4/4 Equivalence of Liouville quantum gravity and the Brownian map
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Miller, Jason
Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has roots in string theory and conformal field theory. The second is the Brownian map, which has roots in planar map combinatorics. We show that the Brownian map is equivalent to Liouville quantum gravity with parameter $\gamma=\sqrt{8/3}$. Based on joint work with Scott Sheffield. September 2016
 surprise 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. 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. So before I do that, let me just quickly review the discrete story and try to motivate 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. 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. And 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 conditions 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. 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. So 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 can 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. And 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, 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. But 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 Carian and John Fonsvaldo Gaul. This was proved by Carian and Gaul. 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 L'Eville quantum gravity using the continuum analog of this discrete picture. And so the continuum analog of percolation is just going to be in SLE6. And the planar map is going to be played, the role of that is going to be played by the L'Eville quantum gravity surface. So we're going to construct this metric by reshuffling an SLE6 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 L'Eville 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 L'Eville 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. You take the projection of H, which is going to be the random field, onto the space of the 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 product variation to be just given by dt, so it has constant speed. So to have a product 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. And just take that to be the same as a Gaussian freefield on the cylinder. And that's the construction of the Leville 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 as what 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, a 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. 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. 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. OK. 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 a 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 sle6. 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 sle6, an independent hole plane sle6, 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. OK. 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. But it turns out that these two objects are just exactly the same. And so what this means is that we have sort of two ways of producing quantum spheres. We can either use our levy excursion or we can use a Bessel 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. But 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 Bessel excursion of dimension 4 minus 8 over gamma squared. And when gamma is equal to the square of 8 thirds, this is just an excursion of a one-dimensional Bessel 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 8-3-0. And this is going to be built out of SLE6 on a square root of 8-3-0, a levy of a 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, well, the growth of metric balls for some metric space. OK. 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. OK. And 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 level 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. 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 SLE6, 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 SLE6. 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 free 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 Lieveville 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 bicoform transformation, then you have some multiplicative factor in the closed 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. So that's right. OK. Right. So let me... Right. 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 make sense to talk about for the SLE 6 on the square root 8-thirds Liouville-Quarton gravity sphere. So you have your SLE 6, 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 SLE 6? And there are... 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. And the reason that this is sort of 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 the first one is the continuum analog of the parameterization, which corresponds to revealing one triangle at a time. So when we talk about parameterizations for SLE 6, 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 about if you want to think of this somehow as being percolation in the continuum on legal 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 SLE 6 on legal 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? So 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. So here's the picture you should have in mind. 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. 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 SLE 6 with levy process time. So the intrinsic notion of time starting from minus infinity. And so I'm going to get my chunk of SLE 6. 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 SLE 6, 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 SLE 6 from here. So maybe it does something like that. And then I will, and then the tip of this SLE 6 is 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 SLE 6 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 SLE 6 rather than something 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 SLE 6 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 kind of 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 a number of different dimensions. They're the same as what you get when you do SLE 6. So they have the same law as in an SLE 6 exploration of a Li-Vo 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 randomizing your tip 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 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 levy or 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 sub-sequence. 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 Levy excursion. And this is the wrong time parameterization for the metric. So this is not the correct time parameterization for the metric. OK, 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 in 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, you 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. The natural thing to do. And so what happens when you do that, your, 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 X c of 1 over X s ds, where X c is the first time t that X t is equal to 0, the first positive time that it's equal to 0. All right, so now I think 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. So maybe I'll start again. 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 X i, 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 X i and H, what we're going to do is we're going to sample conditionally independent QLEs from each X i. So from each of the X i. And actually from each X i to each X j. And then what we're going to do is we're going to set the distance between X i and X j, this is just going to be the amount of time it takes the QLE from X i to absorb X j. And now I've defined this function which is sitting on top of the level of 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. So what we're going to do now is we're going to show that this D is a metric on the points X i. 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 X i and does not depend on the subsequence. So it's not going to matter which subsequence 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. Okay. 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. Okay. Right, 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. Okay, 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. Okay, and here just to emphasize it again, gamma and gamma bar, these are going to be conditionally independent given the underlying surface. Okay, 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, okay, not as time reversal, it takes gamma bar the other QLE to go from Y to X. Okay, and so how is that going to work? I'm going to define a law theta and this is the law of SXY, gamma, gamma bar and some additional randomness U where here U is going to be uniform between 0 and 1 independent of everything else. Okay, 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, we're 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, okay, 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, okay, 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, okay, and as we're going to see this is going to somehow boil down to the reversibility of whole plane SLE6, okay, 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 Hau Wu where they construct a metric on CLE4 somehow using an argument which is related to what I'm going to show you now, okay, okay, 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 let, I'm going to define a stopping time tau and tau is going to be 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, okay, 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 intersects gamma bar of sigma bar is not equal to the empty set, okay, 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, X, 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, okay, so we're first going to check that and then the other part which 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, okay, 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, okay, right, okay, so okay, 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, you know, 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 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, okay, right, okay, so that's the idea and now let me remind you that we have three, well I've described two, 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 eight-thirds, medieval 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, the one which is induced by N where N is the excursion measure for a three-half stable levy excursion, okay, so if you have a three-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, the one which is induced by taking the measure N and then what you do is you add to it another variable T, okay, so there's a second variable here small t, where here dT is Lebesgue measure and t is equal to the length of the levy excursion, capital T, okay, so the 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 zero and the length of the excursion, okay, 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, okay, so it's just a size bias levy excursion, okay, 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 zero up to the length of the levy excursion and then dT again is Lebesgue measure and this is the levy excursion measure, right, okay, and so here in this third construction the marginal on x is just N weighted by the integral from zero to T of 1 over xS dS, okay, 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, okay, so this waiting is like waiting by the distance between two typical points on the sphere, okay, okay, okay, so okay, 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 on 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, okay, 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, okay, 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, okay, 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, okay, so I have this picture where I have drawn one of the SLE6's, 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, okay, so this right here is eta of t and this point down here is eta bar of t bar, okay, 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 plus infinity, sorry 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, okay, 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, is 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, x bar, b, this is equal to the law of x bar, x, b, okay, 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, okay, so let me, let me copy it down here, so the same picture I had a moment ago, so here is eta bar 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 sLE6's 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 sLE6's, you are going to end up with a pair of meeting qLE's, so 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, okay, 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 that, to show what we needed for the first step, okay, so really the heart of the problem is just understand this sort of strange symmetry statement for sLE6 on a levopontum gravity sphere, okay, and so where does it come from? It's going to come from, again, the reversibility of sLE6 and why is that? Well, so for a moment now, let's prove an analogous symmetry statement for, analogous symmetry statement for not m2 sphd but m2 sphw, okay, 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 zero and this time, okay, 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, okay, 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 will let y bar be the surface separated from minus infinity by let's say eta from time t up to 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 m sph law of y bar, y and q, okay, 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 an sLE6 was drawn, okay, and so to finish the proof now, we want to deduce the corresponding thing for the law m2 sph d, okay, 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, okay, 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, okay, 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, okay, so then this becomes eta bar, first hitting eta at exactly at its tip, okay, and if you condition on this to be a cut time, you know, there's some work to make this precise because this is a zero probability event, and 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, okay, 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, okay, and again the reason for this is the following, so we draw the picture again, sorry, so let me just explain that in a little bit more detail, so if this is the strip, then you know, 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, okay, 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 x t, if x t is the boundary length and 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 n sphw to m sphd, and so this symmetry statement falls out, okay, 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, okay. So let me just finish in the last few minutes by explaining a few more things, so at this point what does one have? You have a metric on an accountable dense set, so a sequence of iid points on a square root of 8 thirds, leave a quantum gravity sphere, okay, 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's 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 a, 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 say, 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 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. Okay, so maybe I'll stop there. We don't have a responsible person, so I guess I'll ask any questions. So where's 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, sorry, in the in the Leaville quantum gravity interact with each other, you imagine you've grown a metric ball which goes from minus infinity to plus infinity, this is your your filled metric ball, and what you can then do is then you can draw the geodesics which go, you get some kind of trio 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 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 SLE6's 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 snake appearing in Levy quantum gravity.
 surprise  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.  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.  So before I do that, let me just quickly review the discrete story  and try to motivate 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.  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.  And 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  conditions 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.  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.  So 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 can 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.  And 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, 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.  But 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 Carian and John Fonsvaldo  Gaul.  This was proved by Carian and Gaul.  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 L'Eville quantum  gravity using the continuum analog of this discrete picture.  And so the continuum analog of percolation is just going to be in SLE6.  And the planar map is going to be played, the role of that is going to be played by  the L'Eville quantum gravity surface.  So we're going to construct this metric by reshuffling an SLE6 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 L'Eville 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 L'Eville 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.  You take the projection of H, which is going to be the random field, onto the space of  the 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 product variation to be just given by  dt, so it has constant speed.  So to have a product 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.  And just take that to be the same as a Gaussian freefield on the cylinder.  And that's the construction of the Leville 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 as what 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, a 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.  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.  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.  OK.  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 a 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 sle6.  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 sle6, an independent hole plane sle6, 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.  OK.  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.  But it turns out that these two objects are just exactly the same.  And so what this means is that we have sort of two ways of producing quantum spheres.  We can either use our levy excursion or we can use a Bessel 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.  But 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 Bessel excursion of dimension 4 minus 8 over gamma squared.  And when gamma is equal to the square of 8 thirds, this is just an excursion of a one-dimensional  Bessel 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 8-3-0.  And this is going to be built out of SLE6 on a square root of 8-3-0, a levy of a 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, well, the growth of metric balls for some metric  space.  OK.  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.  OK.  And 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  level 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.  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 SLE6, 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 SLE6.  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 free 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 Lieveville 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 bicoform transformation, then you have some multiplicative factor in  the closed 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.  So that's right.  OK.  Right.  So let me...  Right.  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 make sense to talk about for the SLE  6 on the square root 8-thirds Liouville-Quarton gravity sphere.  So you have your SLE 6, 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 SLE 6?  And there are...  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.  And the reason that this is sort of 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 the first one is the continuum analog of the parameterization, which corresponds  to revealing one triangle at a time.  So when we talk about parameterizations for SLE 6, 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 about if you want to  think of this somehow as being percolation in the continuum on legal 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 SLE 6 on legal 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?  So 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.  So here's the picture you should have in mind.  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.  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 SLE 6 with levy process time.  So the intrinsic notion of time starting from minus infinity.  And so I'm going to get my chunk of SLE 6.  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 SLE 6, 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 SLE 6 from here.  So maybe it does something like that.  And then I will, and then the tip of this SLE 6 is 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 SLE 6 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 SLE 6 rather than something 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 SLE 6 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 kind of 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 a  number of different dimensions.  They're the same as what you get when you do SLE 6.  So they have the same law as in an SLE 6 exploration of a Li-Vo 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 randomizing your tip 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 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 levy or 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 sub-sequence.  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 Levy excursion.  And this is the wrong time parameterization for the metric.  So this is not the correct time parameterization for the metric.  OK, 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  in 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, you 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.  The natural thing to do.  And so what happens when you do that, your, 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 X  c of 1 over X s ds, where X c is the first time t that X t is equal to 0, the first  positive time that it's equal to 0.  All right, so now I think 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.  So maybe I'll start again.  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 X i, 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 X i and H, what we're going to do is we're  going to sample conditionally independent QLEs from each X i.  So from each of the X i.  And actually from each X i to each X j.  And then what we're going to do is we're going to set the distance between X i and  X j, this is just going to be the amount of time it takes the QLE from X i to absorb X  j.  And now I've defined this function which is sitting on top of the level of 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.  So what we're going to do now is we're going to show that this D is a metric on the points  X i.  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 X i and does not depend on the subsequence.  So it's not going to matter which subsequence 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.  Okay.  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.  Okay.  Right, 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.  Okay, 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.  Okay, and here just to emphasize it again, gamma and gamma bar, these are going to be  conditionally independent given the underlying surface.  Okay, 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, okay, not  as time reversal, it takes gamma bar the other QLE to go from Y to X.  Okay, and so how is that going to work?  I'm going to define a law theta and this is the law of SXY, gamma, gamma bar and some  additional randomness U where here U is going to be uniform between 0 and 1 independent  of everything else.  Okay, 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, we're 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, okay, 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, okay, 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, okay, and as we're going to see this  is going to somehow boil down to the reversibility of whole plane SLE6, okay, 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 Hau Wu where they construct a metric on CLE4 somehow  using an argument which is related to what I'm going to show you now, okay, okay, 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 let, I'm going to define a stopping time  tau and tau is going to be 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, okay,  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 intersects gamma bar of sigma bar is  not equal to the empty set, okay, 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, X, 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, okay, so we're first going to check that and then the other part which  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, okay, 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, okay, right, okay,  so okay, 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, you know, 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 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, okay, right, okay, so that's the  idea and now let me remind you that we have three, well I've described two, 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 eight-thirds, medieval 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, the one which is induced by N where N is the excursion measure  for a three-half stable levy excursion, okay, so if you have a three-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, the one which is induced by taking the measure N and  then what you do is you add to it another variable T, okay, so there's a second variable  here small t, where here dT is Lebesgue measure and t is equal to the length of the levy excursion,  capital T, okay, so the 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 zero and the  length of the excursion, okay, 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,  okay, so it's just a size bias levy excursion, okay, 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 zero up to the length of the levy excursion and then dT again is Lebesgue  measure and this is the levy excursion measure, right, okay, and so here in this third construction  the marginal on x is just N weighted by the integral from zero to T of 1 over xS dS, okay,  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, okay, so this waiting is like  waiting by the distance between two typical points on the sphere, okay, okay, okay, so  okay, 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 on 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, okay, 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, okay, 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, okay, 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, okay, so I have this picture where I have drawn  one of the SLE6's, 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, okay, so this right here  is eta of t and this point down here is eta bar of t bar, okay, 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 plus infinity, sorry 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, okay, 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, is 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, x bar, b, this is equal to the law of  x bar, x, b, okay, 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, okay, so  let me, let me copy it down here, so the same picture I had a moment ago, so here is eta  bar 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 sLE6's 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 sLE6's, you are going to end up with a pair of meeting  qLE's, so 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, okay, 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 that, to  show what we needed for the first step, okay, so really the heart of the problem is just  understand this sort of strange symmetry statement for sLE6 on a levopontum gravity sphere, okay,  and so where does it come from? It's going to come from, again, the reversibility of  sLE6 and why is that? Well, so for a moment now, let's prove an analogous symmetry statement  for, analogous symmetry statement for not m2 sphd but m2 sphw, okay, 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 zero and this time, okay, 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, okay, 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  will let y bar be the surface separated from minus infinity by let's say eta from time  t up to 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 m sph law of y bar, y and q, okay, 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  an sLE6 was drawn, okay, and so to finish the proof now, we want to deduce the corresponding  thing for the law m2 sph d, okay, 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, okay, 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, okay, 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, okay, so then this becomes eta bar, first  hitting eta at exactly at its tip, okay, and if you condition on this to be a cut time,  you know, there's some work to make this precise because this is a zero probability event,  and 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, okay, 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, okay, and again the reason for this is the following,  so we draw the picture again, sorry, so let me just explain that in a little bit more  detail, so if this is the strip, then you know, 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, okay, 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 x t, if x t is the boundary length and 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 n sphw to m sphd, and so this symmetry statement  falls out, okay, 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, okay.  So let me just finish in the last few minutes by explaining a few more things, so at this  point what does one have?  You have a metric on an accountable dense set, so a sequence of iid points  on a square root of 8 thirds, leave a quantum gravity sphere, okay, 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's 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 a, 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 say, 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 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.  Okay, so maybe I'll stop there.  We don't have a responsible person, so I guess I'll ask any questions.  So where's 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, sorry, in the in the Leaville quantum gravity interact with each other,  you imagine you've grown a metric ball which goes from minus infinity to plus infinity,  this is your your filled metric ball, and what you can then do is then you can draw  the geodesics which go, you get some kind of trio 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 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 SLE6's 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 snake appearing in Levy quantum gravity.
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