the problem is just to understand this sort of strange symmetry statement for SLE6s on a levopontum gravity sphere. And so where does it come from? It's going to come from, again, the reversibility of SLE6. And why is that? So for a moment now, let's prove an analogous symmetry statement for not M2SPHD, but M2SPHW. So in this measure, what you have is you have a quantum sphere. And on top of it, you have your SLE6, which is going from minus infinity to plus infinity, like this somehow. And the measure is what you get when you take this picture under the original quantum sphere measure and you weight it by the amount of time it takes the SLE6 to go from minus infinity to plus infinity. So what you see here is SLE6 on a quantum sphere weighted by the time to go from minus infinity to plus infinity. And in addition to that, you have a time t, which is uniform between 0 and this time. So the structure you have is you have this sphere. On top of it, you have an SLE6. And then you have a time which is chosen uniformly from the amount of time it takes for the SLE to go from here to here. So you have a marked time. Let's say it's this time. And the point is that this picture here is completely symmetric because SLE6 is reversible. And the time was just chosen uniformly from the amount of time it takes to go from one side to the other. So in particular, under this law, what do we know? We know that if I let y be the surface separated from plus infinity by eta up to time t, I let y bar be the surface separated from minus infinity by, let's say, eta from time t up the time infinity and d, the remaining surface, or sorry, let me call it q, the remaining surface. Then by the reversibility of SLE6, this whole picture is symmetric, we have that the M2 SPH w law of y, y bar q is equal to the MSPH law of y bar y and q. So all that this is saying is that in this picture here, you can't tell in some sense in which direction this was drawn. And this is just because you can't tell in which direction in SLE6 was drawn. OK, and so to finish the proof now, we want to deduce the corresponding thing for the law M2 SPH d. Now remember, in the law for M2 SPH d, we ran the SLE6 up until the special time t, and then we ran the time reversal up until it first hit the SLE6 in the first direction. And the difference between the two laws is that in M2 SPH d, we have this extra factor 1 over xt appearing. And somehow we want to make that appear under the law M2 SPH w that we had over here. OK, and so how is that going to happen? So the idea is very simple. So what I can do is I can assume that I'm working under the law M2 SPH w. And I draw my SLE6 up to the special time t. And then I can condition on the event that this time is actually a cut time. So if I condition eta of t to be a cut time, then what's going to happen is that that's like conditioning the time reversal to first hit eta exactly at its tip. So then this becomes eta bar, first hitting eta at exactly at its tip. And if you condition on this to be a cut time, there is some work to make this precise because this is a 0 probability event, then you're still going to get a picture which is symmetric. So then yy bar q will still have the same law as y bar yq because I took a measure which was symmetric and I conditioned it on a symmetric event. OK, but the point is that the probability that this event happens is just going to correspond to the inverse of the length of eta at time t. So this law, it turns out, is exactly equal to the M2 SPH d law of x, x bar, and v. And again, the reason for this is the following. Let me draw the picture again. So let me just explain that in a little bit more detail. So if this is the strip, then how would you make it precise the event that you're conditioning on something being in a cut time? Well, you imagine you draw the first process up to time t. And then you can condition on the event that the time reversal hits the first process in some interval, like it could hit the interval from here to here of length epsilon. So if that's an epsilon length interval, then the probability that this guy is going to first hit eta exactly in that interval is going to be of order epsilon over xt. If xt is the boundary length, then epsilon is the length of this little thing. And so when you condition on a time being a cut time, it exactly is going to introduce the extra factor which allows you to go from NSPHW to MSPHD. And so this symmetry statement falls out. But of course, there's quite a bit of work that goes behind checking that this is actually the case. So one has to be careful with this conditioning argument. But that's sort of the main idea behind the proof. And this is sort of the heart of the argument that this defines the metric. OK, so let me just finish in the last few minutes by explaining a few more things. So at this point, what does one have? So you have a metric on a countable dense set, so a sequence of iid points on a square root 8-thirds level quantum gravity sphere. OK, and what one wants to show to sort of finish this program is to show that this metric somehow extends continuously to something which is homomorphic to the real sphere, and it is the Brownian map. And then we just sort of very briefly explain in the last few minutes how one checks these things. So to show that the metric continuously extends, there is a particular trick that we've used many, many times, a number of times in these papers, which allows one to control the size of QLE. So you have to control the size and shape of a QLE growth. And there's a very funny trick that allows you to do this. And it's quite powerful because you can also apply this to other things like SLE itself. And that's that if you're working on, let's say, a quantum sphere and you grow a QLE up to a given time, then the unexplored region here is going to be a quantum disk. And one way of thinking about that is that this just tells you that if you apply the conformal transformation phi, which takes you from this complement to, let's say, Euclidean disk, then you know that the field H composed with phi inverse plus Q times the log of phi inverse prime, this is a quantum disk. And let me call this H tilde. And if you somehow want to control how rough the boundary of this QLE is or how big it is, basically, one way of controlling it is trying to understand this conformal map here. And so what you can do is you can just solve, you can use this equation and solve for the log of the derivative. So you have this change of coordinates formula where you know that this comes from a quantum sphere, that comes from a quantum disk. They're coupled in some complicated way. But anyway, it's just going to be true that phi prime is equal to, well, you can solve for the log of phi prime. So Q times the log of phi prime is equal to H tilde minus H composed with phi inverse. And because this is related to a Gaussian free field, and that's related to a Gaussian free field, you can use this to bound the behavior of this map and argue that these QLEs are actually not going to be too large. So this constrains the behavior of phi prime, and that constrains the size of the QLE. And arguments of this type are actually good enough that you can show that the metric space defined in this way does, in fact, it continuously extend to the case of the sphere. And once one does that, then one can also show that this metric space is not only going to continuously extend to something which is homomorphic to the sphere, but it's also going to be geodesic. It's a geodesic metric space. And essentially, to show that the metric space is actually the Brownian map, what one has to do is somehow understand how these geodesics interact with each other. But since now, well, there isn't very much time left. I won't say anything about that. But if anybody's interested, I'm happy to discuss it offline. OK, so maybe I'll stop there. You don't have a responsible person, so I guess I'll ask. Any questions? So where's the Brownian map? Yeah, the Brownian map. Oh, where does it actually appear? So this metric space is the Brownian map. It's the same metric space as the Brownian map. It's not obvious that it is the Brownian map, because the way that the Brownian map is defined is somehow not obviously appearing in this particular construction. But the way that you can see that this is the Brownian map is you try to understand how geodesics in the Brownian or sorry, in the in Leaville quantum gravity interact with each other. So you imagine you've grown a metric ball, which goes from minus infinity to plus infinity. This is your field metric ball. And what you can then do is then you can draw the geodesics, which go with some kind of tree of geodesics going from the boundary of this ball back to minus infinity. And what you can do in both the Brownian map setting and also in the Leaville quantum gravity setting is you can understand the law of this geodesic tree, not only at a fixed radius, but you can understand what happens as you move the boundary of the metric ball back in time and get closer and closer to minus infinity. And essentially to show that a metric space is the Brownian map, you just have to check that this geodesic tree has the right sort of behavior. And it also evolves in the correct way when you move the boundary of the metric ball back to minus infinity. And then you also just need to know that somehow this property is invariant under the operation of sampling these two points. And once you check that, then it's possible to show that it is the Brownian map. So those two properties essentially single out the Brownian map. On the Leaville quantum gravity side, somehow where these properties come from is from the sort of construction itself of QLE. So roughly speaking, you need to understand how these boundary lengths between geodesics evolve. And what you need to know is that they evolve as independent continuous state branching processes as you move the metric ball back in time. And where that comes from is in the approximations to QLE, basically what you're doing is you're somehow attaching little pieces of SLE6 if you want to advance the metric ball either forward or backward. And when you attach a little piece of SLE6, let's say to here, what happens is that this interval is going to get a Levy process increment. And it's going to be independent of, it's not going to depend on these particular intervals. And so when you draw these little SLE6s in, the different intervals are getting independent Levy process increments. And somehow you can extract from that that the boundary lengths really are evolving in the correct way for it to be the Brownian map. So you don't see the Brownian snake appear exactly, but rather you see this sort of geodesic tree structure, which is built into the Brownian stake appearing in Leaville quantum gravity. Yeah.