Courant Events Right: But I'm gonna have the last talk of the afternoon. It's also the last talk of this
Courant Events Right: Three-day workshop. It's going to be, Rick Kedian from Yale, who's going to speak on independent covers.
Courant Events Right: Thanks, Greg.
Courant Events Right: I didn't get a mic. Oh, sorry. Oh, you need a mic? I don't know about that. Is this one working? Yeah.
Courant Events Right: Yep.
Courant Events Right: It's turned on.
Courant Events Right: Is it turned on? Yeah. This value.
Courant Events Right: Oh yeah, okay, thanks for the organizers. I didn't intend to give a talk here, but I guess there was a cancellation, and I got shoved into position, so this is the first time I give a talk on this, and it was…
Courant Events Right: somewhat short notice preparation, is it not… is it not on? It is on.
Courant Events Right: Drop another button.
Courant Events Right: Maybe the other side.
Courant Events Right: How about that? Is that… is that better? You have to press a button. It's on! It's on, apparently. I'll try to talk louder. Is that better? Okay,
Courant Events Right: Yep.
Courant Events Right: Thank you. Anyway, this is a joint work with, Catherine Wolfram, who's at Yale and, Zurich.
Courant Events Right: this year. And, you know, just to start off, you know, pretty much everything I learned in math, I learned as an undergrad. So, you know, when I was a grad student at Princeton, they just didn't have grad classes. They had to… at that time, so, you know.
Courant Events Right: All my talk is going to be linear algebra and a little bit of Fourier analysis, and if there's… if I'm using some… something you don't know, please raise your hand. It should be accessible to everyone, all right? I'm… I'm not going to talk to any experts, because there are no experts on this subject, it's a new subject. Well, okay, maybe there's one or two experts.
Courant Events Right: Okay, here, well, what's the… what's the goal, right? There are… there are certain…
Courant Events Right: stat-meg models, right? Dimer models, fantasy model, Ising model, which we know, and the reason we know them is because they're somehow free fermionic. You can use some linear algebra, determinants, faffians, and so on to study them.
Courant Events Right: And there's an interesting history of those… of understanding their scaling limits, what happens when the lattice basically goes to zero, and the, you know, many of us in the room know that, and
Courant Events Right: There are some other models for which we know less about what's still some non-trivial things about, like percolation and the fibertex model and so on, but the first three are free fermionic in the sense that they
Courant Events Right: you can study them with linear algebra. And… and so I want to define, the new
Courant Events Right: Fairly rich family of other measures that make models.
Courant Events Right: which are determinantal, and they're going to be in ZD in general, although most of the pictures you're going to see are two dimensions, and we're just getting started on three dimensions, so I won't go too far into that. And the tools I'm going to use are simply, you know, very simple tools, linear algebra and
Courant Events Right: some Fourier analysis. But, you'll see that the Fourier analysis questions we ask are kind of sophisticated, so if anybody is an expert on that, I would be happy to talk.
Courant Events Right: Later.
Courant Events Right: Okay, so let me start with the definition.
Courant Events Right: Every talk should start with a definition.
Courant Events Right: I've got a graph. It's a set… you can think of it as a binet set, or if you like, a graph, and a collection of subsets, which I'm going to call tiles, although they're going to be overlapping things, so maybe it's better to call them, like, shingles or something else.
Courant Events Right: But, let me just call them tiles, because that's what we have been calling them. And associated to this.
Courant Events Right: his aid.
Courant Events Right: Matrix, a rectangular matrix, where the rows are indexing the vertices or the elements of the set, and the columns are indexing the tiles, and it's just an incidence matrix, so 1 or 0, depending on whether the vertex is in the tile or not.
Courant Events Right: Okay, so if you like, you can just start with a rectangular matrix, forget that stuff. And what's an independent covering? It's a covering of the graph, or the set, with some subset of tiles.
Courant Events Right: And, if the graph has
Courant Events Right: you know, n vertices, it's a covering of size N, so the one tile per vertex, but the… the union of the satisfies two properties. Union of the tiles
Courant Events Right: Here's the whole graph, so you have to cover
Courant Events Right: And it has to be independent. What does independent mean? It means that the corresponding
Courant Events Right: submatrix of the incidence matrix is a full rank.
Courant Events Right: So the… basically what I'm doing is I'm… I have this matrix, I'm taking a collection of columns of the matrix, which spans the corresponding
Courant Events Right: real space, space.
Courant Events Right: Does that make sense?
Courant Events Right: Let's just do an example.
Courant Events Right: This is just the submatrix of D whose columns correspond to the tiles in F.
Courant Events Right: So D, the columns of the matrix are indexed by the tiles, the rows are indexing by the vertices, and I want to pick a collection of tiles such that the corresponding square submatrix has non-zero determinants.
Courant Events Right: Let's see, let's see an example. They play no role at the moment.
Courant Events Right: edges of the graph play no role, but that adds some geometry, and eventually we'll do some Fourier analysis where the geometry plays a role. Okay, here's a very simple example. My graph is just the interval from 1 to n.
Courant Events Right: Okay, 1 through 10, and my tiles are just gonna be…
Courant Events Right: collections of three consecutive integers, so… so, which can slide… which can be slipped anywhere along there, and… and I'm going to allow the tiles to sort of go past the endpoint, so the… the first tile just consists of a single vertex, the second tile is the next
Courant Events Right: first two vertices, and the third tiles, the first three vertices, and then it slides along. So tiles are all subsets of three consecutive vertices.
Courant Events Right: Including the tiles which sort of overlap the boundary.
Courant Events Right: Okay, so there's a… I can just index the tiles by the numbers 0, 3, and plus 1 if I think about the location of the center of the tile.
Courant Events Right: Right?
Courant Events Right: Everybody with me? So that gives… my matrix now is a… in this case, there's 7 vertices, I think, and there's 9 possible tiles. So as a sound, my 9 matrix.
Courant Events Right: And the, the, right? So, for example, the first pile only contains the first vertex, and that's with the matrix, and that's the incidence matrix, and now you can ask.
Courant Events Right: What's an independent cover? It's a collection of 7 tiles.
Courant Events Right: In other words, well, there's 7 vertices, so I need to pick 7 tiles, and then I want the corresponding submatrix, 7 by 7 submatrix, to have non-zero determinants.
Courant Events Right: And you can ask, that's a fun question to work out, if you're bored, which, maximal miners, which 7x7 minors in this matrix have… have full rank.
Courant Events Right: Anybody want to guess, or know the answer?
Courant Events Right: I'll tell you later.
Courant Events Right: Okay, here's another example. This is kind of the even more motivational example, which is called the Uniform Spanning Tree Model. G is a… now a bipartite graph.
Courant Events Right: I pick up one vertex, these are all called the root vertex, and the set of tiles now are just the edges of the graph, and then I'm going to add one more tile, which is just a single root vertex by itself.
Courant Events Right: Then, the…
Courant Events Right: Well, here's the… here's an example for the… when the graph is a square. You get this 5. In this case, there's, 5 possible tiles. There's 4 edges plus a root vertex, so there's 5 columns, and there's 4 vertices. That's the matrix, and the… the,
Courant Events Right: theorem, which is… well, it's in graph there, it's called the Pointer Ray Lemma, which says that your
Courant Events Right: Minor that is, the, the, the…
Courant Events Right: Miner has full rank, if and only if it's a spanning tree. It's an independent cover is the same thing as a set of V minus 1 edges and V0 itself, which make a spanning tree.
Courant Events Right: And, you know, it's not so hard to prove this lemma, you can prove it on your own. Conversely, any spanning tree is an independent cover, so the set of independent covers is exactly the set of spanning trees. As you can see in this example, there's four possible spanning trees.
Courant Events Right: Of this graph, and each of them has 4… 4 parts, 3 edges and 1 root vertex.
Courant Events Right: Make sense?
Courant Events Right: Who knows the point-grade 11?
Courant Events Right: Now I do. Now you do, okay. I don't know why, I don't know, maybe it really was point Craig, but that's what my graph theory book told me this was called.
Courant Events Right: Is there any connection to the usual pancrella, which is that the closed form is exactly…
Courant Events Right: The way you got in?
Courant Events Right: Oh, I don't know. I don't know. Maybe it is, yeah, must be.
Courant Events Right: Anyway, I… As I was making the slides, I was watching this movie, Dune, with my daughter.
Courant Events Right: Okay. Just in case you're… you were asleep. Okay,
Courant Events Right: There's a natural… Sorry? I had to give this talk in Hollywood. That's right.
Courant Events Right: There's a natural probability measure on independent coverage. Now we understand what an independent cover is. Sorry, I'm kind of looking over here, but you guys are also in, you know…
Courant Events Right: Okay, well, some of them aren't pulled back up. There's a natural probability measure.
Courant Events Right: Where the probability of an independent cover is proportional to that determinant
Courant Events Right: Right, the determinant of the corresponding minor, I called it the upper end before, that's a mistake. Well, okay, anyway, typo. You just take the determinant squared.
Courant Events Right: And then you divide by a normalization constant. The partition function is the determinant of DD star. That's, well, there's a nice linear algebra result, which says that for a rectangular matrix, the determinant of dtt star is the sum of the
Courant Events Right: Maximal miners, squared.
Courant Events Right: Coach keeping in theorem
Courant Events Right: So this, this is the probability, natural probability measure to put on a, on a, independent cover.
Courant Events Right: And the theorem is that this measure is a determinantal measure, and it has a kernel, well, this complicated
Courant Events Right: combination of D, so D star… D star is just the transpose of D, DD star inverse, times D.
Courant Events Right: And if you know about spanning trees, this is a generalization of a theorem of Burton, which was originally to Burton and Pimentel from the 90s, maybe 93, for the spanning tree case.
Courant Events Right: And, you know, this theorem is essentially due to either of them, or… or some… some combination of them, and more generalized by, in this paper.
Courant Events Right: Benjamin E Lyons, Paris and Schramm, 2001. That's where I learned it.
Courant Events Right: Let me explain what that means. What's a determinantal measure?
Courant Events Right: Maybe we, for physicists, we should call this a pre-fermionic measure, but in math, it's a determinantal probability measure. What is it? It's just a measure on binary sequences of length n.
Courant Events Right: probability measure. It's determinant, so if you can find a certain matrix, n by n matrix K, which is called the kernel, such that, has this property, that for any subset of indices.
Courant Events Right: the probability that…
Courant Events Right: for every index in the set S, the corresponding variable is 1, is given by the determinant of the S by S principal minor of that matrix K.
Courant Events Right: So, in some sense, all the…
Courant Events Right: Probabilities that you want to compute for this measure turn out to be determinants of matrices.
Courant Events Right: Constructed from the… from the kernel K.
Courant Events Right: So that's where these measures are sort of defined by linear algebra. And the example, of course, the standard example is the spanning tree measure. Edges in a uniform spanning tree form a determinantal measure.
Courant Events Right: Right? If you… if you take a graph and you ask, is this edge in and this… is that edge in? This is a 2x2 determinant of the… of the kernel that you associate to the… that graph.
Courant Events Right: Another example is the edges in the dimer model for a planar bipartite graph. This is a theorem of myself.
Courant Events Right: Maybe it was known earlier. And there are some continuous examples of determinantal measures as well, where instead of probabilities, you get probability densities as determinants. And one of the
Courant Events Right: ones we'll use in this talk is this, example of the Gaussian analytic function.
Courant Events Right: This is due to Yuval Perez and Valen Birag in 2005. What we do is we take a…
Courant Events Right: Power series, just a one variable power series, complex
Courant Events Right: function, complex analytic function, where the coefficients are independent, IID, standard, complex normals,
Courant Events Right: And this is a power series which converges inside the unit disk.
Courant Events Right: And you can ask, where are the zeros of this function? They're random zeros, random locations of zeros, and the… that set of zeros, the locations of those zeros, is a point process which is determinantal, and the density is given by,
Courant Events Right: The determinants of some… of some…
Courant Events Right: Well, currently some, yeah, operator in this case.
Courant Events Right: And there are lots of other examples of determinantal processes and random matrices, like GUE, CUE, Ginibra Ensemble, you've probably heard of those.
Courant Events Right: Are they frickle, German?
Courant Events Right: No, well, nevermind.
Courant Events Right: Not in this talk, for sure. Okay, all my determinants are going to be defined, here. So here's a nice linear algebra fact, right? I said we're going to use some linear algebra. This is maybe something you didn't learn in your linear algebra class, at least. I didn't learn it in mine, but I learned it recently that, if you have a
Courant Events Right: K-dimensional subspace of Rn. W is a K-dimensional subspace of Rn, and suppose they have a basis for W, W1 through WK,
Courant Events Right: Then, you make this rectangular matrix, K by N matrix, whose rows are the vectors W1 through WK. Then this expression here is the matrix of the orthogonal projection to W.
Courant Events Right: Okay? Is it… who knows this result? Who learned this in linear algebra? All the Europeans.
Courant Events Right: I… I don't think so. I double-checked, but I could be wrong. Okay. The rows, the rows of the vector… You're correct, you're correct, right. No problem.
Courant Events Right: Okay, so this is the thing which comes out in the kernel for my independent covers, and so the corollary is that the kernel of the independent cover determinant will measure, right, that matrix which defines the probability measure is, in fact, the matrix of orthogonal projection to some subspace.
Courant Events Right: Right? Remember, our matrix is indexed by the set of all tiles.
Courant Events Right: And the subspace is the… I mean, the corresponding substrate is for… the matrix is the matrix of projection onto the kernel of D,
Courant Events Right: Right, D is the… D is the…
Courant Events Right: Remember, the mapping from the space of tiles to the space of vertices.
Courant Events Right: Oh, okay, so, let's see.
Courant Events Right: Yeah, let… okay, so…
Courant Events Right: What's a comic? A comic is the complement of an ick, an independent cover, right?
Courant Events Right: So, it's this…
Courant Events Right: It's the set of tiles which you don't use in your independent cover, right? If you have an independent cover, you have a certain set of tiles, you take all the tiles which you don't use, that's… that's the complement, that's to call the comic, and…
Courant Events Right: That's a property of determinantal measures, is if you exchange the zeros and the ones, then the new measure is also determinantal, but the kernel just becomes the identity minus the original kernel.
Courant Events Right: So the natural determinantal measure on the
Courant Events Right: for comics, as the kernel i minus K, where K is this previous kernel.
Courant Events Right: And so it represents orthogonal projection to the perpendicular
Courant Events Right: subspace to kernel D. Kernel D perk.
Courant Events Right: And the nice thing about a comic is that in many cases, it's smaller than the original, independent cover. Since the size of the comic is the complement, that's the number of tiles minus the number of vertices of the grid.
Courant Events Right: Okay, so then, if you only care about the comic, then you can…
Courant Events Right: rather than use the matrix D, you can use the matrix which forms the basis for the kernel of D, and then the probabilities all work out the same. The probability for a particular comic S, maybe I should have called it something besides S, but it doesn't matter, the… is this same kind of expression.
Courant Events Right: Okay, so now we can go back to our original example, just to…
Courant Events Right: summarize what I just said, this was our original example, this was our matrix D,
Courant Events Right: This is a 7x9 matrix of full length, so its kernel is only two-dimensional.
Courant Events Right: And the basis for the kernel is just given by the… the… I put them in rows here rather than columns. It's just given by the… these functions such that the sum of any three consecutive entries is zero, and I could choose my basis. Well, there's several ways to make a basis, but just using some complex numbers.
Courant Events Right: cube roots of 1, that's a basis. The 2 by n plus 1 matrix.
Courant Events Right: And,
Courant Events Right: Therefore, you know, my independent cover is going to use 7 tiles, but the complement of that is only going to use 2 tiles.
Courant Events Right: It's got the two, right, so two locations.
Courant Events Right: And, the… you can ask, what's the probability that those two locations are at two points, X1 and X2? It's just that…
Courant Events Right: is just the corresponding minor squared of that matrix B that I wrote on the previous slide, which is just the…
Courant Events Right: You know… What do you call it, Matrix?
Courant Events Right: It's just a 2x2 matrix. You take the… you take the determinant, it just comes out to be 0 if X1 is congruent to X2 mod 3, and it's some constant otherwise.
Courant Events Right: some constant, like, Yeah, anyway, whatever the constant is. So, if you, if…
Courant Events Right: If you're interested in which maximal minors of this matrix have full rank, it's exactly those. Well, you have to remove two columns, but the two columns you remove cannot be a multiple of three apart.
Courant Events Right: So if you're… if the two columns you remove are a multiple of three apart, then the minor is 0. Otherwise, it'll be… it'll be some constant, plus or minus some constant.
Courant Events Right: Okay, no big deal. That was just kind of a baby example. We want to do this in higher dimension, but let me tell you, just to pique your interest, let's try a slightly slight variant. It's still in one dimension.
Courant Events Right: Alright, so this is a…
Courant Events Right: I'm going to take the same graph consisting of endpoints in a row, and now my tile is going to be… I'm going to put some weights in my tile, some integer weights, so that translates of the tile 14641, so it's going to be 5,
Courant Events Right: vertices, consecutive vertices, but they'll have weights 4, 6, 4, 1, and those go into the matrix here. And so my D matrix looks like this, looks very similar to the previous one.
Courant Events Right: just all translates of the 1464.1. And… but now the… the kernel is more interesting,
Courant Events Right: The kernel of this matrix consists of vectors, which are
Courant Events Right: Now, f of x, where X is a polynomial degree at most 3.
Courant Events Right: Except for this sign. Like, for example, if I take, you know, 1 minus 2, 3 minus 4, 5, and so on, that's in the… that's in the kernel of this, and so is the same. Each… each row of this matrix
Courant Events Right: That is in the kernel, and therefore, the kernel, because this is of all polynomials of degree at most 3, except for this, minus… except for the alternating, oscillating
Courant Events Right: index minus… minus 1 to the i.
Courant Events Right: So, the probability that… that… so now my… in this case, my… my comic has 4 tiles.
Courant Events Right: I mean, my independent set is, has size, 5, and the total matrix has size 9, 9 rows, so my independent set has 4, tiles, and you can imagine this being very long, it's the…
Courant Events Right: You know, I can make N here as large as I want, and the comic will still have only 4 points in it. That's the advantage of doing the comic rather than the independent cover.
Courant Events Right: And the probability that I get, that X1, X2, X3, X4 points are…
Courant Events Right: My comment consists of exactly those four points.
Courant Events Right: is proportional to the… so what I do is I take the V matrix, and I look at the column, the X1 column, the X2 column, the X3 column, the X4 column, and I take the determinant, but that's exactly a Vandermont.
Courant Events Right: matrix whose… whose determinant is this, you know, Vandermont squared. I mean, when you square it, the determinant is the Vandermont squared, so the probability proportional to the difference of the
Courant Events Right: Xi's squared. Product over all I's and d's.
Courant Events Right: So, it's not a coincidence.
Courant Events Right: No, that's not a coincidence. What's… what's going on is that I, you know, I took a… I took a tile here whose, right, whose coefficients were binomial, so it's… it's, you know, Fourier transform has multiple roots on the… on the unit circle, and that's what caused this interesting behavior here in the… so… so…
Courant Events Right: So it's an interesting determinantal process on four-point subsets of a… of a big interval with… with, but the density is a very familiar, you know, beta, beta equals 2 density.
Courant Events Right: Does it count?
Courant Events Right: Bigger numbers to find me?
Courant Events Right: I get more points.
Courant Events Right: If I… if I took a, you know, 1, 5, 10, 10, 5, 1, I'd just get 5 points with the same… with the same kind of expression.
Courant Events Right: Any questions so far?
Courant Events Right: Because I'm going to go to a higher dimension now. Yeah.
Courant Events Right: Yep.
Courant Events Right: Dude, just pick up, I can't really hear it.
Courant Events Right: Right, the entire was 5 consecutive versees, 114641.
Courant Events Right: And the… the…
Courant Events Right: Oops.
Courant Events Right: I went back too far.
Courant Events Right: Yeah, so this dot represents the center of one of my tiles.
Courant Events Right: Right? So this dot really represents a tile sitting there in… inside the interval from, you know, well, minus 1 to F plus 2, or something like that.
Courant Events Right: Was that your question?
Courant Events Right: Nothing is quite early.
Courant Events Right: I really can't hear you because… What's the role of the weights? What's the role of the weights?
Courant Events Right: Well, they changed the kernel out of matrix, and they changed everything, right?
Courant Events Right: Right, I guess in principle, you know, you could use, other non-integer weights, but,
Courant Events Right: This problem originally came from a different problem where we needed positive integer both.
Courant Events Right: So… Basically.
Courant Events Right: If you want, you can ignore this example and then ask for all the weights to be 1 or 0. It could just be the incidence matrix.
Courant Events Right: Okay, let's go to two dimensions now, and we see some more phenomena that we didn't see before. So here, I just took the… the… G is now a subset of Z2,
Courant Events Right: And my tile is just going to be the L tile. All translates of the L tile, you know, just translates, I'm not allowed to rotate, so the L tile is just the… the polyomino, or the subset 001001 in Z2. And so you can see a kind of a picture over here. What this is, is I took G to be a big square.
Courant Events Right: And it took a random coming.
Courant Events Right: So that I'm not… I'm not drawing the independent cover, because the independent cover has… has N squared… almost n squared tiles.
Courant Events Right: In fact, it will have exactly N squared times, but the… but the comic, the complement of it only has, like, two n times.
Courant Events Right: And what you see is that the tiles mostly, you know, near the boundary, this is a red… this is a perfect sample. The tiles are mostly near the boundary, there's a few in the middle. What did I do for this picture? I just, you know, I just tilted that picture with a linear map. First of all, I took a triangle.
Courant Events Right: Rather than taking the square, I took a triangle and then tilted it so that you can see a little bit of extra symmetry here. If you tilt the… if you tilt the L title, then it looks like a little equilateral triangle. But then I took a comic for that one. So this one has some nice three-fold symmetry, which is not evident in that previous picture.
Courant Events Right: And so what's… what's happening now, in the scaling limit?
Courant Events Right: to this process. It's a determinantal process, and the scaling limit is that it's looking like the zeros of a Gaussian analytic function. Here's a picture of the actual Gaussian analytic function, right? As I defined before, you take a power series with
Courant Events Right: independent Gaussian entries, you look at the zeros, they look kind of like that. This is an actual simulation of the zeros.
Courant Events Right: And, Yeah, so the…
Courant Events Right: theorem of Paris and Bragg should show that the zeros are determinantal process on the unit disk with a certain kernel, which is called the Bergman kernel.
Courant Events Right: It's a… it's a function of Z and W, which is on the… on the unit disk, it's given by this, but this Bergman kernel exists on any domain in the complex plane. It's the projection kernel from
Courant Events Right: L2 of the domain to the
Courant Events Right: L2 holomorphic functions on the domain. So you take an L2 function on the domain, and there's a… there's a projection operator to… to the… you find the closest analytic function on the domain, which is an L2, and that's the… this is the kernel of that projection operator, and that's the theorem. The theorem is that,
Courant Events Right: You take the L-tyle.
Courant Events Right: Here, on a domain in U, you take a lattice approximation to it in epon-C2, I changed coordinates a little bit. In the limit, in the scaling limit, in the limit as the scaling scale goes to zero, the point process for the comic
Courant Events Right: for this L tile, converges after, after that linear math that I discussed to the zeros of the Gaussian analytic function on the domain U.
Courant Events Right: So, you know, if I… if I… as the scale… as the mesh size goes to zero, the scaling limit of this tonic process
Courant Events Right: will still be on the triangle, but if I apply the Riemann map to the circle, I will get the same process in law as the zeros, the Gaussian LA function.
Courant Events Right: This, this, the nice thing about this Gaussian analytic function process is that it's conformally invariant. If you apply a conformal change of domain, it goes to the Gaussian analytic function on that domain.
Courant Events Right: I spent with Jaime, how many,
Courant Events Right: How many points are there in that triangle?
Courant Events Right: Well, it's linear… the number of points is linear in the size of the triangle. So it's like 2… twice the side length.
Courant Events Right: Roughly. I've forgotten exactly the amount, but it's a linear number outside. Most of them go to the boundary, but in the scaling limit, you know, they'll sort of accumulate on the boundary, but there'll be still some inside, and the density of them inside is given by this… this, the same as the Gaussian analyzed function.
Courant Events Right: So it's kind of a nice, reasonably simple… yeah. What do you mean by up to, you know? Well, the linear map which takes the square grid to the…
Courant Events Right: See, if I did it in the square grid, it wouldn't converse to the Gaussian… the conform… the correct conformal structure is obtained by… by this… by this
Courant Events Right: coordinate rather than the square root coordinate. So I have to take the L tile and then tilt it by the… whatever that linear map is.
Courant Events Right: Any more questions about that?
Courant Events Right: Blow up the triangle to a circle, will they go.
Courant Events Right: Yeah, that's right. In order to identify this process with this process, I have to use a conformal map from the triangle to the disk.
Courant Events Right: Yep.
Courant Events Right: Okay, so, well, corollary, the comics for the L-tyle are conformally invariant, because we know… we know what they're… we… well, we've identified their limits.
Courant Events Right: There's no choice of victims were identified as a piece of land.
Courant Events Right: Yeah, well, see the next slide. He asked, what happens if I change my tile from an L to some other tile?
Courant Events Right: I mean, that's… that's pretty much the next obvious question. Sorry, that… but, okay, sorry, let me tell you about how the proof goes, and then I'll get to your question, Martin, right?
Courant Events Right: This… this proof is kind of is not… not…
Courant Events Right: so easy, right? So, you know, just because I fit it on one slide doesn't mean the whole proof is there. This is the… this is a very rough idea of the proof. well…
Courant Events Right: there's two things you need to do, right? The… the…
Courant Events Right: Remember that the kernel of the project… of the kernel of the terminal process is projection on this particular subspace of functions.
Courant Events Right: And that subspace is,
Courant Events Right: Well, kernel of D. What does that mean? It means that your function on Z2 is in the kernel of D if, when you sum the value at a…
Courant Events Right: I didn't make a… I didn't make a slide, but I have a function on… Right?
Courant Events Right: There's the lattice… there's two lattices. There's a set of files, which is also a Z2 lattice, and there's a set of vertices, which is Z2 lattice. And if I have a function on Z2, it's in the kernel if, when I sum it at these three locations, I get zero, because
Courant Events Right: for each vertex, V, I want this tile, the value of this tile, plus the value of that tile, plus the value of that tile, to add up to 0.
Courant Events Right: So I'm looking for functions on G2 which have this property that this value plus this value plus this value equals 0.
Courant Events Right: That's the kernel.
Courant Events Right: least, at least, you know, and, and then there's the boundary conditions, which, I'm just taking, sort of.
Courant Events Right: three boundary conditions, I'm allowing the tiles to overlap the boundaries, but let's not worry too much about those. So, one example of such a function looks like this, right? If I just take omega to the X minus y.
Courant Events Right: Where omega is this, cube root of… cube root of… Give root of 1.
Courant Events Right: Right, so omega plus omega… well, 1 plus omega squared plus omega. That's one function. And the… and the…
Courant Events Right: Lemma is that any function in the current… any function which satisfies that on a large domain, will… will…
Courant Events Right: look like this function here times a smooth function, unless it, it… any function which stays bound, which, you know, is bounded in L2.
Courant Events Right: Right? If, if, if I, if I…
Courant Events Right: If I… if I… I can define any function in the kernel by just defining the values on a vertical strip and a horizontal script, and then… and then the value here will be the sum of the previous two, and this value here's the sum of the previous two. But all those functions will sort of blow up exponentially fast, unless… unless they are of this type.
Courant Events Right: This particular periodic function tied to smooth function. So that's something you have to prove. And in addition, once you have this form, and you apply the D operator, you see that F has to be actually an analytic function
Courant Events Right: of the… just by Taylor expanding the D of F, you see that F has to be an analytic function of this… the tilted coordinate, X minus omega squared y, the coordinate when you get tilted like that.
Courant Events Right: So that's a little… that requires some… some, 40… a little bit of 40 analysis and the uncertainty principle, to prove that step.
Courant Events Right: But then you see that, after, you know, after, you know, conjugating your function by this oscillating prefactor, this prefactor, then it looks just like a smooth function, and then the projection from this
Courant Events Right: On this smooth function, you know, is, it, it tells you that the,
Courant Events Right: when you predict on this lattice, the projection is a discrete version of the Bergman kernel, and then you can kind of wave your hands a little bit, at least that's what I'm doing now, and show that it converges to the… in the limit, the scaling limit, to the actual
Courant Events Right: Bergman kernel, the projection from L2 to the analytic functions.
Courant Events Right: Okay, that's all I want to say about the proof, but there's some, there's a lot of details there.
Courant Events Right: How do I get what?
Courant Events Right: I have the impression that we have a U for a bow, or going to disabled.
Courant Events Right: Yes. Yes, from your county.
Courant Events Right: Yeah, well, we have a… we have a…
Courant Events Right: Let's talk afterward. That's… it's somewhat technical, and and I want to get to the fun stuff.
Courant Events Right: Okay, so now… now to, Martin's question. What… what happens if… what property of the L tile did we use, and can we use that property for other tiles?
Courant Events Right: Right? Well,
Courant Events Right: If we think about the Fourier transform of the L tile, the L tile has this… has these three vertices, 0, 0, 1, 0, and 0, 1, and if we… if we call, you know, Z the Fourier transform of translation to the right, and W the Fourier transform of translation left, then the Fourier transform of L tile is just 1 plus Z plus W, just multiplication by 1 plus Z plus W.
Courant Events Right: And this…
Courant Events Right: polynomial of two variables, 1 plus E plus W, has… has the property that it only has simple zeros on the unit torus.
Courant Events Right: You know, if I… if I…
Courant Events Right: Well, certainly that's true. I mean, if you look for the zeros of this thing on the Uniforce, they're just omega, omega squared, and omega squared omega, and that's the property that
Courant Events Right: leads to this conformal invariance, leads to this Bergman kernel.
Courant Events Right: And,
Courant Events Right: Basically, because of the same story on the previous slide, if you just have simple zeros, then you can multiply by this oscillating factor and find a smooth function.
Courant Events Right: If you just have a pair of simple zeros.
Courant Events Right: So, similar results are going to hold for any comic, as long as you're… when you have only one tile, and that tile has the property that it's Fourier transform, P, has only a single pair of complex conjugate simple zeros on the unitores. For example, and here's some examples.
Courant Events Right: So, some tiles are nice. They lead to conformally invariant processes. In fact, they always lead to linear scales of this Bergman kernel process. Gaussian analytic function.
Courant Events Right: However, there are plenty of other tiles which don't have that property. For example, this one, this tile has, multiple simple root pairs.
Courant Events Right: And so what we get is actually a linear superposition of these Bergman kernels, and it's not conformally invariant.
Courant Events Right: Or rather, there's two conformally invariant things, but each one is shifted and, by a different linear factor.
Courant Events Right: the sort of a superposition of two different shifted and oscillated Bergman kernels. Sorry, that's… I should have made a picture here, but I didn't.
Courant Events Right: But then,
Courant Events Right: Some tiles, like the square, that was Form's idea, intersect the torus in more than just a tiny set of points, and those have weird behavior, which we don't really know how to classify. So, for example, this square tile, the Fourier transform is 1 plus Z times 1 plus W,
Courant Events Right: It intersects the unitaurus in the union of two curves, Z equals minus 1 and W equals minus 1.
Courant Events Right: here's a random sample of that. It's still a determinantal process, it's pretty easy to get your computer to spit out a random sample for the triangle, but the sort of covariance function between these tiles is kind of complicated. If I… what I did was I asked, you know, what's the…
Courant Events Right: covariance between a tile here and a tile at a different location in the triangle, and you can see that the covariance function has the sort of delta functions along the horizontal and vertical, and then it's got this analytic part, and then it's sort of only piecewise analytic. In fact, the
Courant Events Right: the…
Courant Events Right: The square over here is independent of the square over here. The locations… the density of squares here and over here are independent of each other.
Courant Events Right: for mysterious reasons. But, I mean, we can prove it, but it has something to do with the fact that these… these curves are…
Courant Events Right: This polynomial factors.
Courant Events Right: So we get kind of weirdish behavior in this setting, Here's another example with…
Courant Events Right: also weird behavior of a completely different type. Our tile, we just tweaked the tile just a little bit by adding some weight tubes. This is the new polynomial.
Courant Events Right: Now, the zero cent on the torus, is also not a set of points, it's a curve.
Courant Events Right: But a different kind of curve.
Courant Events Right: And, you know, this… here, this… this… this sample was on a… on a… on a disc again, but… but, you know, apparently the disk's not so important as it's on a disk. The… the… the interesting thing is that the covariance, functions for this, for tiles here, has this kind of,
Courant Events Right: Chronicle behavior, kind of like a… interesting…
Courant Events Right: Minkowski or Werenci in geometry. If you have a tile here and a tile out here, they're sort of exponentially uncorrelated, but tiles which are costly related, have some polynomial correlation.
Courant Events Right: Kind of weird, I don't understand, but this is what the… what you get from the Fourier analysis of this tile.
Courant Events Right: Again, I'm just doing linear algebra, right? It's just, you know, to get this covariance function, I just have to invert the certain matrix.
Courant Events Right: Which is based on the, you know, where the… basically finding the Fourier coefficients of 1 over P.
Courant Events Right: Here's… here's… here's another polynomial, the plus polynomial. It's also got a curve, but it's a closed curve in this case, and I… there, the covariance function is… is pretty weird, sort of oscillates in every location, but it's definitely got some long… long-range structure, which you don't really understand.
Courant Events Right: None of these are, sort of, conformally invariant.
Courant Events Right: But they… but we do, sort of.
Courant Events Right: Can… we can compute their scaling limit.
Courant Events Right: A scaling limit of the kernel.
Courant Events Right: Well… At least that we know how to do some Fourier integrals.
Courant Events Right: Yeah, but what happens in 3D? Oh, I had some… I had a picture for 3D. Well, let me… let me read the slide first, and then I'll show you a picture.
Courant Events Right: If you have a… if you have one tile in three dimensions, typically it's not going to intersect the torus in a finite set of points. It'll… it's, it's a… it'll intersect the torus in a one-dimensional
Courant Events Right: You know, some variety.
Courant Events Right: And, and… that's the case where the Fourier analysis is complicated. We don't have any sort of general
Courant Events Right: description of the… because it depends on sort of a case-by-case basis, what that curve is. I don't have a good, I'm a, you know, baby when it comes to Fourier analysis. I don't understand how to do these integrals yet.
Courant Events Right: Still thinking about it, working on it, or looking for an expert.
Courant Events Right: When… on the other hand, if there's two tile types, then typically you get isolated points just by dimension counting. For example, if I take 1 plus Z plus W in the XY… one tile in the XY plane, and one sort of vertical dimer in the… in the Z plane, then the… then you get a funny set of points. I think this… there's only two points here, U equals
Courant Events Right: minus 1 and Z and W are the Q roots of 1, then you get some nice
Courant Events Right: 3-dimensional point process, which has this
Courant Events Right: Well, I can't say conformal invariance, but I can at least say it's sort of stretched rotational invariance. Once you apply the appropriate linear map, you get a rotational invariant process.
Courant Events Right: And again, going back to what I said before, if… if you take your…
Courant Events Right: graph to be Z cubed, and
Courant Events Right: all… you take all the edges of Z cubed plus one extra
Courant Events Right: a large region in Z cubed and one extra vertex, but along with the edges, then you get… do get the 3D spanning tree as the independent set, independent cover model, and
Courant Events Right: Well, we know to what extent we…
Courant Events Right: That's a sort of big open question. There are questions about the spanning tree for which we know the answer, and there has some rotationally invariance of the
Courant Events Right: edge covariance functions, but the… the other questions, like, you know, what is the… what are the branches of the spanning tree look like? Those are still open questions.
Courant Events Right: So just because we can prove that certain things are conformally invariant, or know the scaling members of certain observables, there are other… there can be other observables for which we don't get information, directly from this, procedure.
Courant Events Right: Let me see if I can, oh.
Courant Events Right: Let me… let me pull up the other, just to show you that the one… oh, where did it go?
Courant Events Right: Here it is.
Courant Events Right: I made a picture of a… Don't see that?
Courant Events Right: This is a 3D point process, and you know, in 3D, you can't really see a whole lot, but this is a 3D process, point process on a cube, which comes from the tile, which is the three-dimensional version of the L tile.
Courant Events Right: And, you know, I don't know what you can really tell from this example, but there it is. It's not so hard to make these simulations. These are all exact samples, because there is an exact sampling algorithm for determinantal processes.
Courant Events Right: Any questions? Sorry, what? What's the analog of the barbacco?
Courant Events Right: What's the analog of the Bergman kernel?
Courant Events Right: Is there? Is there? No, this is not the…
Courant Events Right: This is… there's… yeah. I don't know… I don't know what to… what to say about this. I mean, I can write down a formula, but there's no conformal invariance, conformal invariant object in three dimensions. There is no Bergman kernel in three dimensions, so…
Courant Events Right: Well, you could study some truths of the rock equation.
Courant Events Right: the analog of polymorphic function is a solution to the rock equation, basically.
Courant Events Right: L2 speeders and projectile to another ball, projectile to the… Yeah, yeah. Whether that arises in such a model is not clear to me, but
Courant Events Right: Right, so the question is, what's… what's the analog of the Bergman kernel in 3D?
Courant Events Right: Is there some direct… direct operator which, I don't know. Good question.
Courant Events Right: And that… thank you very much, that's all I had to say.
Courant Events Right: Are there more questions?
Courant Events Right: Yeah, Mark. So, like, most of them do then just fail to basin. You still have all the…
Courant Events Right: This one is kind of a non-isotropic, model. Definitely, definitely the points are accumulating on three of the faces, but not the other. I mean, it's some sort of a…
Courant Events Right: You still have the same model of points on top of that? Yeah, yeah, I think that the density of points inside is some continuous function of the distance.
Courant Events Right: Yeah. Does it make sense to introduce some parameters in your model, like maybe weights, then study how they… Oh, for sure, yeah. How they measure depends on the weights.
Courant Events Right: Yeah, that's right. I… for this talk, I just chose the constant weights, but it's kind of like the spanning tree. You can add conductances to the spanning tree, and then you get slightly different behavior, but the global behavior is roughly the same. That's what I think happens in these models.
Courant Events Right: Basically, you know, the underlying Laplacian, which is DD star, you can add some conductances in there and put parameters. Is that what you're asking? Well, there is an interesting connection in the space of parameters.
Courant Events Right: Which is like a generalization that you can actually be shown
Courant Events Right: engine is written in terms of nose operators, and…
Courant Events Right: One of the examples of that would be the…
Courant Events Right: aviation construction for the… for instant dots, for emergencies.
Courant Events Right: Sure.
Courant Events Right: Okay, maybe we can talk. If you have some good ideas, this is great. It's a new… it's a new subject for us, too, so if you have good ideas, please bring them to me, and we talk afterwards?
Courant Events Right: Can you come tomorrow? Yes, tomorrow, okay.
Courant Events Right: Any other questions?
Courant Events Right: Oh, there's one that's back here.
Courant Events Right: Can you speak out loud? Wilder!
Courant Events Right: Often.
Courant Events Right: Can you repeat the question? Yeah, he's asking,
Courant Events Right: Is there a… can we prove that there's a scaling limit in 3D, even if we can't describe it?
Courant Events Right: Yes. Well, no, okay, I'm 90% sure that I can, but, you know, until things are written down, which they are not at the moment, I'm not gonna, you know, be definitive about that. So, yeah, it's a good question.
Courant Events Right: But I believe that the linear algebra all works out, and there will be a… there will be a scaling limit.
Courant Events Right: If there are no further questions, let's thank, Rick for his comic relief.
Courant Events Right: And this concludes this first part of this week's activity.