 OK, so why don't I get started now? So what I'm going to do today to begin with is just give you a quick recap of what we did last time. So before, we had a Gaussian free field on a domain D in the complex plane. And for each value of gamma between 0 and 2, we could define the gamma-Leville quantum gravity measure. And what this was is it's the limit as epsilon goes to 0 of epsilon to the gamma squared over 2 times e to the gamma h epsilon of z d z. Where here, h epsilon of z, this is the average of our Gaussian free field on the boundary of the ball of radius epsilon centered at z. And d z is Lebesgue measure on our domain D. And this normalization factor that we see here is the correct one because the variance of h epsilon of z of the circle average turns out to be log epsilon to the minus 1 plus something which is bounded. And we also spent quite a bit of time last time driving the following fact. And that's that if you look at a Leville quantum gravity surface near a typical point, and by this we just meant a point z, which is sampled from the Leville quantum gravity measure, then this looks like a Gaussian free field plus gamma times the Green's function centered at z. So here, g is just the Green's function. And this is the same thing as a Gaussian free field minus gamma times the log centered at z plus a function which is harmonic. And once we have that, we then derived a particular object called a quantum cone. And a quantum cone is just what you get when you zoom in in this construction near z in the appropriate way. And I don't want to go through the definition of the quantum cone again because it's not going to be important for what we're going to do now. But let me just say that slightly more per-slicy, a quantum cone is the same thing as a whole plane Gaussian free field minus gamma times the log function where you fix the additive constant in the whole plane Gaussian free field in the right way. And in terms of random planar maps, when gamma is equal to the square root of 8 thirds, if you take a quadrangulation of the plane, then this converges in the limit to the Brownian plane. And the Brownian plane is what's going to correspond to the quantum cone. And then the last thing that we did last time is I very briefly explained what a quantum sphere is. And a quantum sphere is constructed from a quantum cone by pinching off a bubble somehow. So somehow the picture here is that you have something which is supposed to be like your quantum cone, which is a surface which is homeomorphic to the plane. And on it, you have a marked point. And what you can do is you can condition on the event that near this marked point, you actually have a large bubble. So the picture somehow looks like this. And if you condition on the mass of this bubble relative to this bottleneck being very large, then what you get in the limit is a sphere, the quantum sphere. But I don't want to go through the definition of the quantum cone or the quantum sphere right now, because it's not going to be, again, super relevant for what we're going to do today. So today, what we're going to do is we are going to look at SLE curves on Leville quantum gravity surfaces. So we're going to look at the relationship between Schramm's SLE and Leville quantum gravity. And before I jump into it, I want to explain to you what the general idea is. So the general idea is that, and this is sort of what you do in all of this stuff, is you just make a guess. So you make the following ansatz. And that's that Leville quantum gravity plus SLE, these things describe the scaling limit of random planar maps. So RPM is my acronym for random planar maps decorated by a statistical physics model. So you imagine that you have a random planar map, and you can imagine that on top of it, you have something like percolation. You could have the easing model, et cetera. And somehow SLE and Leville quantum gravity are the right thing to describe the large scale behavior of something like this. And then what you do is you just sort of see what properties your discrete models have. So by analyzing them, you see that certain things are obviously true. And then once you've sort of determined what the natural properties of the discrete models are, what you can do is you can then try to show that Leville quantum gravity and SLE, these things satisfy the same properties, or at least the continuum analogs of the same things. And this is very important to keep in mind when we think about what we're going to do in a moment, because there are many things that seem very, very strange that happen to be true for Leville quantum gravity if you think about what's going on purely in terms of the way that things are defined. But if you instead imagine these things somehow as describing scaling limits of planar maps with statistical physics models, then these properties that we're going to see are not miracles, and they seem like they're very natural. And this is what allows one ultimately to start to make the connections between, well, scaling limits of planar maps and Leville quantum gravity and SLE. OK, so to make this a bit more concrete, I want to give you two examples that I want you to have in mind. So in the first example, I want you to imagine that you have two independent quadrangulations of the upper half plane. So you have a picture that looks something like this. So you have one quadrangulation. And this is a quadrangulation of the upper half plane, so it has boundary. I'm going to assume that it's actually simple boundary. And so you might have a quadrilateral that looks like this. You could have another one that looks like this, maybe one that looks like this, et cetera. And I have a second one over here. And maybe this one has a quadrilateral. It could look something like this, et cetera. And then what you can do is you can glue them together. So you can glue these things together by identifying edges along their boundary. So you can glue these things together along their boundaries. And what do you get when you do this? You're going to get a quadrangulation of the upper half plane with a distinguished path. OK, so here in this picture, I'm not gluing the entire boundaries of my quadrangulations. I'm only gluing, let's say, the positive ray of this quadrilateral, this quadrangulation, so everything to the right of here, together with the negative ray of this quadrangulation. So everything to the left of here. And when I glue these things together, I get a new quadrangulation of the upper half plane. And I'm going to get a distinguished path. And it will look something like that. And if these two quadrangulations that I picked, these are uniformly random quadrangulations of the upper half plane, then the path is going to be a self-avoiding walk, in some sense, an infinite volume limit of a self-avoiding walk. And so now on Z2, on a planar lattice, it was conjectured by Lawler, Schremm, and Werner that the self-avoiding walk is supposed to converge to SLE 8 thirds. So it's natural to expect that the scaling limit of this self-avoiding walk on a random lattice is related to SLE 8 thirds in some way. And you can also make the guess that the glued quadrangulation, this converges to some form of square root 8 thirds lethal quantum gravity. And so what's happening over on the discrete side is you have your glued quadrangulation over here on the right. And because of the way that this was constructed, if you draw the self-avoiding walk on top of the glued quadrangulation, then it cuts it into two sub-quadrangulations which are independent of each other, because that's exactly how we constructed the right-hand picture. And so what this tells you is that if you believe that random planar maps are related to square root 8 thirds lethal quantum gravity, and the self-avoiding walk is related to SLE 8 thirds, then you should be able to square root 8 thirds lethal quantum gravity surfaces into independent chunks, into independent surfaces using SLE 8 thirds type paths. And so as we're going to see a little bit later, it's going to be perfectly natural to take a lethal quantum gravity surface where gamma is equal to the square root of 8 thirds and then cut it with an SLE 8 thirds. Why do you expect that the glued quadrangulation of sub-quadrangulation is too much? Well, because it's a gluing of two uniformly random quadrangulations. And if you believe that, I mean, this is all very heuristic, but if you believe that the two sides are converging to square root 8 thirds lethal quantum gravity, then somehow this is the natural welding, would be the discrete analog of the natural welding that you're going to have. But these are all just heuristics and motivations for continuum statements we're going to work towards proving later on. Yep. And if you glue just your stuff from one and you glue from the middle, would you expect to get a finite ECD? No, if you take one, so there are many different gluing operations that you can perform and the different gluing operations are going to give you different types of SLEs. So one possibility is that you can take one quadrangulation of the upper half plane and you can assume it has a marked edge like this and you can also glue this way. And when you glue this way, you're also going to get a square root 8 thirds lethal quantum gravity surface. The difference is that the distinguished path is not going to be a chordal SLE. It's going to be a whole plane SLE from 0 to infinity. So there are, and there are many different variants of this. So this is like the, that's like the, sorry. If you glue everything, but if you just glue a finite part of it, would you get a finite SLU? Oh, yeah, yeah, sure. That's right. Yeah, yeah. So any of these gluing operations will produce something like an SLE 8 thirds lethal quantum gravity surface, right? And then all of the discrete paths are just going to be variants of self avoiding walks. OK, and there are various statements of this type that you can actually prove, but I don't want to go into that now. I just want to talk about this at the level of heuristics to motivate the continuum theory. OK, so that's the first example. The second example is going to be the one which is actually going to be much more important with regards to the construction of the metric for lethal quantum gravity. So here's, actually let me do this on this side. So here's the second example. So here what I want you to imagine is that you don't have a quadrangulation of the upper half plane, but you have a triangulation of the upper half plane. So maybe it looks something like this. And well, this is the boundary of it. And then what I'm going to do is I'm going to color the positive boundary ray. I'm going to make these vertices blue. OK, so I'm going to assume that I have a marked edge on the boundary of my triangulation. And to the right of it, I'm going to color all of my vertices blue like this. And then I'm going to color the vertices which are on the negative boundary ray yellow. So I get something that looks like this, et cetera. And the vertices in the interior, I'm going to color by performing site percolation. So I color the vertices inside of the triangulation either blue or yellow with equal probability a half. And then let's just imagine what happens if we start to explore this. So I'm going to start doing my percolation exploration from the interface between yellow and blue. And maybe the first triangle that I see, it might look like this. That's possible. And maybe this is a blue vertex, let's say. And I'm going to keep blue always on the right. So I'm going to turn this way. And then maybe in the next step, I see a triangle that looks like that. This is a yellow vertex. And then maybe I see one that looks like this. And I have another yellow one, which means my percolation does this. But every once in a while, you're going to see triangles that look like this. And when this happens, I'm going to cut off an entire region from infinity with this percolation exploration. So you get something like that. And as you continue exploring, this is going to happen a lot. So it could be that you see another triangle that looks like that. This could be blue. And you continue. And maybe you see one more of these big triangles, let's say on the left-hand side, et cetera. And if you choose your random planar map model correctly, so it's a uniformly random triangulation of the upper half-plane, then as you explore your percolation, then the disks that you cut out. So these regions here, these sort of chunks of triangulation. Actually, let me say that the triangulations cut out, these are going to be independent. And they're going to be independent triangulations of the disk. And this just follows from the fact that if you set things up properly, the law of the unexplored region where you haven't been yet doesn't change as you start to explore your percolation. And not only that, but it's also going to be true that the net change in boundary length when you perform this exploration, this will have IID increments. Because again, you don't change the law of the unexplored region once you reveal a triangle. So now you start to make guesses. And first of all, the percolation, this should converge to SLE6, to an SLE6 curve. And the triangulation, this should converge to a square root 8 3rds, levial quantum gravity surface. And actually, we know that this is going to be a particular type of levial quantum gravity surface. This will be the quantum wedge, which is coming a little bit later. And what this tells you is that if you draw, if you put, let's say, explore square root 8 3rds, levial quantum gravity with an SLE6 curve, then first of all, what's that going to look like? So you have some kind of half planar surface. You draw on it your SLE6. And as the SLE6 goes, just like the percolation cut out holes, this is also going to be cutting out a bunch of holes. Then it should be true that the holes that are cut out are a Poissonian collection of quantum disks. So this is just the continuum analog of a triangulation of the disk. So this is the continuum analog of the first observation that I made. And the second observation in the continuum translates into the guess that the net boundary length change, this is going to be a levy process. This is some kind of a levy process. Oh, so I haven't defined. So I'm going to define in a moment the way that you measure length. So there's going to be a quantum analog of the area. Sorry, there's going to be a boundary length analog of the area measure that's going to come in just a moment. And if you know these things very well, so if you're sort of an expert on random triangulations, then you know sort of the exact tails for how these disks behave as you do this type of peeling exploration. And it's very natural to guess that in the setting of percolation, not only is it going to be a levy process, it's a very particular kind, it's going to be a 3 half stable levy process. You could even guess exactly what type it's supposed to be. And what I'm going to do today for you is I'm going to prove some of the basic math that goes behind making statements like this precise. And if I have enough time by the end, I will hopefully arrive at this statement here, which is that the holes really are conditionally independent quantum disks. And you will see the levy measure for a 3 half stable levy process appear. So that's kind of what we're aiming at. OK, so now before I start to actually make precise statements and prove things, I want to take a step back, because probably not everybody here is an expert in SLE. So I'm going to do a very quick review of SLE. We're going to review just enough of it so that we can do the calculations necessary to prove some of these results. OK, so let me remind you a little bit about SLE. So this is a random fractal curve. And it lives in simply connected subsets of the plane. And actually the way that, well, subsets of the plane, and it's going to connect two boundary points. And following Schramm's original definition of it, it's much nicer to take your domain just to be equal to the upper half plane. And we're going to take our two boundary points to be 0 and infinity. And so how is it defined? So you imagine that you have your SLE, whatever that is. And this is some curve, which is living on the upper half plane like that. So let's suppose that eta is my SLE kappa. And if I run it up to a given time t, then I can always apply, I can always uniformize the complement of the curve with a conformal map g sub t. So g sub t is going to be a conformal map, which goes from the complement of my curve back to the upper half plane. And it's going to be normalized so that it looks like the identity at infinity. So if you have your curve, you can define this family of conformal maps. And the point is that there's a theorem due to Lovner. And what Lovner's theorem tells you is that these conformal maps satisfy a differential equation. It's a very simple one. So the time derivative of the conformal map is equal to 2 over the conformal map minus something called w of t, where the initial condition is equal to the identity map. And here, the input into this differential equation is w of t. And this w is just a real valued continuous function. And how is it related to the curve, where w of t just turns out to be the image of the tip of the path at time t? And so what this theorem gives you is you have this correspondence, well, approximately, between curves in the upper half plane and continuous functions, which are real value. And what is SLE? SLE is just going to be what you get when you take your function w of t to be equal to square kappa times b sub t, where b is a Brownian motion, a standard Brownian motion. OK, and let me just remind you of one last fact about SLE before we continue. And that's that its behavior depends strongly on kappa. And so it's simple when kappa is between 0 and 4, including 4. It becomes self-intersecting, but not space filling for kappa between 4 and 8. And finally, it's a space filling curve when kappa is greater than or equal to 8. OK, but for what we're going to do, I mean, to actually prove the statements about relating SLE and Lieville quantum gravity, we're just going to need to know the definition. It's just going to follow directly somehow from the definition of SLE in terms of the Lovner equation. And of course, the importance of SLE is that it was developed to describe scaling limits of statistical physics models like the self-avoiding walk in percolation that I described earlier. OK, so in order to make these statements precise, I need to now tell you about Lieville quantum gravity services with boundary. And this is actually parallel with what we did in the case of Lieville quantum gravity services without boundary. So there are sort of two important examples. There's the one that corresponds to what you get when you take a quadrangulation of the upper half plane, which remember converges to the Brownian half plane. And this is going to be, on the Lieville quantum gravity side, the quantum wedge. So this is sort of the infinite volume surface. And then the finite volume one is going to be what corresponds to a quadrangulation of the disk in the limit, which on the Brownian side is the Brownian disk. And on the quantum gravity side, this is a quantum wedge. Sorry, quantum disk. OK. And the way that these things are constructed, I'm not going to torture you again with the full details of that. But they're going to be constructed in the same way as the cone, so as the quantum cone and sphere, except we're going to use the free boundary Gaussian free field as the starting point. And I'm going to review the basics of the free boundary Gaussian free field in just a moment. So just to be slightly more precise, a quantum wedge, this is just going to be what you get when you describe the local behavior of a Lieville quantum gravity surface near a boundary typical point. And a quantum disk, there are different ways of constructing quantum disks. But a quantum disk can be constructed by pinching off a bubble, pinching off a disk in some sense from a wedge. So these are the objects which are actually going to appear in the more precise continuum analogs of the discrete statements I described a moment ago. But in order to, so now what we need to do in order to start to work towards proving these theorems is talk about the Gaussian free field with free boundary conditions. So what is that? It's going to be defined almost in exactly the same way as the ordinary Gaussian free field with zero boundary conditions. And so the way it works is that if you have a domain in the complex plane, I'm now going to take my space h of d to be the Hilbert space closure of those functions, which are c infinity inside of my domain with respect to the same inner product as before. So I have the Dirichlet inner product. And the free boundary Gaussian free field, h, is just given by the sum alpha n phi n, where the alpha n's are iid normal 0, 1. And the phi n's are an orthonormal basis of this space. So that's how you, that's the construction of the Gaussian free field with free boundary conditions. Let me make one quick remark. And that's that this construction, this is the same as the Gaussian free field with zero boundary conditions or Dirichlet boundary conditions. The only difference is that we take the closure of a different space. So we take the c infinity functions in place of the compactly supported c infinity functions inside of our domain. OK. So the definition is quite simple. But there are some really annoying things that you have when you work with the Gaussian free field with free boundary conditions as opposed to the Gaussian free field with zero boundary conditions. And that's that the free boundary Gaussian free field, h, this is only defined modulo a global additive constant. And the reason for this is that the constant functions are in c infinity of d and they have norm 0. So for this reason, the free boundary Gaussian free field actually doesn't have well-defined values. So this means that you can't integrate it against a smooth function and have it make sense. But you have to restrict your space of test functions. In particular, the integral of the free boundary Gaussian free field against a test function, this is going to make sense only when your function phi is mean 0. So this just means that the integral of phi is equal to 0. So kind of what's going on is that the free boundary Gaussian free field doesn't have well-defined values, but it does have well-defined differences. So you can integrate, you can make sense of the difference of the integral of the free boundary Gaussian free field against two test functions. And that's translated into this statement here. OK, I'm coming to it right now. That was exactly where I'm going. So equivalently, the free boundary Gaussian free field h, this is Gaussian. And the covariance of h integrated against phi and h integrated against psi, this is going to be the integral of phi of x times g of xy times psi of y, where here g is going to be the Green's function with Neumann boundary conditions. And here the test functions that you're allowed to integrate against are mean 0. You have mean 0 test functions. So let me make this a little bit more explicit. When you work on the upper half plane, which is the case that we're going to be interested in, so when d is equal to the upper half plane, this function g is just equal to minus the log of x minus y minus the log of x minus y r. So this is just in contrast with the zero boundary case. So in the zero boundary case, the Green's function is minus the log of x minus y plus the log of x minus y. OK. So that's the free boundary Gaussian free field. And just like the regular Gaussian free field, this has some nice properties, which are easy to derive. And the first one, which is going to be very important for what we're going to do, is that it's conformally invariant. And what this means is that if you have domains d and d tilde in the complex plane, a conformal map, which goes from d to d tilde, and a free boundary Gaussian free field on d tilde, then h tilde composed with phi, this is going to be a free boundary GFF on d. And this is proved in exactly the same way that the conformal invariance of the zero boundary Gaussian free field is proved. It just follows because you can check that this Dirac-Ley inner product up there is conformally invariant. OK. And the free boundary Gaussian free field also satisfies the Markov property. And in this case, what this means is that if you have an open subset of your domain, then you can write h as the sum h1 plus h2, where h1 is a zero boundary Gaussian free field, h2 is harmonic, sorry, zero boundary Gaussian free field in U, h2 is harmonic in U, and h1 and h2 are going to be independent of each other. OK, so this is exactly the same Markov property as in the zero boundary case, except for one very small thing. And that's that you have this slightly annoying issue that you're only working modulo and additive constant. And in this case, this translates into the fact that h1, this is going to have well-defined values, but h2, this is only going to be defined modulo additive constant. And the Markov property for the free boundary Gaussian free field is also proved in the same way as it is for the zero boundary Gaussian free field. I don't want to go through it now because of time constraints, but if you want to see the proofs of any of these things, you can always ask me and I'll explain later. OK, so this is the object that we're going to be working with. And this is, again, the starting point for constructing the Liouville quantum gravity analogs of scaling limits of planar maps with boundary. And since we're going to be working with planar maps with boundary, we're going to need a boundary measure. And so now let me explain to you what the Liouville quantum gravity boundary measure is. So I want you to imagine that you have a domain in the complex plane. And I'm going to assume that the boundary of the domain contains a linear segment. So the picture that you should have in mind is that you have a domain and maybe it looks something like this. So this is d and this right here is l. This is the linear segment. And what is the Liouville quantum gravity boundary measure? This is going to be the measure nu sub h, which is the limit as epsilon goes to 0 of epsilon to the gamma squared over 2 times e to the gamma h epsilon of z divided by 2 times dz. And here, what is h epsilon of z? This is going to be the average of h, not on a circle, but on the semicircle, which is the boundary of the ball centered at z of radius epsilon intersected with d. So this looks something like this. So this might be your semicircle. And dz is the big measure on l. Yep? No, epsilon to the gamma squared over 4. Ah, sorry, epsilon to the gamma squared over 4. Thank you. Now, the reason that you see epsilon to the gamma squared over 4 for the normalization term and you have an extra factor of 1 half here that did not appear when we constructed the Liouville quantum gravity measure is that for the free boundary gas in free field, when you're near the boundary, the variance is different. So it's going to be true that the variance of the semicircle average is not going to behave like the log of 1 over epsilon. It's going to behave like 2 times the log of 1 over epsilon. And that's just reflected here. So let me just record that. So the variance of h epsilon of z, this is going to be equal to 2 times the log of 1 over epsilon plus something which is bounded. OK. Great. So let me just tell you a few more quick things. And then maybe we'll stop for a break before doing the SLE stuff. So when you construct the Liouville quantum gravity boundary measure, it's very convenient to assume first that you're working on a linear segment because when you work on a linear segment, it's easier to define things like the semicircle average and control its variance. But once you've constructed it on linear boundary, then you can define it on general boundary just by applying conformal mapping. Now, one other kind of annoying thing that's sort of underlying this is that if we're working with a Gaussian free field with free boundary conditions, then this implies that our field h is only defined modulo additive constant. And so this measure that we've defined, new h, this is only defined modulo multiplicative constant. So this is kind of an annoying issue that comes up in some places. So it's important to keep this in mind. Let me point out two more things. First of all, if you run through the same proof that we did last time, then you can see that near a boundary typical point, so in other words, near a point sampled from the boundary measure, what does the field look like? The field looks like a Gaussian free field with free boundary conditions plus gamma over 2 times the Green's function at that point. And because the Green's function near the boundary doesn't look like the log function, it looks like twice the log function. This is the same thing as the Gaussian free field minus gamma times the log centered at z plus something which is harmonic. So that's what happens near the boundary. So you still have a gamma log singularity. And so again, I don't want to torture you with the definition of a quantum wedge. But let me just say that the way that you construct the quantum wedge now is that you just zoom in near a boundary typical point in the right way. So the quantum wedge is just going to be what you get when you zoom in near z in the right way. And in other words, I think I said this earlier, the quantum wedge is the same thing as a free boundary Gaussian free field with the right log singularity where you fix the additive constant in the right way. But I don't want to go through the details of that because the exact way it's done is not going to be that important for what we're going to do what we're going to do next. OK, so maybe now we stop for a short break before diving into the relationship between LiVo quantum gravity and SLE. OK, so let me get started again. So now what I'm going to do is I'm going to prove to you the theorem which is this is really the first connection between SLE and LiVo quantum gravity. And what you're going to see in a moment is that this is what's going to lead to the fact that when you put an independent, let's say, SLE 6 on LiVo quantum gravity, the holes that it cuts out are actually disks, quantum disks, and they're independent of each other. And this is a theorem which is due to Scott Sheffield. And he calls it the quantum gravity zipper. And let me take a moment to explain. I'm first going to state the theorem. And then I'm going to take a moment to explain what the actual interpretation is. So you start off with a free boundary Gaussian free field, H. And then I have an SLE. So eta is going to be an SLE kappa process. And it's going to be independent of H. And it's going to go from 0 to infinity. And I'm going to call it Sleven revolution G sub t. And throughout, I'm also going to let F sub t be equal to G sub t minus W of t. So the reason for introducing F sub t is that it's kind of nice, because then if you apply F sub t to the tip of the path, you're going to get 0 instead of W sub t. OK, so what does the theorem say? So I'm first going to write down the formula, and then I'm going to explain it with the picture. So what it says is that for each t greater than or equal to 0, H composed with Ft inverse plus 2 over square root kappa times the log of Ft inverse plus q times the log of Ft inverse prime. This is equal in distribution to H plus 2 over square root kappa times the log function. And here, what is q? q is 2 over gamma plus gamma over 2. And gamma is going to be the minimum of the square root of kappa and 4 over the square root of kappa. So that's the theorem statement. Now let me try to make this more concrete by explaining it with a picture. So what you imagine is that you have the upper half plane like this. And on the upper half plane, you have your free boundary Gaussian free field plus 2 over the square root of kappa times the log function. And I'll explain in a moment what the 2 over square root kappa is supposed to be telling you. And you have this Gaussian free field, and then you sample your SLE eta independently of the Gaussian free field. So this is now an independent SLE, and you draw it up to some time. Let's say this is eta of t. And then what you do is you imagine that you cut along your SLE and then map back. So you can apply the map F sub t. And F sub t just takes the complement of the SLE back to the upper half plane. And then what you do is you consider the quantum surface which corresponds to the complement of the SLE. And what is that? By the change of coordinates formula, it's h composed with f t inverse plus 2 over square root kappa times the log of f t inverse. And then you need the correction, which is q times the log of f t inverse prime. And what the theorem is telling you is that if you start off with your free boundary Gaussian free field with the right log singularity, and then you cut along an independent SLE, then the law of the unexplored region, which is what's described by this field here, is exactly the same as what you started with. So it doesn't change. So let me write that down. So this just tells you that the law of the unexplored region, which is what the yellow stuff corresponds to, this is invariant under the operation of cutting by an SLE. And if you haven't seen this before and you think about this purely from the Gaussian free field side, this is quite a striking theorem because it tells you that if you start off with a free boundary Gaussian free field and you cut it with an independent SLE, then what you get when you map back miraculously is again a free boundary Gaussian free field on the other side. And that's kind of weird because you would guess that maybe somehow the fractal behavior of the curve would give you some other type of boundary behavior. But it turns out that if you match the parameters up properly, you get exactly a free boundary Gaussian free field again. And this is very natural to expect. And this is very encouraging. If you think that this particular picture is somehow describing the scaling limit of a random planar map with a distinguished curve, because this is exactly the property that the discrete model has. OK, so that's the theorem statement. And what is the 2 over kappa? What is this sort of telling you? This term, 2 over the square root of kappa times the log, this is basically encoding the typical behavior of the tip of the SLE curve at a typical time. So what do I mean by that? So in this formula here, you can always solve for the derivative of the conformal map, because you can just represent that as a difference of two Gaussian free fields. And somehow this 2 over square root kappa log term allows you to determine how the derivative of the conformal map is behaving near the tip of the path. And so that's the reason that you have this extra term is that somehow the tip is sort of special. You have some special behavior there. OK, does anybody have any questions about? This is a very important theorem, and this is sort of what underlies everything in some sense. And again, where it's coming from is that the way that you think of this is that not in terms of the formulas, but rather in terms of just being continuum versions of operations that you can perform on planar maps. Just one question. So before you started to motivate these kind of things with independent calculation, for which very special values of kappa. And so what you're saying is that even for models with more dependency, this decoupling would work. Yeah, that's right. So you just changed the value of kappa for your model. So the story that will correspond, for example, to the easing model, you're going to have kappa equal to 3 or 16 over 3, and it will just change the value of gamma. And still, everything is exactly the same. Or for, you can actually get all of the different values of gamma between 0 and 2 with a different natural statistical physics model. So for example, the FK models will give you all values of gamma between the square root of 2 and 2 with the square root of 2 corresponding to the uniform spanning tree. And then there are other natural models which will give you gamma below the square root of 2. And you see exactly the same thing in all of them. Does anybody have any other questions about this? OK, great. So what I'm going to do now is I'm going to just give you the proof of this, because this is actually very nice. And it's surprisingly simple to prove, because all that we're saying here is that this distribution here on the left-hand side, this just has the Gaussian distribution with a certain mean, that's in some sense what the theorem statement is. So to prove this, what do you have to do? Well, first of all, to light notation, I'm going to let h tilde be my free boundary Gaussian free field plus 2 over the square root of kappa times the log function. And using this notation, what we want to show is that h tilde composed with ft inverse plus the coordinate change correction, this is equal in distribution to h tilde. And because we're trying to show that the left-hand side here is just the Gaussian distribution, in order to show that something is the Gaussian distribution, all that you have to do is show that for each test function phi, so for each mean 0 test function phi, you know that h tilde composed with ft inverse plus q times the log of ft inverse prime integrated against phi. You just need to show that this has the Gaussian distribution with the right mean and variance. This needs to be a normal random variable with mean mu of phi and variance sigma of phi squared, where here mu of phi is the integral of 2 over the square root of kappa times the log. And the variance sigma of phi squared is equal to the integral of phi of x g of x y phi of y dx dy, where let me remind you that of what g is. So g of x y is equal to minus the log of x minus y minus the log of x minus y bar. So we just have to check that this is true for each fixed test function. And once we've done that, then that determines the Gaussian distribution. And it shows that we have the correct Gaussian distribution. So the log singularity doesn't contribute to the line? Yeah, that's right. Yeah, that's right. OK. And the proof is just going to be by Ido calculus. I'm not going to do the final part of the calculation. I'm just going to show you how you set it up. And it will be to the point where all that you have to do is calculate some Ido derivatives and see that something is a martingale. And it is. And that's the proof. So how does that work? So again, we're just going to prove this using Ido calculus. And right. So I think it's a little bit annoying here when you try to do calculations with Ido calculus, is that you have this FT inverse map. And we want to make this a bit simpler. So it's convenient to replace the forward Lovner flow, which are these maps, F sub t, with the reverse Lovner flow. Because then our Ido derivatives are much easier to calculate. And what is the reverse Lovner flow? So this is just exactly the same differential equation except for you just reverse time. So the time derivative of g tilde of t is going to be minus 2 over g tilde of t of z minus w tilde of z with initial condition given by the identity. And here I'm going to take w of tilde to be equal to square root kappa times a Brownian motion. And then one can check. This is just a small exercise using time reversal symmetry that g tilde of t is equal in distribution to gt inverse. And this holds for all t. And if you let f tilde of t be g tilde of t minus w tilde of t, then f tilde of t is equal to f t inverse for all t. And so just to make the quantifiers clear, this just says that for each fixed t, f tilde of t is equal in distribution to f t inverse. OK? So this is just like a forward reverse symmetry. It's just sort of a deterministic fact about love revolution plus some time reversal symmetry for Brownian motion. But this is much nicer now because the differential equation for g tilde of t is much nicer than the differential equation for gt inverse. All right, and this is also a very common trick in SLE whenever you want to work with the inverse map. It's much better to work with the reverse love ner flow instead. OK, and so what do you do? So what we want to show now is that h tilde composed with f t tilde plus q times the log of f t tilde prime integrated against phi. This is a normal random variable with mean mu of phi and variance sigma of phi squared. And to show this, we're just going to calculate the characteristic function. So let's calculate e to the i theta times h tilde of t composed with f tilde of t plus q times the log of f tilde of t prime integrated against phi. And we just need to show that this characteristic function takes the correct form for the Gaussian distribution. And now what you want to do is you want to use the conformal invariance of the free boundary Gaussian free field. So what I'm going to do is I'm going to condition out a sigma algebra, which is given by the driving process for the reverse love ner flow up to time t. And with this sigma algebra, what do you know? So you know that 2 over the square root of kappa times the log of f t, this and q times the log of f tilde of t prime. These are f t measurable because if you observe the Brownian motion at the time t, then you can construct the conformal map. And you also know that given f sub t, h, this h that you're going to see here, this is a, hold on just a second. So you also know that if you condition on f sub t, then h tilde composed with, sorry, sort of this. So what you know is that given f sub t, h integrated, h composed with f sub t integrated against phi. OK, so h, remember, is h tilde, but without the 2 over square root kappa times the log. This is going to be a normal random variable. And it's going to be a normal random variable with mean 0 because our test function is mean 0. And it's going to have variance, but I'm going to call sigma of phi squared of t, where sigma of phi squared of t, this is equal to the integral of phi of x g t x y phi of y dx dy. And what is g sub t? g sub t is just the Green's function for the free boundary Gaussian free field on the domain minus the SOE. So here, so g sub t of x y is equal to g of f t of x, f tilde t of x, f tilde t of y. Sorry, there should be a tilde here. And this is the Green's function for the free boundary Gaussian free field on h minus eta up to 10 t. And so what this means is that this equation here star by conditioning on f sub t, you see that this is the same thing as the expectation of e to the i theta. And the Gaussian part, the part that's Gaussian gives of f sub t disappears. And that's going to become minus theta squared over 2 times sigma sub phi squared of t. And then the part which was f t measurable remains, and that's 2 over square root kappa times the log of f tilde of t plus q times the log of f tilde of t prime integrated against phi. So that's why that didn't quite fit. So you do this calculation, and you get this formula here. And now everything is explicit because the Green's function is explicit because f tilde of t satisfies a very nice STE. And everything else is just in terms of f tilde of t. And so you can calculate the e-to derivative of what appears inside of this expectation. And the way that the result is proved is you just show that what you see inside of this expectation, so this whole thing, is a martingale. So the quantity inside of the expectation is a martingale. And to do that, again, it's just a couple of applications in Edo's formula. It's not something that you want to see somebody do on the board. But you can find all of these calculations in Sheffield's paper on this on the quantum gravity zipper. You could perhaps say that the first term is the martingale. The one in front of teta. And the second term is the quadratic variation. Yeah, exactly. That's right. So what Bertrand is pointing out is that this is really just an exponential martingale. And this term here happens to be a martingale. And this is just the quadratic variation of that martingale. And you just check that. And then you're done. Now, if you've also seen people talk about the flow lines of the Gaussian free field or the level lines of the Gaussian free field, the way that you construct those and couple them with the field is essentially exactly the same proof. It's exactly the same calculation as this here. OK. Right, so that's the quantum zipper. Let me make a couple of remarks about this before I derive at the very end the relationship between SLE6 on leave of quantum gravity and 3 half stable Levy processes. So actually, the story is not really done after you've proved this theorem for a few reasons. Because remember, I said that we want to think of this as a continuum analog of graph gluing, planar map gluing. But there are a few things that are sort of wrong with this theorem statement if you want to interpret it in this way. So this is in the case that, let's say, capital is between 0 and 4. And let me just point those out. So you need to show a few things. Number one, you want to show that in this picture where you have your quantum gravity surface and you have your SLE on top of it, you want to somehow say that this is a welding according to quantum length. So what does that mean? Well, you can take this picture and you can take the left-hand side and you can map it down and you get something like this. And you can take the right-hand side and you can map it down and you get something like that. And what you want to think of this picture is somehow a welding of these two sides like this. And one thing, which is actually not at all obvious and it's something that you have to prove is that this really is a welding in the sense that if you take an interval here, a piece of the curve, this is going to get mapped to two intervals under these two maps. You're going to have an interval over here and you're going to have an interval over here. And you want to say, for example, that the quantum length of this interval is equal to the quantum length of that interval. So in other words, if you measure length from the left-hand right-hand sides, you want to say that these things are actually the same. So you need to show that lengths as measured from each side are the same. So that's one issue that you want to deal with. The other quite serious issue is that here we have the wrong time parameterization. So when I described in the statement of Sheffield's theorem this cutting operation up there, the time that we're using is the capacity time for the curve. And capacity time is not good. And the reason is that it depends on the embedding, in some sense, of the surface. So it's somehow not an intrinsic notion of time. So for example, if you take that picture and you scale it by the factor 2, then the capacity is also going to scale. So that's not the right notion of time because it depends on how you chose to embed the surface into the upper-half plane. And so what you want to do, instead of using capacity time, is you want to show that there's a statement where you replace capacity time with quantum length time. Because quantum length is the intrinsic notion of time. So you want to say that somehow you have a version of the quantum zipper where you're not cutting according to capacity time, but you're cutting according to a given amount of quantum length time. That's another kind of issue. And then the last issue that I want to point out is that you also want to show that you can reconstruct the picture, which has the SLE on it, from the two sides. So what do I mean by that? So if you see this picture up here where you have the whole surface and the SLE on top of it, then you know how to reconstruct the left side and the right side because you just can formally map the two sides away. And one other question you can ask is if you just observed the field down here and the field down here, is it possible to reconstruct the top picture, including the SLE, in a unique way? And of course, you can do that with planar maps because if you glue two graphs together, you get a new graph and you know exactly what it is. And you can ask in the continuum, is this welding operation well-defined in that sense, too? So lastly, can you reconstruct surface with SLE from the two sides? And yeah, so all of these issues, they can be addressed and dealt with. And the answer to this last question is yes. And this is all explained in Scott's paper on the quantum zipper. I don't want to focus on them in the remaining time today because the theory, which is actually most relevant for constructing the metric on L'Eville quantum gravity, is understanding what happens when you draw SLE curves when Kappa is bigger than 4 on top of L'Eville quantum gravity. So in particular, SLE 6. So let me not say more about these things, but let me point them out that these are actually quite serious issues and it does take a little bit of work to deal with each one of them and get the correct statement. And let me also say that once you get sort of the correct statement, you translate the capacity statement into the quantum length statement. You end up with something which involves quantum wedges rather than free boundary gas in free fields because somehow you need to fix your additive constant, really, even to talk about quantum lengths. So now let me finish in the last 20 or so minutes today by explaining. Yep. Yeah, the question about this. Under this picture, you have two sides that you understand to explain this picture. Yep. To object to quantum ratio to the half length. Yep. And here in the theorem of the ZIPA, you have just one, you know, the right. Right. Well, it depends. So you have two options. So you can draw this up until a fixed amount of time like I did there. And this picture is a bit deceptive because I drew an arrow here. But what I meant to do was draw this all the way out to infinity. And then you cut it into two pieces. So in the quadrangulation side, this is like giving you the UI HPQ, yes? Yeah. But you're feeling this for a finite time. You're just like stated about the quantum data. You have a version which is equal to infinity. Yeah, so I mean, so first of all, it's much more, OK. So in the graph case, you can imagine that you have, I mean, there is still a finite time analog in the quadrangulation case. Because if you have a quadrangulation of the upper half plane with a self-avoiding walk along it, let's say, you can always cut to a finite time and then unzip to there. And that will also leave the picture invariant. So the analogy of this sort of version of the quantum zipper is there. But you also, of course, have, perhaps in a more natural way, the analogy of the form of the quantum zipper where you draw it all the way to infinity and then you just ask about the two sides. Right, OK. So let me now, I want to, in the last 20 minutes, somehow make things like 3 half stable Levy processes appear out of SLE 6 on Leville quantum gravity. OK, and more generally, this is going to apply for the case when kappa was between 4 and 8. So let me also say that in the case when kappa was between 4 and 8, this statement up here, there's nothing special about this statement when kappa was between 4 and 8. It's just that this curve is intersecting itself. So it's also cutting out bubbles. But you can still talk about the law of the unbounded component. But now what we're going to do is we're going to be interested in the bubbles that are actually cut out. And these are going to be a Poissonian collection of quantum disks. And what I'm going to do now when I derive this is I'm going to calculate at some point the law of a bubble being cut out. And this is actually a second way of seeing that the sort of definition of a quantum disk or a quantum sphere that I indicated is correct. So you're going to see the same definition appear but in sort of a different way. OK, so going back to the quantum zipper. So you can always think of the quantum zipper somehow as a Markov process on random fields decorated with a path. And again, the way it worked is you start off with a field, which is h tilde given by free boundary Gaussian free field plus a certain log singularity. And then what you do is you cut your free field with an SLE. So what you do is you sample your SLE Kappa process. And then you let f sub t be at centered Lovner flow. And then you replace h tilde composed with f t inverse plus q times the log of f t inverse prime. And this operation where you cut, this is what Scott refers to as the zipping down operation or unzipping. So this is zipping down. And the inverse of this operation, this is zipping up. So if you imagine that you start now with the new field that you got after you performed the unzipping operation, you can always sort of invert it. And then you go back to the original field, net zipping up. And to make that precise, what you do is you can imagine just using abstract nonsense as you can define a time stationary process, h tilde of t, eta tilde of t, where this is defined for all t and r. And you know that for each time t, this is equal to a free boundary Gaussian free field plus 2 over square root Kappa times the log. And eta tilde of t, this is an SLE Kappa in h. And the way that you can kind of construct this time stationary process to find for all t and r is you just keep applying this operation. And then you take a limit as t goes to infinity. And you get this process. And for this process, again, you have two operations that you can perform. When you go from h tilde of t, eta tilde of t, to h tilde of t plus S, eta tilde t plus S, when S is positive, then you're unzipping or zipping down. And when you invert this and you go from h tilde of t, eta tilde of t, to h tilde t minus S, eta tilde t minus S, then you are zipping up. OK, so you have this sort of infinite process. On top, you have your Gaussian free field. You have some kind of SLE. It could be hitting itself. And you can kind of do two things. You can either cut down. You can either map back or you can sort of map things in and sort of zip this thing up. OK, so now let me try to show you how you derive the Pasadian structure of the bubbles. What do you mean by you don't have access to the limit? Yeah, exactly. That's right. So it depends whether or not kappa is smaller than 4 or bigger than 4. When kappa is smaller than 4, that's right. The operation is defined just by welding. But when it's bigger than 4, there's some extra information because there are these bubbles which are going to be sort of pushed in all of the time. And what we're going to do now is we're going to describe how you make sense of the extra randomness from these bubbles. So actually zipping down is not well different. I mean, when you add the bubbles, you don't know how to. So zipping down is the one which is well defined, zipping up. Yeah, exactly right. So zipping up right now is just defined for this infinite process when you just time reverse. You just go back s units of time. So it just refers to the shift map at this point for this sort of time stationary markup process. But what we want to do now is we want to understand the law of the bubbles that sort of appear when you're doing the zipping up. We want to figure out what type of Liouville quantum gravity surfaces they are. OK. So the statistic of the bubble would be the information you need to add in order for certain things. Yeah, exactly. That's right. And it turns out that although this is quite a lot of work that goes into it, you can actually reconstruct this entire picture if you're just given the bubbles somehow, the Poissonian collection of bubbles. Yeah, and that's actually related to a big set of questions which always appears in this stuff. And that's that if you have sort of the different chunks of Liouville quantum gravity surface, which appear when you cut somehow, is it possible to reconstruct what the original picture was? And that can be a very difficult problem because it's related to what can be a very hard problem in complex analysis, which is whether or not a set is removable. OK, so one annoying thing here is that this is a time stationary collection of fields, but this h tilde of t is a free boundary Gaussian free field, which means it's only defined modulo additive constant. So let me fix the additive constant once and for all. So I'm going to fix the additive constant. And I'm going to do this by setting, I'm going to just declare the average of h tilde of 0 on the boundary of the unit ball of the unit disk in the upper half plane to be 0. So once I declare that this average is equal to 0, then the additive constant for my field at time 0 is fixed. And this, in turn, fixes the additive constant for all times. So it's always fixed. OK, and now let's see what the law of the bubbles are. So I'm going to fix some parameters, epsilon and delta. And I'm going to assume that these are very small. I will first send epsilon to 0 with delta fixed and then eventually delta should be thought of as being sent to 0. OK, now you have this picture. So this is the time 0 picture. And you have on top of it your SLE kappa between 4 and 8 process. And because the additive constant is fixed, you can, what you can do is you can perform the zipping up operation until the first time a bubble appears with area. So with quantum area greater than or equal to delta. And I'll explain what I'm going to write down in a second with a picture. And quantum boundary length of the top being equal to epsilon. So let me now redraw my picture on the other side. So you have your initial field. It looks something like that. So this is h tilde of 0, eta tilde of 0. And the way that you think of the zipping up operation is that sort of below this thing, you have a bunch of bubbles. And you're somehow going to push these bubbles in when you're sort of zipping up. So new bubbles are going to be appearing. And here I've defined a time, which is the first time when you're zipping up that you see a bubble appear so that the length of the top, which I'm going to indicate now in, let's say, orange, I want the length of the top to be epsilon. So orange is equal to top. And then I want the area of what's inside the bubble, this blue stuff, to be at least delta. Now, one question you could ask is, if there really is a time where the top of a bubble is going to be equal to epsilon, an answer is yes. Because when you perform sort of the zipping operation, you're pushing bubbles in. And as they get pushed sort of further and further in, more and more of the boundary length of the bottom sort of gets pushed into the top. So there really will be a time where you see exactly epsilon boundary length in the top. So you can do that. And now what you can do is you can look at the law of this bubble at this time. So at this time, the law of the field, let me call this just h hat. So the law of h hat in the bubble can be described explicitly or sort of explicitly. So if we look at the law of the field inside of this bubble here, given its boundary values on the top, by the Markov property for the Gaussian free field, is a Gaussian free field with given boundary values on the top. And it's conditioned on a positive probability event. And that's that the mass is greater than or equal to delta. So you can write down, well, up to not knowing the boundary values of the field along the top of the bubble, you can describe the conditional law of what's inside. And it's just a Gaussian free field with free boundary conditions down here, fixed boundary conditions here, and then conditioned on the event that it has a certain amount of mass. Now, we want to use this somehow to extract what the law of this bubble has to be. And the way that we're going to do it is that we're going to try to understand what happens when we condition on less and less information. We want to condition on somehow less and less of the boundary values on the top. Now, when you do all of this stuff, it's very convenient to perform a change of coordinates. So you don't want to work in a random domain. It's much easier if you work in a fixed domain like the strip. OK, so let me draw a picture now of what this looks like. So you have, let me just copy that picture from above. So you have your bubble that's formed. OK, and now what I want to do is I want to perform a change of coordinates, which is going to take this random domain to a fixed domain. And my fixed domain is just going to be the strip. So this is the strip. And the strip for me is the real line, the positive real line times the interval from 0 to pi. And I'm going to pick my conformal map so that the top, this orange stuff here, gets mapped to here like that. And that fixes two points on the boundary. To determine the conformal map, I need to fix a third point. And the way to fix a third point is to just choose one at random from the boundary measure down here, OK? So it's natural here to then weight the law of H hat by its boundary length, OK, by its boundary length of the bottom. And this corresponds to adding a marked point with a minus gamma log singularity centered at that point. So I've just performed actually a small change of measures, which I'm going to get rid of at the end. But this is just for the purposes of adding a special marked point, let's say, here. So this is z. And z is going to go to plus infinity. OK, and now what do I have over on the right-hand side? So over here, so H hat composed with phi inverse plus q times the log of phi inverse prime. This is what I get after performing the change of coordinates. This is going to be a Gaussian free field on the strip with given boundary conditions on the orange part, so from 0 to i pi. And because I have this log singularity here, that's going to translate into having a downward drift of gamma minus q times the real part of, say, w, OK, because when you take a minus gamma log singularity and you map it to the strip, you get sort of a downward drift of size gamma minus q, q coming from the coordinate change formula. And you have free boundary conditions on the rest of the strip, so on the strip minus the orange part. OK, so we know. You have a whole line, whole line, whole line on the top, from the fix to. No, no, it's always free, because it came from here. So this was free here. So I have. I have. Oh, I'm just imagining I'm conditioning on that. I'm conditioning on it, because this is coming from the Gaussian free field boundary values. Fixed on the orange part. Yeah, exactly. It's on the. That's right. It's fixed on the orange part. That's right. So I'm conditioning on that. And what I want to do is I want to argue that when I condition on less and less of orange, somehow the extra information you're going to get by conditioning on it is going to go to 0. So that's the idea. OK, so let me just quickly, so now since I'm a tiny bit over time, let me just quickly just explain the last point. I'm just looking at the conditional law given orange. Yeah, so I'm just looking at the conditional law given orange. So I'm fixing it. And then once you fix it, then you have just free on the rest. OK, but the important thing is that the downward drift that you have is gamma minus q. And that's what determines the sort of Poissonian structure of the bubbles. Because in this picture, it's natural to consider the process e to the gamma over 4 times a sub u, where u is equal to the average, sorry, where a sub u is the average of the field on the line with real part equal to u. OK, so you have your strip like this. You have the orange part. And you can look at this process, a sub u, where for a given line with real part u, a sub u is just what you get when you average your field on u. And a sub u, you can write down exactly what it is. It's just a Brownian motion because it's a free boundary Gaussian free field that's twice as fast. And it has drift gamma minus q times u. And the reason that it's natural to consider this process here is that you can calculate its quadratic variation. And the quadratic variation of this process, this is like e to the gamma over 2 times a sub u times a constant we don't care about. And this is approximately the same thing as the boundary length of the bubble. The reason it's the boundary length is that I have gamma over 2 here and not gamma. OK, but now people who know about Bessel processes, you know that when you take the Brownian motion with drift, because the boundary length is like e to the gamma over 2 times something like an average of the, I mean, if you do anything which is kind of like e to the gamma over 2 times some sort of regularization of the field, you're going to get something like boundary length. In this case, you're doing something kind of like that. Something kind of like that. It can actually be described as the conditional expectation of the boundary length given a sub u. That's one way of thinking about it. It's just like a conditional expectation of the whole thing. So the top and the bottom. And then I integrate this from 0 to infinity. There we go. That's better. Right. OK. That's right. Sorry about that. OK, great. Now people who know about Bessel processes, you know that when you take a Brownian motion with negative drift and then you do something like this, then this is just going to correspond to the length of the corresponding Bessel excursion. And you can calculate what the corresponding Bessel process dimension is. So this is going to correspond to a Bessel excursion with dimension delta. And delta here, if you do the calculation, is 4 minus 8 over gamma squared. And the excursion measure for this type of Bessel process is t to the delta over 2 minus 2 dt, which if you plug in this value of delta it's going to be t to the minus 4 over gamma squared dt. Now that doesn't look quite right because we somehow want to see when gamma squared is equal to 8 thirds, we want to see a 3 half stable levy process. And this does not look like the levy measure of a 3 half stable levy process yet. But the reason that it's different is that in this construction, I needed to add a new marked point. And that adding the new marked point corresponded to weighting by the boundary length. And so to get rid of it, I need to unweight. And unweighting just corresponds to dividing by t. So if you unweight, then you divide by t and you get t to the minus 4 over gamma squared minus 1 dt. And now if you plug in gamma squared equal to 8 thirds, you see exactly the levy measure for a 3 half stable levy process. And that's actually the reason that when you explore the right kind of levial quantum gravity surface with an SLE6, the boundary lengths evolve as 3 half stable levy processes. And this also is how you get the Pasodian structure of the bubbles. Because whenever you condition on orange in this picture, what I see inside of the bubble is conditionally independent of what I see sort of outside of the bubble. And so if it happens that the amount of information you get from orange tends to the trivial sigma algebra, as you send the size of the top to 0, then you see that you also get the right kind of conditional independence to have a Pasodian structure. OK. Right. So I think I will stop there for now. And sorry for going a little bit over.