 These are probability measures, right? A is a subset of probability measures. So now you can minimize IV. It's a different A. A is a subset of probability measures. And you're asking, what's the probability that I'm going to land in there when I take the limit as n goes to infinity? Yes? So the rate is this, IV minus IV of mu V. That's the rate. I wrote it here. It's the functional IV minus its minimum. You have to subtract the minimum. So the rate is always a functional, OK? All right, so now we have information for the setting of random matrices, for example, because the points, typically, will also arrange themselves according to the circle law or the semicircle law. By the way, I should quote names for this theorem. So this was proven by Benarus and Guione in the 1D situation, Benarus and Zetuni, and for the real Geneva ensemble, Petz and EI. But you can easily generalize the proof to all these situations. In fact, you can write the proof for any G, any nice G that decays. It's not really dependent on the Coulomb nature. So far, if you want, everything I've said is not really dependent on the Coulomb nature. There's always an equilibrium measure, even if G is not Coulombic. There's always a large deviation principle, even if G is not Coulombic. So the Coulomb nature is going to be exploited later, in fact. OK, so at this point, we can pause and ask ourselves the main questions that we want to ask, which is, now, we know this thing. We know the sort of macroscopic behavior. We know it for minimizers, but also for situations with temperature. So let's say for typical configurations. Another way of writing it is I'm going to be interested in what you can call the fluctuations, which is the difference between the sum of Dirac's minus N mu V, so called this fluked N of a configuration. So this fluctuation from what we've seen, we expect it to be much smaller than N, in some norm that needs to be specified. Because we know that this converges to mu V, so it means that when you multiply by N, the result is much smaller than N. And the question is, can we do better? So we know now that the distribution of the points approaches the equilibrium measure, but how well does it approach it? And in fact, the striking fact is it's going to approach it much better than this. So we really want to understand these fluctuations. And for example, we want to look at these configurations. You have many, many points now, N goes to infinity. So you can imagine that the typical distance between points, there's a line scale here that is N to the power minus 1 over dimension, the typical nearest neighbor distance between two points if you're in dimension D. So there is this line scale, and you want to ask yourself, if you look with a microscope, and you come with a microscope and you put a little ball, smaller and smaller ball, and you look at how many points you have in that ball, is it true that the number of points in the ball is well approximated by the integral of N mu V? That's a natural question. And is it true down to what you call the microscopic scale? So the microscopic scale is the smallest one at which you expect things to happen. It's N to the minus 1 over D. And the mesoscopic scales are all the scales in between. So there are all the scales N to the alpha, where alpha is between, so let's say minus alpha, between 0 and 1 over D. So if you look at small boxes there, is it true that the configuration is well approximated by the equilibrium measure? Can we find a better order than N for the difference? Can we even find limiting laws, like central limit theorems? And can we describe the configurations after zooming at the microscale? So if I zoom with my microscope, I rescale things. So let's say I multiply everything by N to the 1 over D. And suddenly my points are distance 1 apart from each other. So I see a typically infinite, when N goes to infinity, I see an infinite configuration of points that fills up the whole space. Can we say something about these configurations of points? What do they look like? Are they ordered, disordered? And remember the triangular lattices that I talked about in superconductivity, this would be the microscopic patterns you would see in those instances, which correspond to minimizers. So I can tell you right away, we expect minimizers after blow-up to look like triangular lattices or something like that. But we don't know how to prove that. OK, but we can try to find ways of describing these microscopic configurations and characterizing them. This will be the object of the last lecture. In the meantime, I will try to show you results that are of the type of a central limit theorem for fluctuations in these two settings, one and two logarithmic cases. So these types of questions have been examined by a number of people. Let me say right away, the one dimensional situation is the best understood by far. It's actually completely understood, essentially, thanks to the works of Burga, Dardo, Shiao, and also Boroguione, Charbina. Dimension two is a little less understood. So I will describe the works that we did. There is also results of Amur and Delman Makarov, Ryder Virag, Bauer Schmidt, Burga, Nikola, and Shiao. And dimension bigger than three is a largely unexplored field. I mean, there's old classical results of Le Bovis, Shunko Vici, et cetera. But in the directions that I'm going to describe, not as explored. OK, but if you want more complete references, of course, I invite you to look at the notes. All right, so I think I've explained the motivations and what we are trying to do. And now I want to present to you what I call the sort of electric approach or the energy approach, where we are really going to take advantage of the Coulomb nature of the interaction. So as I said so far, everything I've discussed is not really using the Coulomb interaction. So the first thing is, in order to access these information at the microscales, we are actually going to have to expand the energy further. And you remember, I told you that the minimum of hn divided by n squared is converging to IV of mu v. Now I want to go further and expand to next order using the expressing things in terms of the fluctuations. So we have a little formula that is actually not difficult to prove. Call it the splitting formula. It says that for any configuration, so I will call xn, xn will be a shortcut for the n top all x1, xn. So for any xn, the energy I can split as, well, the minimal one, this n squared IV of mu v plus 2n sum of zeta of xi. And I will define all these terms plus fn mu v of xn. And this function zeta is something that already appeared except I forgot to define it. You remember the characterization of the equilibrium measure, maybe? It was that h mu v plus v over 2 is either minus a certain constant, is non-negative, or is 0 in the support of the equilibrium measure. Well, this whole function I called zeta. So it depends on v. Once you know v. So it's going to be 0. Looks like that. It's 0 in the support. And then otherwise it's non-negative. And you can even see that it's going to grow like v. So it's going to be confining. And in fact, this term behaves like an effective confining potential that is only active if the particles try to escape sigma. So if they're in sigma, this is 0. If they try to move away from sigma, the outliers, if you want, you will pay a price here. This term is constant. It's the minimum value. So we just subtract it off. And now this term is the interesting term. It's defined as follow. So for any probability measure mu, I define this as the double integral. This notation means the complement of the diagonal of g of x minus y, d-flugged n of x, d-flugged n of y. And remember, flugged is the fluctuation measure. So it's defined here. So it's the difference between the Dirac masses and the equilibrium measure. So if you think again in terms of electrostatics, these Dirac masses, you can view them as positive singular charges of mass 1. There are singular charges. And this minus n mu v, you can also view that as a charge which is negative, but which is diffuse. Which is non-singular, because mu v, we have assumed, is a measure with a density. And they both have mass n. So here you have n positive charges. Here you have a probability. So this whole thing has mass n. So we have a globally neutral electrostatic system formed of positive singular charges and a negative charge. And here we are essentially computing the self-interaction of the system. So the Coulomb self-interaction of the system. Except we have to remove the self-interactions of the charges, otherwise we would get infinities. So I can always define this. And then I have to prove this formula. So this formula is not really difficult to prove. I can do it. OK, so prove. I start by rewriting hn as a double integral. OK, so hn is going to be the double integral on the complement of the diagonal of g. OK, I don't write g of x minus y, but it's sum of delta xi, sum of delta xj of y, plus n integral of v, sum of delta xi. So up to now it's just a language, a little change in language. And now you see the idea is that we're going to expand, like Taylor expand, around the minimizer, around n times muv. So we're going to write that sum of delta xi is n times muv plus the fluctuation, which is just the definition. So you take this splitting and you plug it into here, and you find that hn is equal to what? So there is going to be some terms in n muv, n muv, n muv. It's going to be n muv, n muv. So this term, if I look at it, put everything together, it's nothing else than n squared iv of muv. That's the definition of iv. OK, here there is the complement of the diagonal. But muv, I assume, is a measure with a density. And so it does not charge points. And in particular, the diagonal here is not charged, so I can put it back in the integral and I find this. So this is the first term. All right, then I'm going to get terms that involve the difference, the fluctuation. So I'm going to get g, fluct, fluct. Well, that's easy. That's actually the term I announced. It's fn mun of xn. And what's left is the sort of cross terms. In these terms, they have to simplify because I didn't choose muv randomly. I chose it because it's the minimizer of iv, so it satisfies a certain equation. So OK, let's rewrite. Let's write what we expect here. We expect g of x minus y, fluct, n of x, n d muv of y, plus the symmetric term where x and y are reversed. So actually, I have two of those. And then I have n integral of v, deflect. And now this term, I can recognize if I integrate g of x minus y against muv, it gives me this potential h muv. And actually, this is the same as h muv of x, deflect of x, plus it has an n. So I have 2n plus n integral of v, deflect. But 2h muv plus v, by definition, that's 2 zeta plus a constant. So it's 2 zeta plus 2 times a constant. OK, so I get zeta. I get 2 times zeta times the fluctuation. So I'm almost there. So I get 2n integral of zeta, deflect, plus 2c integral of the fluctuation. But the fluctuation is integral 0 because there's mass n and mass n. And zeta, deflect, this is zeta sum of delta xi minus zeta muv. But then I remember that zeta is precisely equal to 0 on the support of muv. So this one disappears. And so what I have found is exactly 2n sum of xi. So this is just a simple computation that exploits the fact that muv is a minimizer of iv. And so now we find ourselves with this formula. And we're going to be able to subtract off this leading order term, which is always going to be fixed. It doesn't depend on the configuration. And the idea is that what's left here is lower order. So what's left is in these two situations, it's going to be of order n plus an n log n term that's going to come out. And the analysis now is going to focus on this guy. So how much do I have? Maybe it's five? All right. So now it's time to start studying the interesting term is this one. So this, I said, is something that's active only for outliers if you decide to go outside of sigma. Now what about this guy, which is computing, as I said, the self-interaction of this neutral system? Well, I'm going to rewrite this term in a different way, where now I really take advantage of the Coulomb type nature of the interaction. So I'm going to form the potential generated by this fluctuation density. Let's call it hn. So of course, it depends on the configuration. But I'm going to omit this dependence. So it's what I obtain when I integrate g against this measure of y. If you prefer, it's the convolution of g with this. And if you remember the description before, it's the Coulomb potential. Or in 1d, it would be the logarithmic potential generated by this. And so in the Coulomb cases, it will solve minus Laplacian hn equals this constant cd. So it's nice because it solves an elliptic PDE when we know stuff about elliptic PDEs. So what is fundamental here is that g is the kernel of a local operator. You can always write, you can always form this quantity for any g. But if g is arbitrary, typically you get some sort of Fourier integral operator. But you don't get a local operator. Laplacian is local. If you are doing log in 1d, what comes out here is not Laplacian. It's what's called the half Laplacian. So 1d log. Again, the half Laplacian is a little trickier than the Laplacian. But this can be fixed by going into the plane, so by looking at the extension of your function hn to the plane. I will discuss that next time. So the idea is you want to make a formal computation. And I'm going to show you the computation, which is wrong. But then we'll see how to fix it. You want to look at fn, if you remember, of xn. It's this double integral of g x minus y, fluked some delta xi minus n mu v of x and the same of y. So if you forget for a moment this business with the diagonal, you might be tempted to say that this is the integral of hn times the Laplacian of hn up to a minus sign and a 1 over cd. Because you see this is minus cd Laplacian and this is h. Formally, it's this. And then once you're there, you know Stokes formula or Green's formula. You want to integrate by parts. And this becomes the integral of gradient hn to the square. So the total interaction, total Coulomb interaction, it's actually nothing else than a well-known and functional, the Dirichlet energy of the potential generated by the points. And that has an advantage is that it sort of makes the energy local in some sense. So you know you have these Coulomb interactions. You have to look at all pairs of particles. This guy interacts with all the other guys, even the guys that are far. But when you look at it from this point of view, the energy in a box is about the potential hn in that box. And you can try to add energies in boxes as opposed to making all these pair interactions interact. Come into play. But of course, making that precise is much trickier than it seems. So the other thing is can someone point out why this is cheating? Do you see why? What's the problem with the computation I made? The diagonal, yes. So the diagonal, you see? Because here the diagonal is removed, here it's not. And the result is I get to a place here where this integral is actually infinite. These integrals are not convergence. Because think about it. What is hn looking like? hn at each point xi, it looks like g of x minus xi. Also hn near each point is blowing up like g. So this is infinite. And the gradient is blowing up like the gradient of g. So for example, with the log, it's 1 over x minus xi. So if you integrate 1 over x minus xi squared, in dimension 2, for example, it's infinite. 1 over r squared is precisely infinite. And so this integral is infinite. And this comes from the fact that you forgot to deal with the diagonal. So the way to do it is there's a procedure to truncate, I'm going to introduce a parameter eta here, which is going to truncate these infinite parts and compute these integrals sort of infinite parts. OK, so I will explain this tomorrow. Thank you. Maybe Alice, you know. I don't know. What is it, the term that your code didn't prove? Yes? Yeah, oh, I see, I see. So I think it's about how you want to specify the notion of convergence, right? They didn't. Right, so I. Yeah, yeah, yeah. So. I'm not sure. With me, it doesn't work. So I think with Bayer is looking at the. Yeah, yeah, this is everything is Gaussian here. This is the Gaussian. It's, yeah. So here, if you want, the direction is opposite to what people are doing this morning, right? You start from Gaussian. Everybody starts from Gaussian. And then you want to deviate to non-Gaussian. And here, it's rather deviating to non-quadratic, right? Instead of having quadratic potentials, you want to. So it's orthogonal. There are other questions or comments, OK? So if not, so we will have the lecture by Peter Forrester in 15 minutes or a bit less, maybe now. And the problem sessions would be in the tent. So it would be first. The first part would be on the second part on the second part on the CIVIA's CIFATIS course. OK, so let's thank again CIVIA.