 Merci beaucoup pour l'admitation. En fait, c'est en anglais, il y a une main place que j'en parle pas français, c'est bien pour donner cette exposition en français. So let me switch to English from the very beginning. And of course, it's a pleasure to open such an event and to speak in such a famous place. So I was asked to give a kind of an expository talk on what is going around of say my research of my colleagues. So it's going to be about two-dimensionalizing model, which is a famous example of a statistical mechanics model into dimensions that is in some sense completely solvable. I mean, it carries a lot of structure. So here is the plan. Actually, I'm going to spend a considerable amount of time just explaining this structure. So it is well-known indeed, it was revealed, in fact, even 70 years ago by Anzager. But still, I think it deserves to be really known. I mean, there is something in this model which allows us to analyze it. And a kind of an analogy I suggest you to have in mind is the structure of harmonic functions, which underlines all the problems related to random walks. So the message is that in this model, if you never see the definition, it's going to appear in the next slide, there is something similar. So here it's going to be sometimes spent on combinatorics. Then what is important at the critical point, well, I'm going to comment on this on this phase transition, is that similarly to random walks, where just discrete harmonic functions appear, here the structure is the one of discrete holomorphic functions. So this Y, one should expect that in the scaling limit, when you think about the behavior of your whatever events on the verifying latest, you should expect a kind of conformal invariance. Similarly to the conformal invariance of the Brownian motion. So, okay, this holomorphicity. And then I will show some results. I mean, how much one can actually advance using this structure. I mean, what type of results are available nowadays. So this is basically related to two acronium CFT, which is the conformal field theory. And it's a shame to speak on conformal field theory here. But okay, and CLE, which is a conformal loop ensemble, which is a more geometric viewpoint on what's going on. And then there is a kind of, well, conclusion remarks. There are conclusion remarks. And the message is that, okay, it was a big buzz and a lot of progress exactly at the critical point. But the structure of, okay, not holomorphic, but massive holomorphic, the structure of functions, it exists all the way. I mean, it doesn't rely upon the fact that the model is critical. It doesn't rely upon the fact that you work on a regular grid. So this is just 2D, any planar graph. You can see the rising model there. And there is a structure that allows you to study it. So that's a kind of, the kind of an advertisement. So what is the model? Basically you take a lattice or whatever graph. I would prefer to switch from vertices to faces of this graph. So it doesn't matter because of the obvious duality. So you assign two colors to the faces of your graph randomly. And then there is a question. So I heard this called the percolation on which ramp is going to talk about at the end of this day. Not this one, actually, I remember the kind of. So the answer is that, okay, this picture is uniformly at random. And of course, the most difference is that here you see blue and yellow. And there it was on the previous slide, blue and red. So that's really different. So, okay, here your color faces uniformly at random. And in the eyes and model, in the nearest neighbor eyes and model. So basically what you do, you choose them configurations according to some law. And the rule is the following. So to every edge, separate into faces, U and V, there is some price you must to pay if two spins are disaligned. So basically your system are going to, well, would like to be aligned. All the spins would like to be of the same color. So here is the price. So the probability of the configuration basically is proportional to the product of all these prices. This constant, okay, well, here is some physical, more physical notation. What you should keep in mind is that there is a parameter in the model, which is related to the temperature, this is inverse temperature, such that when it increases from zero to infinity, basically those prices, those multiples, they increase from zero to one. So when the temperature is very small, you pay a lot for spins to be disaligned. If temperature is very big, you pay almost nothing. If it is infinite, you pay just nothing, and then this is the percolation, side percolation model. Well, if you think about the model geometrically, you might wonder about some more convenient parameterization. So here there is one, actually never mind. I'm not going to appeal much to this parameterization. And of course, if we are on the regular grid, you can just think about the homogeneous model. So just take Z2, take some subdomain of Z2, put all x's equal to each other, and then there is just a single parameter at this temperature. So you vary x from zero to one. Okay, seems to be rather naive, but then it turns out that there is a non-trivial phenomenology, even in such a simple situation, which is called a phase transition phenomenon. By the way, here is a remark. So I think most of you are PhD students. So if you want a real success, I mean in science, here is an advice. What you should do, you should ask your PhD advisor to advise you a good model. This is exactly what Lance did in 1920. So he gave this model to his PhD student, and then, okay, he studied it in 1D. It turned out that there is no phase transition. So because of that, actually for 10 years, people thought that there is no phase transition in higher dimensions as well. So, and it was clarified only a decade ago. And then, actually, he didn't do any serious science afterwards. So, okay, he was in trouble during the Second World War, but escaped to US and became a school teacher there. And only many years ago, he learned that the model is extremely famous. If you Google, this is like hundreds of thousands of references. Okay, so just ask your PhD advisor for a model, and you have a chance for a really great success. So, okay, there is a phenomenology which is the phenomenology of the phase transition. Actually, it turns out what you would expect is that, okay, if X is small, then the model is more aligned rather than if X is big. But what's interesting is that, well, okay, there's a quite common phenomenology, is that it happens just at a single point. It happens instantly. So, yet again, if you take X slightly less, just 1% less than some critical value, then take the limit when the mesh size tends to zero, then basically the picture you will see from distance looks like that. Here, there is a kind of a cheating. So, every concrete spin still has a quite reasonable chance to be colored blue. I mean, it depends only on the states of four neighbors. So, it has some concrete probability to be colored blue. But if you make a kind of an averaging, say, if you consider a majority in a block 100 by 100, then it turns out that there is a quantitative majority. So, this is why, okay, here it's drawn all red, though every concrete site still has a reasonable chance to be blue. So, okay, if you do some averaging, then basically what you see are just this ocean of blue and this piece of red land. Okay, also there is a question of boundary conditions. I never told you what are boundary conditions. It depends. So, for instance, in this picture, there are so-called double ocean boundary conditions. You color everybody here by blue and everybody here by red. So, then if you are above criticality, then actually what you see is a total chaos. So, at least at first glance, you do not see any structure. It is supposed to be at least after appropriate averaging, the same picture as for the percolation, though it is not a theorem yet. So, this is an open question, just if this is another way to become famous. I mean, just prove this result. So, we now have two ways to advance your career. And then there is this critical situation, which I will mostly discuss in this talk. So, the material introduction is going to be very general. So, in this critical situation, many conformal invariant quantities but that's a covariant quantities appear. Okay, just a conformal invariance phenomenology. Okay, here are also some history. Okay, there were doubts about phase transition, then it was clarified. Then the concrete value was found by some combinatorial argument. Okay, this pi over four actually is the half angle of the square. If you sample the same model on the honeycomb grid, for instance, it's going to be pi over three and on the triangular grid is going to be pi over six. And then it was proved actually by some tater's computations. I'm going to comment maybe a bit on them later on. So, right now, okay, we think about this picture. Our goal is to describe it. Something interesting happens and we want to understand what. Okay, so what are possible questions? The very first possible question is the following. Okay, I told you that there is a phase transition and actually even at the critical, already at the critical point, if you think about the probability to see spin a concrete point red or blue plus or minus one, this probability tends to zero when the size of your box tends to infinity or the mesh size tends to zero. So, the first question is what is the scaling exponent? And the scaling exponent appears to be non-trivial. So, this is, well, quite famous in this business, the scaling exponent one over eight. So, one of the goals for today, for instance, is to understand why one over eight. I mean, how such a number can be actually derived from, at least guessed from rather general considerations. Okay, and then if we want to prove it, we need to compute something and that's exactly what people did. So, they did computations in the full plane. So, you take the full plane limit and then you take two spins here and there and you, okay, there are some tools to compute these expectations. Anyway, this suggests that okay, this is a good scaling, but then you can ask the further question. So, what is the further question? What are the scaling limits? Imagine you fix the shape of your box. So, like this rectangle or maybe something more complicated, you rescale the correlation functions and you ask, okay, if I'm, well, could one prove such a convergence or not? And if the answer is yes, what is the structure of the limits? In particular, I told you that there is a conformal invariance in the model. So, it means that if you apply conformal transform to the shape of your box, so just every one-to-one homomorphic bijection, then the answers, the scaling limits in this domain and in the other domain, something like here, maybe disk, which is discretized in a completely different manner, they must be related, well, by a very simple rule. And this one over eight, well, it should be consistent with this rescaling procedure. Otherwise, you immediately arrive at the contradiction. Okay, so, right now there's a theorem. Then, of course, you can ask more questions of maybe more probabilistic nature. Okay, can I say that if I consider random distributions of this kind, so I just consider those functions plus minus one, then I rescale them, so I have something weird, but this is still Schwarz's distribution in the limit, probably. Well, is it possible to prove that there exists a limit in the space of Schwarz distributions? The answer is yes. And, okay, it gives you a kind of an interesting example. So, this random field in a limit is not Gaussian at all. Well, it has some completely different properties, so you can study it. So, for instance, okay, there was a paper to study the disorder relevance for this field when you add some random magnetic field to the model, so ask what is the response, and so on and so forth. So, this is a kind of an interesting object. So, those are questions, first questions to study. Another question is the following. Okay, here we speak about correlation functions, but there is another viewpoint, possible viewpoint. So, let me try to think about this picture geometrically. So, okay, this is an assignment of plus minus ones, but I can think about boundaries separating them. So, this is called domain walls in the physical literature, and try to understand what is the limit of these objects, what is the limit of these curves. For the abrush and boundary conditions, it's going to be a curve running from here to there, which separates blue, which keeps blue to the left and red to the right. Say, for all plus boundary conditions, it's going to be just a collection of loops here and there, and you can ask, okay, if I'm able to prove the convergence of such an object. And because of this conformal invariance phenomenology, it was supposed for a while to be conformally invariant limit, and now it is proven it is indeed a conformally invariant limit, something very concrete. So, this is also a theorem now. So, that's another example of a question. One can study at criticality. So, coming back to the plan. So, this is like a conformal field theory side. This is like a conformal loop ensemble side. Yeah? Is it related to the boundary condition? Nope. No, this is just... I mean, it's symmetric, right? Yeah, that's true, but that only means that n must be even. It means that already in discrete, this quantity is zero unless you have the even number of speeds. I was thinking about the rules. Okay, yeah, actually it cited badly. So, what he did, it is two-point correlations. So, all these results, those are two-point correlations. I'm going to comment on them. Yeah, that's a very good question here. It might be misunderstood here. So, what I'm saying is that, well, what people did, they computed two-point correlation functions, and they found an exponent one-quarter there, which intuitively means one over eight per point. Also, okay, I wanted to skip it, but nevertheless, here there are two different computations. And a priori, so this computation, you take two-point correlation functions at a very big distance, but below criticality. It gives you a sum of zero number, which turns out to be, okay, of order x minus x critical to one over eight. And here is another computation. You take the infinite volume limit. You take two spins at fixed number of latest steps. You compute what is the correlation, and then send this separation to infinity. And it's again one over eight. A priori, they are not the same. So, this is the kind of effect about the Isom model that they are the same. This is related to the fact that energy density has scaling exponent one. That's exactly this fact. Okay, let us spend now some time on combinatorics. So, the goal is to explain why this structure of functions appears in the Isom model. So, this is just the partition function. Remember, we sum over all configurations of pluses and minuses. And here is the price to have a concrete configuration. And one of the reasons, one of the ways to see the structure is the following. Actually, there is this many combinatorial correspondences with what is called the dimer model. So, on sum, actually a graph. So, what is the dimer model? Given a graph, it's much better to have a planar one, but a priori it is not. You are interested in perfect matchings of its vertices, meaning that you want to find a subset of edges such that it covers every vertex exactly once. Subset of non-intersecting edges that covers all the vertices. And okay, what is the link? As I mentioned, there are many such correspondences. This is my favorite one. Given the configuration of spins on the original graph. So, what you do, you just keep all the edges, all the long edges, separating those pluses and minuses. So, you take this one, but not that one, for instance. And then what you see on the green graph is that, well, but what remains is just to fill the cycles. Those are even length cycles. There are two ways to match vertices there. So, basically, this is one to two to power v correspondence. Quite transparent. And for the dimer model, there is a famous result due to Castile. And this research follows. So, the partition function actually can be computed is a determinant. Okay, not a determinant, it's fathom. If you never heard what the fathom is, okay, here is a definition. You sum over all permutations product of n entries. This factor, basically, it reflects the fact that when you permute, say, sigma one and sigma two, then the sign changes and so there's two terms and the matrix is anti-symmetric. So, the answer is the same. So, these two terms are just the same. And okay, basically, it says that only the pair in the structure of the pairing matters. But if you never saw it before, just think about determinants. This is a square root of a determinant of some anti-symmetric matrix. So, the message, highly non-trivial message here is that if you have a planar graph, there is an assignment of weights to this agences in matrix, such that this formula becomes true. A priori, this is absolutely non-trivial, but this is a classical result in combinatorics. So, what is the message? Okay, at least the partition function is computable. Well, by simple means, by the way, there exists another formula for the partition function which is called the Cuts Word formula. I'm not going to comment on it much. So, now the matrix is labeled by oriented edges, not this green vertices, but oriented edges. And the non-zero entries are essentially those where the second edge prolongates the first one. And there is some complex factor here. Okay, yet another formula. In fact, this is just a equivalent to what I explained above. You might notice that vertices of this type are in one-to-one correspondence with oriented edges. So, this is why there is not such a big surprise because of the equivalence. If you are interested in details, here is a reference. So, all this is a very, very old story. Unfortunately, we didn't find a place where it is written in, well, I mean, in complete manner. So, okay, we wrote a kind of an exposition. So, we don't pretend on having actual new results here, but this is a sort of an exposition. Okay, so, there is such a formula. And then, why is it, I mean, for what is it good? For instance, you might notice the following. Imagine you want to find the probability that these two spins are different from each other. This is something what is called energy density in the Isian model. Why? Because the probability of a full configuration, somehow, is built up on the probabilities of two nearby spins that are being disaligned. So, imagine you want to find this probability. But then, what you can do, you know, that basically, on the dimer side, what's that? This is just a partition function of those configurations, which do not contain this edge. So, I mean, that is just partition function of a configuration on a graph with these two vertices removed. And because of that, there's just an entry of the inverse matrix. Okay. What is the message? Some quantities relevant for the model that can be expressed in terms of k minus one. But what is k minus one? Just by definition, this is an inverse matrix to k. So, you can write this identity and say, okay, but k is a kind of an adjacency matrix. It is a very sparse one. I mean, in the typical row, only, say, three or maybe four elements are not zero. So, because of that, if I think of k minus one as a function of its second argument of this E, which is an oriented edge of this C, which is, okay, green vertex of this type, so a corner of the original graph, then I have some identities for those entries. And come back to the analogy with random walks. This is exactly the discrete, well, an analog of discrete harmonicity identities for every hidden probability, for a green function, for whatever reasonable object for random walks. Okay, not whatever reasonable object, but for many interesting objects for random walks. Okay, this is something that satisfies local identities. And this is exactly the structure which underlines the model, but this is not the end of the introduction. So, here is yet another way to see it, which was advertised by Smirnov in, okay, 15 years ago, I would say. So, you can ask, okay, but is there any combinatorial way to represent such entries? And here there is one. So, this is what you can do, might be a bit technical. You see some strange complex factors, so what is that? So, you define a function on edges of your graph, and to give a definition, what you do, you sum over all possible configurations, start of this kind. You start at some point, and there is a curve running into this edge, and maybe many, many loops. And you sum some contributions, okay, product of weights with some complex factor. Actually, we already saw this factor here in the CutsWord approach. So, okay, this is the same. Okay, this is some function, which is now complex valued. It can be written in terms of diners. What is, for what this can be, can this be good to consider such a representation, just because local relations becomes very, very transparent. So, still it satisfies some local relations. One of the ways to see them is just a corollary of those ones of relations for k minus one, but you can give an independent proof. So, this is a function, which is now, at least for the critical model, it satisfies local relations, which are a form of Kashiriman identities. So, here, okay, some bell rings. We arrived to some discrete homomorphic structure, discrete complex structure, okay? Moreover, okay, inside of the domain, this is a discrete homomorphic function, but what are the boundary conditions? And then, on this picture also, the boundary conditions can be visualized. Actually, they are the following. So, if we are at the boundary, then the rotation, the total rotation angle of this path is fixed. And because of that, okay, for your function, the complex phase is fixed. This is a bit weird boundary condition, but you can have a hope to handle solutions to such boundary value problems. Actually, this was an idea of Smirnov, that to understand the model, the scaling limit of the model in finite regions, one should think about these boundary value problems. One should represent quantities of interests, interest in terms of solutions to boundary value problems of such a kind, okay, which we prefer to call Riemann type boundary value problems. So, the message is that, okay, there is such a structure. In particular, as I already explained, the energy density, the product of two nearby spins can be expressed in terms of k minus one, so it can be expressed in terms of these observables. And, okay, it gives you an access to the scaling limit of those energy densities. What about spin correlations? So, spin correlations, well, they are more complicated, so there are some non-trivial exponents. So, how should you think about spin correlations? Imagine you are interested in just two-point correlation between here and there. So, what should we count combinatorially? Combinatorially, if we think about interfaces separating pluses and minuses, effectively, we should just compute the parity of the bonds taken in the configuration along this path. So, we should link them by path and compute the parity of the edges along this path. Here is exactly what I explained. So, this expectation, it becomes just the ratio of two 5-pheons. So, this is a path of the original matrix and this is a path of a matrix where you change plus one to minus one on these edges, ratio of 5-pheons. Okay, how would I analyze it? It's not completely clear because the ratios, they do not satisfy any identities, but what I can do, I can ask the following question. What happens if I shift U1 to some nearby phase? Effectively, it means that I slightly change the numerator. I slightly change this matrix. And because of that, the ratio is just a concrete combination of those entries. And it gives you an idea. So, if you would like to analyze a spin field, what you must do, you should consider such matrices now with some cuts, maybe. And they give you an access to special derivatives of spin correlations to the ratio of those when one of the points moved one step. And again, this is fairly, fairly general. I mean, it works on whatever planar graph. I mean, it's just the structure of the model. Okay, and of course, instead of slits, you might think in a more invariant way, you might think about, or should maybe think about, double covers of your graph, branching over these marked vertices, I mean, to make everything independent of the slits. Okay, you also can define these entries combinatorially, so this is a headache and I'm not going to show you a definition. So I tried at least 10 times and there is no chance to explain the combinatorial constructions during the talk. Yeah, sure. Double covers. Okay, that's like a rim and surface of a square root of a polynomial. So you just, they take the double cover branching over all these vertices. This is essentially unique, right? And then, okay, instead of this change in signs, what you, I'm going to discuss this in more details. Think about rim and surfaces of square root of a polynomial. This is a good example. Okay, and instead of giving you a sort of an analog of this picture of this formula, which is boring, I would like to mention yet another combinatorial approach, which else is very classical, so which is called the spin disorder formalism. And again, this is like 40 years old. So what is that? So, okay, we had spin correlations. Now, and spins they leave on faces. Remember, I work in a slightly non-classical notation. So now let me pick several vertices of my graph. Here there are two. And then let me do the following. So let me draw a path, choose a path on a lattice. And replace all the interactions along this path. So this is not a path drawn on the picture, it is somewhere. And replace all the interactions, okay, by inverse. It means now that along this path, I prefer the spins to be disaligned instead of preferring them being aligned, okay? And consider the new partition function. And the more invariant way in this situation is just to think as follows. So I take a double cover. Now I assign plus minuses to faces on this double cover with a built-in symmetry. So if on one of the sheets I have plus one, on the other I must have minus one, okay? So I consider such a model. So those are like assignments of spins to this faces with this agreement, this constraint. And then, okay, after I said all these words, I can't think about spin correlations on this double cover so I can define those functions. By the way, here there is no path after all. So this is an invariant definition. I mean, I started with saying, okay, I draw some paths and do some modification here and there, but this definition is now invariant. So okay, I can think about these correlation functions. Those are correlation of spins here and there with those disorders inserted. Okay? You see, it's a nice picture due to Clément Anglère. So something strange happens when, with the colors, when they go around. This is exactly this double cover structure, how he sees it. Okay, so I can try to study these functions. And by definition, they change the sign when I move around the branching point. What is the link with what I told you about before? The link is the following. So here there is no holomorphicity at all. This might be thought as a kind of a generalization of spin correlations. But nevertheless, it pops up in the following way. Now let us take a corner of our graph. So just what is a corner, I take a face and this is my U and I pick one of the vertices. This is my V. And let me try to consider those V and U just nearby, as I explained, just a face and a vertex of this face. And try to plug them into this correlation function. What happens is that you arrive exactly to the same quantities. So those are exactly thermionic observables I introduced before through dominoes. Okay, why? Okay, why? You should do something to prove it. But effectively, the reason is that when you expand everything combinatorially, you just see the same combinatorial expansions. So this is not something you expect a priori, but nevertheless, it is there. And okay, so it means that you can play with all the correlations of that kind. So you can think about mu mu mu sigma sigma sigma and one psi and every such a correlation gives you a discrete holomorphic function just because we already knew that all the identities for psys, they can be thought of as discrete holomorphicity. And this is again a very, very old idea. So this observation on the discrete holomorphicity goes back at least to 80s, so maybe even to 50s. Okay, this is yet another viewpoint on the combatorics. So what is the punchline right now? The punchline is that, okay, there is a structure of functions in the 2D nearest neighbor model that can be used to analyze, I mean to answer questions. Okay, so for instance, here is an example again how one could use this formalism to analyze spin correlations. Okay, imagine we want to shift one of the points to a nearby point. So we want to analyze this ratio. What do I do? I just consider this formal correlator of spins and disorders. And then, okay, I have a holomorphic function and this disorder is placed exactly here, nearby of you. And then I say, okay, what is a particular value at this corner? At u plus half delta. Okay, by definition this Pc is sigma multiplied by mu, so mu and mu cancels out, so this is sigma, sigma, sigma over sigma, sigma, sigma. So this is effectively one, actually i because of this factor. What is its value here? So again, this is sigma by mu, mu cancels out, so this is the correlation written above. So what is the message? Here is a concrete example of a holomorphic function. Such that it is normalized in some way. It has also some singularities at other points. And such that local values, this local value, give you an access to the quantity of interest. Here is an example of how this philosophy works. Just a very concrete, okay. And again, functions of such type, they all can be thought of as solutions to boundary value problems in discrete. This is the underlying idea. Okay, so just two slides on again illustrating how one could use this idea in a very classical setup when we want to compute the two-spin correlation function along the diagonal in the full plane. So just imagine you want to compute this function. And here there are results I already mentioned. By the way, this is a really revised version, so I mean, this is an old computation, but now it can be packed in two pages effectively. So what should we do? We should find a function. Let us focus on this type of corners. So we should find just a real valued function in the full plane, which branches around this and that. So it has a structure of like the square root of Z minus A, Z minus B. Which satisfies local relations. And I'm doing this a way of criticality. So my X is just something, just below criticality. So what are the local relations? The function on all the corners, it was a version of discrete homophicity. The function here is like a real part. It is a version of harmonic functions, actually. You just see that, okay, the question is, you need to find a function which satisfies those local relations. And you already see why the value X critical is special because those relations become just harmonicity. And which decays at infinity because of, okay, along the limit we take, the boundary conditions decay at infinity. But this function must be more or less unique, right? It must be something very, very concrete. Actually, this is not hard to say that there exists only two parameter family of those, and they can be constructed more or less explicitly. And the quantities of interest, well, they're exactly two point correlation functions. Okay, not only at this temperature, but also at the dual temperature, okay, at this tangent power four minus theta. Think about criticality. So at the critical value, those D stars are just D. So you see it is even symmetric. So these values are, well, it contains D and being multiplied by something very concrete, you do a kind of Fourier analysis, they become, I mean, this polynomial actually becomes a function which does not contain any term here. And instead, you see the correlation Dn plus one. And this gives you a recursion. So you can do all these computations more or less explicitly using orthogonal polynomials technique. So you already see here is an orthogonal polynomial. So this is a polynomial which is multiplied by some weight, real symmetric weight, and becomes orthogonal to all the modes in between zero and n. So it allows you to express just Dn plus one via Dn. And then you can study whatever you wish. You can write all the asymptotic expansions, for instance, and so on and so forth. So what is the message here? The message here is that the question of interest is reduced to finding a very concrete function, like a discrete harmonic function, in a plane with two branching points. And that's it. And as for scaling limits, okay, as I told you, here we work in a full plane, so there are techniques like Fourier analysis available. But if we are in a bounded domain, we can't think about all these functions as solutions to boundary value problems. And then instead of Fourier transform, what we probably need is like conversion theorems. So we need to be able to say that the solution to a boundary value problem in discrete converges to a solution of a boundary value problem in continuum. Similarly, for instance, as you could analyze problems related to random walks. So if you want to say something about a random walk in a domain, I mean about hitting probabilities, for instance, of this random walk, one of the approaches is the following, is to say that this discrete heat, this hitting probability in discrete is a harmonic function. So it is a solution to a Dirichlet boundary value problem. So because of that, it converges to a solution of a continuous Dirichlet boundary value problem. So you have an information. Similarly here. So now I'm not going to give you many details on these results. Now the converter expert is finished. I even said a word on some classical computations. And now I'm going to present you just what one can gain out of it. Actually, there are many, many results proven during the last decade. So okay, it's going to be more like a description of this field. So the first result I already mentioned. So this is about the energy density. Remember, energy density is essentially the product of two nearby spins. Okay, you should subtract a constant because if you take a product of two nearby spins in the limit, it's going to be just some concrete number depending on the lattice. Okay, they have some probability in the limit to be aligned and some probability to be disaligned. So to have this result, you subtract a constant and then this is going to be, is going to vanish in the limit and the rate of decay is exactly delta minus m. Compare this with the result of spin correlations I showed you. Here it was m over eight. Okay, so this is another scaling. So one over eight is a scaling exponent for the spin and one is the scaling exponent for the energy density. And here is a theorem. So you can analyze all the correlation functions of those energy densities in the limit. So you pick some points, you fix a shape of your domain, you consider better and better discrete approximations and here is what you see. And those guys, they are conformally coherent, similarly to the result on spin correlations I presented with the exponent one. And because of that, it is enough to give an answer in some concrete domain and in the half-plane, the answer is extremely simple. There's a faffion of some simple metrics. We shouldn't be surprised that the faffion appears, right? Because the energy densities, this is something about k minus one for dominoes, this is something about fermions actually. So, okay. And how do you prove it? You consider thermodynamic observables introduced at the very beginning. So those essentially entries k minus one or sums over those pictures, you prove a convergence of those observables to conformally coherent limits and then you recover everything back. So this is the idea. I'm going to say more on spins, not on energy densities. For spins, I'm going to show you just a concrete boundary value problem one should consider. And here, okay, the corresponding result for spins, the setup is exactly the same. So you take a domain, you fix several points and then, okay, you have a convergence of such a kind and again in a half-plane, there is an explicit formula which is not new. Of course, it was known in theoretical physics for years, but it's not that simple. Okay, yet again, the spin field in the limit, it doesn't have any simple structure. Just if you think about correlation functions, it's not Gaussian at all. I mean, this is something much more complicated. Let me also mention that, okay, this is our theorem, but there is another approach to spin correlations, notably due to Julian de Bidet, which is called exact positionization. Okay, what's the question? Okay, what is the general strategy to prove such a result yet again? So we know how to encode special derivatives of these quantities in discrete. So we can prove convergence of solutions to discrete boundary value problems to similar problems in continuum. It is not trivial but technical. Then, out from these convergence, we can derive the answer for special derivatives. Again, this is not trivial because you need coefficients at singularities, you need a careful analysis near singularities, but okay, it's doable. And then, okay, you have an answer for special derivatives and you can recover correlations themselves. This is a question of one multiplicative constant. Yet again, not trivial. So you need some probabilistic techniques for that like FKG and GHS inequalities, if you know what it is. But I'm not going to give you details. Instead, I'm going to give you an example of a boundary value problem to consider. So if you just want to handle one spin, then you take your domain and you want to find a function which branches over at this point, which has these boundary conditions I briefly mentioned at the beginning. So a weird one. So you take a domain and you want to find... So here is your domain, let it be smooth, for instance. Here is the branching point and the function is analytic inside. But on the boundary, what you know is that it has an argument complex phase depending on the tangent vector at this point. So okay, it's written on the slides. And then it has a prescribed singularity at the branching point and turns out that such a function is unique. Okay, this is not like the green function you see for random walks, but anyway, this is some boundary value problem. So it is unique because of that, it defines this boundary value problem, it defines you the next coefficient in the expansion. So it has some coefficient in the expansion. So let us denote it by A. And then the theorem is that special derivatives they do converge to something related to A. And if you want to recover 1 over 8, just to understand why 1 over 8 appears, you say the following. So how a solution to such a boundary value problem depends on the conformal... How does it behave on the conformal transforms? This is simple. So it is just multiplied by the derivative of the conformal map. 1 over 8, it must be consistent with boundary conditions. But then the question how this coefficient behaves on the conformal maps is a question of simple calculus. This is an exercise for the first year students. So because you know this is some computation near singularity and what you gain is this rule. And then when you integrate it back, okay, this is conformal invariant, but this leads to a multiplication by 5 prime to 1 over 8. So effectively, if you understand what is a nature of boundary value problem, needed to be considered, effectively, you immediately guess that 1 over 8 should be there. Okay, then you can do... This is just a single slide, okay, not two slides. Then of course you can ask more. So you can say, all right, now I have two fields, spin and energy density. But also I mentioned disorders. I mentioned fermions. How about all the mixed correlations? Do they converge to something? What are the rules for the limits? This is the subject of the conformal field theory for which this is just a kindergarten case. I mean, so this is a very simple example of a conformal field theory. So okay, you can prove convergence of all the mixed correlations. This is still in progress unfortunately, but we are right and okay, we already have 50 pages. So it is there. I mean the only question is how to make it readable because all the methods are there, but as usual, kind of a problem is that if you do it in straightforward manner, you end up with 200 pages that are impossible to read. That's it. So okay, what is important for CFT? What happens when you collide two points? So for instance, you can say, okay, I have these correlation functions which are now defined as solutions through solutions to some boundary value problems. I remember that if you and you were close to each other, then it was like the energy density. And for energy density, I also have a correlation function. Can I prove this in continuum? Can I see this in continuum? And so on and so forth. So you can ask questions about the structure of this CFT. Of course, this all is known in the physics literature, but somehow there, the existence of such correlation functions is assumed from the very beginning. So what conformal field theory starts with is just the claim that it must be some fields that you can understood them through correlation functions. It must be objects of that kind that satisfy conformal covariance and this and that fusion rules. So and if you assume that, you can recover all the structure. What we are doing here is totally different. So we start with a discrete model. We define all these objects, you might say like limits of discrete objects and then prove that, okay, they satisfy all the assumptions of the conformal fields. Okay, there are some projects on more advanced stuff. I'm not going to go into more details. Instead now, probably I'll be tired because of all this notation. Also this ideally one should do this on Riemann surfaces. Here we work with multiply connected domains and general boundary conditions. This is already quite general. Maybe it's even better to write on Riemann surfaces. Okay, this is a project for somebody, essentially. All the tools are basically there. And now I'm going to switch to spend remaining 15, maybe minutes or 10 minutes, how many of the talk discuss in the geometric side of the story. And then briefly, it's going to be several directions, several open questions, what one can still ask. Okay, so just forget everything I told you about correlation functions. Now we are purely geometric. So we want to understand the low of the scaling limit of these loops of boundaries between clusters. So if it was plus boundary conditions, then there's just a collection of loops. If it was free boundary, if it is free boundary conditions like here, then also there are some interfaces running from boundary to the boundary gain. What is the intuition? The intuition is that loop ensembles of this kind think about plus boundary conditions. So plus boundary conditions, what you have is a main. And many, many loops floating inside, actually they are nested. So here there is also something. But let us focus on outermost loops. Let us try to guess the answer. What should be the low of these loops in continuum? We assume it should become formally invariant because it must become formally invariance in the model. And also what we know is the so-called domain what we expect, not know, but what we expect is the so-called domain mark of property. So what does it mean? It means that if I just take some domain and observe everything which is inside of this domain, including the loops which intersect this domain, then in the remaining part, indiscreet at least, what I have is again a sample of the Isen model on which I know only the fact that along the boundary the boundary conditions are plus. So it must be an independent loop ensemble of the same kind. So let us assume these two axioms and try to construct such an object. Not thinking about lattice models at all. So what is the most natural for in probability just the most natural conformally invariant object? This is Brownian motion. So, okay, let us think about Brownian loops floating around. So you should put an appropriate measure, conformally friendly measure on those loops. And then, okay, there are many, many loops. And let us take a Poisson process, a Poisson cloud of these loops. So here is a kind of an intuition drawing. And we need something non-self-intersecting. So because of that, let us take the outermost boundaries. So here they are. This is a very basic property of Poisson processes is that the result, if it is well-defined, the results satisfy conformal invariance because of the construction because of the Brownian loops. It satisfies the main Markov property. So here is a natural candidate. The philosophy is that, okay, here I have an object constructed by some other means, but this is a very natural candidate for the answer. And there is an amazing theorem due to Scott Scheffelt and Wendel and Wernher, which says that actually those two assumptions model of the loops are simple. They are indeed sufficient to claim that the loop ensemble must be, must be one of those loop super constructions. That the only parameter is just C, which is the intensity of the Poisson process. So what is the bottom line? We have a natural candidate for the geometric description of the model. Okay, but then of course, how one could prove a convergence and what is the value of C? Okay, I'm going to comment on this. So here is just a definition of an object in continuum. And this is a theorem, recent theorem, due to Stéphane Benoît and Clément Anglère, that indeed the loops are rising in the criticalizing model. They indeed conversion low to this loop sub construction, to the output of this Brownian loop sub construction. Okay, either you should consider doubt or most loops and then this is the construction I explained, or you consider all the loops and then you iterate the construction inside of each of those clusters. Both variants, okay. Okay, this is not an easy walk. This is really a tip of the iceberg. So many people were involved and it took some time to develop some necessary ingredients. So here is a list. Okay, Hugo is in the room, so I ask him. Those gentlemen that are in Helsinki, Clément is in Lausanne, Stéphane is in MIT I guess now, somewhere in US. Stas is everywhere, so okay. So I'm just going to give you an intuition on the strategy of proving such a result. So first, you can simplify. For a while, you can think about much simpler problem. So what is the law of a single curve? Remember the Brochian boundary conditions, for instance. So they fixed a long curve from one boundary point to another. So you can focus on it. This was an idea of SRAM and ask what is the law of this interface in the limit. Then you can try to glue this picture just by saying, okay, I'm going to observe my model, just tracking say this and that interface and thinking what remains and so on and so forth. So I can try to build a kind of iterative exploration of my model where every time I track a single curve. Something like that. To be able to do so, everything is rough. So these curves are not as smooth curves in the limit. So you need also some technicalities, which we call Roussen-Seinberg-Welsch type bounds, which mean that, okay, whatever even bad domain you are given, the probability to have a curve, I mean, if it is very thin, for instance, the probability to have many crossings, to have a crossing, I mean, from top to bottom is very small, something like that. So you need a priori estimates that these curves are not weird in the limit, that you can still speak, describe them in terms of continuous curves, that there is no, they are not chaotic enough in a sense. They are regular enough. Okay, so some details, but again, well, okay, this is a very, very this is only a glimpse of ideas behind. So how to track a single curve? This is the first step. So how could I describe the law of this curve if I already know the behavior of correlation functions? Here is a simple observation, just take a point inside. So for instance, consider a domain with plus, minus, and free, meaning I'm not saying anything on the spins, boundary conditions. Let me take a point inside z and consider a random variable, which is just the value of the spin there. And then let me start tracking this interface. Effectively, okay, if I had conditioned this random variable on the filtration generated by the beginning of the interface, this is trivially martingale. I just conditioned a random, I take conditional expectation of my random variable with respect to a growing sequence of sigma algebras. This is a martingale. But those conditional expectations, they all actually of the same kind. So because if I conditioned on this beginning of the interface, the only information I gain on my model is that the boundary conditions are still plus here and minus there. So again, the conditional expectation is again a quantity of the same type. So because of that, okay, so I have a martingale and then this martingale is a correlation function. So I explained that you can prove a conversion statement. So you have a martingale in the limit, which you're supposed to be a martingale for the limit of the discrete curves. So the message is that, okay, it is enough to describe the law. Okay, spins is not the same technically is a hard example. You should work with other martingales, but never mind. Okay, to be able to perform this program, you need more ingredients. So you need to say that, okay, you want to say that the limit of martingales is a martingale for the limit of curves. First, you need to know that the limit of curves exists. This is a sort of a tightness statement. Okay, here are some links. Then I'll say you should be able to interchange the limits. Okay, those are again some estimates, but all that can be done. So you have conformal invariant limits for the curves. And then as I said, you build up a really complicated and clever exploration procedure, which, okay, basically, you can think of it as that you track all these curves from all the points to all the other points. So, and then you have a law of at least these parts, and then you switch to a different representation of the model. I'm not going to speak about that. Really clever exploration algorithm, which allows you to prove that there exists something conformally invariant in the, there exists a conformally invariant limit of loops as the mesh tends to zero. And finally, for this limit, which maybe not, I mean, not fully understood through this construction because the construction is complicated. But for this limit, you know that it satisfies the two axioms. I started with. You know that the whole procedure is conformally invariant, and you know that the Markov property is still there. And then you use the theorem due to Scheifeld and Werner and say, okay, but then I know the answer, and the only question is to identify C and okay, it can be done. Okay, so the punchline is that starting with observables, you can track single curves because of this Mertingale ideology, because you have enough information in a sense to characterize the law of the curve if you observe it conformally from many, many points Z in the domain. And then you glue them together and you have some loop ensemble, so okay. You prove this really nice theory. So okay, let me now indicate some open questions. The first open question is directly related to what I told right before. This theorem, the proof of it, it goes across the whole universe. And okay, you need some holomorphic functions. You need identification of single curves and so on and so forth. But at the end, the very end, the result is okay, this construction, it admits some discretizations as well. The combinatorial question is, is it possible to see a kind of a discrete analog, a precursor of this construction inside of the Isen model? It would be very nice if it works because as I tried to explain, this is really hard theorem. I mean, it needed a lot of work. So okay, nobody knows. So on the one hand, the result is simple and one can hand wave that, okay. Why shouldn't we try the same discrete? On the other hand, it might be that the answer is just so universal, like the Brownian motion, that it appears in very, very different manners and the priori it could be that there is no link between this and that procedure. I mean, just a message that the answer is extremely universal. Okay, the second question is the following. We have two sides of this story, even at criticality. One is about the correlation functions, this is CFT, the other is about loop ensembles, this is CLE. The question is, can we forget about the discrete model and construct correlation functions just out of these loops? Because traditionally the story was that people studied lattice models because of that, it was a motivation to study CFTs and geometry appeared later. Now the question is, can one go back? And okay, nobody, people, well, I know that people are working on it, but there are no definitive answers. I mean, there is only very partial progress. And this, okay, should be very, very probabilistic because you forget about all the tricks in discrete. So there's just questions about Brownian motions in a sense. Then the question is, okay, how universal are all these convergence results? I briefly mentioned that you can play with different lattices, regular lattices, square, honeycomb, diagonal. Also, there is a family, a special family of what's called isoradol graphs, but the result itself is supposed to be extremely universal like convergence to the Brownian motion. Let me indicate that even if you start with a general periodic graph, it is not null. My impression is that it should be doable, but okay, it is just not the questions, just to prove this convergence for, I mean, in a reasonably general setup. You can ask about not nearest neighbor model. This might be infinitely hard. So because of what? Because remember, I told you that for the nearest neighbor model, there is a structure, underlying structure of functions, and here you immediately lose it. So there is no fashion structure in discrete, but nevertheless, there is some recent progress. It appears in the limit, so okay. I would say that as a result, well, the first day about energy densities, I mean, spins are not treated yet. They are rather hard, but okay, it might be possible. Also, okay, as I mentioned, you can play with whatever planar graph you wish, so you can imagine a lot of different setups like regular interactions, or just go downhill to our say, there is a big activity on planar maps, studying model on planar maps. There are some projects on that. And the very last question is the following. I'm coming back to the phase transition. I told you that, okay, if you take the temperature above criticality, then what you conjecturally see is the same as percolation. This was the second way to become famous and more constructive, rather than asking PhD advisor. Just prove it. Just prove that for a fixed temperature above criticality, the limit has the same geometrical description as percolation. So you still can think about the interface keeping blue to the left and red to the right, and it is known what is the limit for the percolation. So this is a cellistic curve. So just prove it for a supercritical model. That's it. Thanks. And good luck, of course, with this question. There is maybe time for one question. Are there similar works for using modeling in three dimensions? Well, that's a very good question. Very natural. The answer is it is not similar in the sense that in three dimensions, there is no such structure of functions. So because of that, there's a totally different story. So what you can do rigorously, I mean, mathematically rigorously is, okay, I'll ask you to go on. Just continuity of phase transition, for instance, a famous result, which was done two years ago, right? Three years ago. I mean, just recently. But the message is that mathematically, you do not know how to proceed, say, for scaling limits. So right now, the results are much weaker. Even on the theoretical physics side, if you ask what is the exponent of the spin field, it is not supposed to be a rational number or even algebraic, I think, number. So there's just some number. And the predictions for a while were based just on Monte Carlo simulations. And only recently, due to work of Slava, Richkov and collaborators, some still mathematically non-rigorous, but okay, some method to compute approximations to them numerically, but not by Monte Carlo simulations appeared. And it was also a breakthrough. So in 3D, this is a very important question to understand what's going on. But I should say that the progress is rather limited. I mean, both continuity of phase transition was a very much appreciated mathematical world and this work of Richkov and collaborators was really appreciated in the theoretical physics to give you an impression. Another question. So maybe I have a question. So suppose I'm a master student and I look at this and I think, wow, so it looks like if I want to work in that field, I have to read 70 pages, 70 years of physics literature and many decades also of mathematics literature. Can you? Well, the rest service first. I mean, for combinatorics, I actually gave you a reference. This is still not a short paper, but okay, if you are just interested in formalism, I mean, how this appears, it is there. So this is our paper with David Simazonia in Atreian Castle. So for the structure of correlation functions at criticality, well, okay, for the curves, there are lecture notes by Hugo, which are a bit outdated now, but still, okay, you can read them. Unfortunately, we still do not have a book on that, but maybe at some point, I don't know, I wouldn't like to make any claim, you know, for the record, so please switch off. But maybe at some point it appears. So there are surveys, this is the message. I mean, if you are interested in concrete, in a concrete part of the story, typically now there are some sort of papers. Okay, so let's now have another coffee break and thank Dimitri again.