 The next lecture series is going to be given by Laszlo Erdisch from Institute for Science and Technology in Austria. Laszlo is one of the key contributors to the recent progress in universality. And he'll probably talk about it. The title of his lecture series is The Matrix Dyson Equation in Random Matrix Theory. Okay, thanks very much. Can you hear me? My microphone is a little bit strange. My ears, but hopefully to be aware. Okay, so first of all, thanks to the organizers for inviting me and giving this opportunity to talk about this project, about these works here. Now, as everybody else, there will be lecture notes or there are already lecture notes. And of course I cannot cover everything within these four hours. The notes are quite extensive. And I try to write them in such a way that they are user-friendly. So hopefully they are understandable. I'm not going to follow line by line lecture notes. The notes are actually longer and they contain many more background material, but more or less I will try to follow that philosophy. Most of the notes do not contain the strongest results. This is eventually an educational program. So the goal is to convey some basic ideas about these proofs. The goal is not to go into the strongest or the deepest or the most general conditions. Also the notes are not written in that way. At least I gave references to the best available strongest results, but the philosophy of the notes is more user-friendly and sort of down to earth. You will appreciate it. Also here I will discuss the simplest case, which sort of represents or demonstrates the main point. One more thing is that I will start because there is this idea that there are exercise sessions and also lectures, so I will skip now the motivations. Motivations will come tomorrow basically because I would like to reach a point where my TA can give the exercise session in the afternoon. So maybe some things will be unmotivated at the beginning, but tomorrow there will be more motivations. So let me start with a little bit advertisement, which has nothing to do with random matrices, but I feel almost obliged to do that. I would like to say a word about my current institution. This is this Institute of Science and Technology in Austria, which is a brand new institution. It's a multidisciplinary research institute financed by the Austrian government. We have mathematics, but apart from mathematics there are computer science, physics and biology, also various interdisciplinary research directions. It's a graduate school. We do not have undergraduate students, but we have graduate students and we are desperately looking for good graduate students who may be interested in pursuing their graduate studies at our institution. We are fully funded and everything is fully English-speaking, although we are somewhere in the suburbs of Vienna, but everybody speaks English on campus. So if you are interested, then on the webpage you can find us. There are job opportunities on every level. So we are looking for graduate students, but we are also looking for postdocs and faculty members. We have just some pictures about the place in case you want to find them. This is the brand new building where the mathematics groups are found on the top floor, and this is some internal discussion area, so very cozy. You can easily relax there. Okay, so if you are interested, then let me know. Just look at it directly. It's a nice place. Okay, so now here is roughly the plan of these four lectures. I hope I will be able to get to the end. So there will be a short introduction, then I will give a little bit of a review of the results. And today I should also reach this third chapter about the tools. I would like to introduce the Steele-Chest Transform and Resolvent, which are the basic mathematical tools of our study. And then tomorrow I will start with, not tomorrow, but Wednesday, tomorrow is holiday. Then tomorrow I will start with the motivation. That's what I said that, in principle, motivation should have come earlier, but if I do that, then Torben has nothing to do with you in the afternoon session. And then I will discuss the Resolvent method in more details. And then there will be various Dyson equations on various levels. There is a vector version, there is a matrix version of that, which we will be analyzing in full details. And then around the end I will explain the basic structure, how one proves the local law, because I will tell you before that what the local law is. And then there will be some recapitulation at the end. Okay, so Torben, let me start very basic, although some of these things have been already mentioned by Teddy Tau's lecture. And also those who were here in Jau's lecture. But anyway, I start from scratch and I assume that you don't know too much about random matrices. Although you were supposed to be here last week as well, so there were some random matrices there. Anyway, so the basic question goes back to Eugene Wigner, who physically motivated us the following question, what can be said about the statistical properties of eigenvalues of large random matrices? It's a very, very general question. And also it's a very natural question, which I keep on saying that it's a question which mathematicians failed to ask. We needed a physicist, Eugene Wigner, who actually asked this very natural question, which I'm glad, and from a mathematical point of view, it really should have been asked at least half a century before Wigner started it in the 50s. But all the necessary concepts, what a matrix is, what an eigenvalue is, and what randomness is, they were available many, many decades before Wigner asked this question, but mathematicians failed to ask it. We needed a physicist to do that. But anyway, the question is very natural. So how do the... So the statisticians did? Yes, I should. There is Wischart, thanks very much, but Wischart actually was almost a biologist. I think biostatisticians, so in some sense it's even worse for mathematicians. Yes. So now, so here's the question. You take a large random matrix in all our talks, capital N will be the size of the matrix, we're talking about squared matrices, every matrix an n-biome matrix, and it has eigenvalues, which will have n eigenvalues lambda one through lambda n. And typically the matrix is random, there are many, many ways to impose, to put a random distribution, to put a distribution, a space of matrices, we will discuss them in more details, but you can just think of it for the moment that, for say, the usual Wigner matrix, which was around or at the last week, which means that you just have IID, independent identical distributed entries, in the matrix elements. And for most of the studies, we will study Hermitian matrices, which means that there's a natural Hermitian symmetry, so the h21 and h12, these are complex conjugates of each other. So of course, when we say that they are independent, the matrix elements are independently distributed, implicitly it's always understood that independent modular Hermitian symmetry. So of course, it means that elements above the diagonal, they are independent and anything which is below the diagonal, they are determined by what you have above the diagonal. Okay, so now the basic question is, is there some universal pattern which emerges? So put a randomness, put any kind of, or not any, but a very general distribution in the space of matrices, and then you look at how the eigenvalues, how the statistics of the eigenvalues look like. Now if you put a distribution in the matrix, a probability measure in the space of matrices, it naturally induces a probability measure in the eigenvalues, that's obvious, but whatever simple probability measure you put on the space of matrices, for example, what could be simpler than IID, or even even more simple Gaussian IID, even if you do that on the level of the matrices, the eigenvalues will become very complicated, the distribution of the eigenvalues will become complicated, simply because the functionality, the map going from the matrices to the eigenvalues to its eigenvalues is a complicated map. You learn it in linear algebra, you have to write down the characteristic polynomial, find the roots and so on, so it's a complicated story. So in particular, the measure which may look simple on the level of the matrices under this map will be turned into a very complicated, could be turned into a complicated measure on the eigenvalues, so that's the complication here. But still we are looking for some kind of universality on the level of the eigenvalues, so the eigenvalues are random objects, we are looking for a universality in terms of distribution, so we are hoping that the eigenvalues are some natural question, some natural statistics of the eigenvalues will exhibit some kind of universality. And the analogy what you should have in mind is the central limit theorem, which in spirit is similar, but of course it's a much simpler situation, you don't talk about matrices, you only talk about an array of random numbers, so you take x1 through xn, these are say IID random variables, a scalar-valued random variables, they can have any distribution what you wish, okay maybe there is some moment condition, an essential arbitrary distribution, and then you suppose that they are centered and when you add them up, and of them divide by square root of n, it's a random object, it still remains a random object after all this scaling, but then it converges in distribution to the normal distribution, no matter what the original distribution of the axes were. So this is a type of universality, you start with something, for example axes represent some Bernoulli distribution, coin flip or dice or something, no matter what you had here, once you form this sum, this normalized sum, the original distribution evaporates, it disappears, and what emerges instead is the Gaussian distribution, so that shows that the Gaussian distribution is a very universal object, which emerges naturally out of no very few wish, and something like that we are expecting or we are looking for on the level of the eigenvalues for random matrices. Okay, so here's the Wigner ensemble which you have seen already last week many times, so let me not spend much time on it, so it's simply the n-by-n matrices n is our IID, independent identical distribution, but up to the Hermitian symmetry, and we fix a normalization which is, which depends on the schools, our normalization is that the expectation is zero and the variance is one over n. So the typical size of the matrix element is one over square root of n, variance is the square of the typical size, and we chose this normalization because under this normalization the typical size of the eigenvalues is a order one that governs our normalization. So to see it, at least on a very, very naive and simple way, is that first that you can compute easily the expectation of the sum of the squares of the eigenvalues divided by n. So this is an average, this quantity here will tell you an average size of the eigenvalue in a very crude way. Average means, average is meant in two different senses. First, you take not just one single eigenvalue, but you take the average of the eigenvalues divided by one over n, eigenvalue squares divided by one over n, and on top of that you take the expectation of that to get the number. And this you can easily compute because the sum of the squares of the eigenvalues is the same of the trace, but the trace you can compute in another way, you can compute the trace of h squared is just the sum of the matrix elements squared. So you have a trivial relation, but on that side you have some quantity, you have some information, some quantity about the eigenvalues, and on the other side you have some information about the matrix elements itself. So that's a very useful relation between these two quantities because this establishes some explicit formula between the known object, the eigenvalue, the matrix elements are known before the distribution of them and then establish the relation between that and the eigenvalues. This is behind this very basic instance of the moment method which you have seen last weekend in Jan's talk. And now if you impose the condition that variance is one over n, then here you have a double sum, each of the term is one over n, but then there's another one over n from the fact that you took the average, so you get one. So you have an average diagram that is out of order one. Okay, there's one more thing which I would like to emphasize here that there are two different classes of random matrices. One of them is the real symmetric, the other one is a complex Hermitian symmetric class. The names basically tell what they are. I will not really distinguish in these talks between the two because all the results what we do and everything basically holds for both of them, and this is who many of you probably already heard about the various symmetric classes and eventually the final result, the university, the precise form of the limiting distribution depends on the university class. So maybe you should just keep in mind that there are these two different university classes. Okay, and then, so this is for Wigner matrix, so here the entries are her arbitrary distribution, and of course if the entries are Gaussian, then this is one single, very distinguished matrix ensemble which is called the Gaussian unitary ensemble if you talk about the complex Hermitian case and the Gaussian orthogonal ensemble for the real symmetric case. Okay, so now here is the first question what you can ask about the eigenvalue. This is not yet the question on the level of the central emits here and it's more like the question on the level of the law of large numbers, if you take this analogy, and here the question is what is the eigenvalue density, what is the density of the eigenvalues and that of course leads to the semi-circle law which you have seen over the last week, so it's just under the capitulation. So here are the eigenvalues, random eigenvalues of n equals 60 by 60 Wigner matrix and then the semi-circle law indicates the limiting density as n goes to infinity of the eigenvalues. Now, the typical distance between neighboring eigenvalues is 1 over n in my scaling because we had n eigenvalues and as I showed before typical size of the eigenvalues is order 1. So the typical distance between neighboring eigenvalues is 1 over n, this little 1 over n shows you the local scaling. So immediately you see that the problem leaves at least on two different scales. One of them there is a global scale which is basically a scale here, an order 1 scale, a scale on which you see all the eigenvalues and in particular this is a scale on which the semi-circle or the shape of the semi-circle emerges. But then there's another scale, there's a microscopic scale, a local scale. If you zoom it out, you take a microscope which zooms out everything by a factor of n and suddenly you start seeing really individual eigenvalues and then you can ask how these individual eigenvalues behave what the local statistics of the individual eigenvalues look like. So these are these two natural questions here. The first was the global density for the semi-circle law. Now, once you have this question, so this is on the very global scale and then the other one, this Wigner-Diesel meta-universality is the local scale when you ask the question the 1 over n scale. Now, there is something in between. Namely, you can ask what happens between the scales order 1 and the scales 1 over n. Maybe in principle it could be at some interesting new statistics emerges say on the scale 1 over n, but that's not the case. Actually what happens here is that if you start reducing the scales you try to take a better and better, stronger and stronger microscope and see what happens locally then you will see that the semi-circle law in a local sense still holds right down to the smallest possible scale of order 1 over n or more precisely a little bit above that. So I will write it up more precisely what this means in terms of formulas, but here you just should remember that there is basically no fluctuation before if you zoom it out you go down to the smallest possible scale basically there is no fluctuation in the density, no sizable fluctuation in density before you scale the before you reach the scale 1 over n and on the scale 1 over n of course you still already see individual eigenvalues and individual eigenvalues are fluctuates on that scale order density fluctuates. Okay, so now here is Wigner's revolutionary about these two scales he noticed that the global density may be more than dependent but this look the eigenvalue statistic, the look the microscopic statistics is very robust it really exhibits a universal phenomenon and it depends only on the symmetry class whether it's real symmetry or complex Hermitian. Now you may not have seen this so far, I think in the last week's talk there were no examples for that maybe they were in the free convolution story but the typical random matrix studies usually start with the semicircular and you almost take it for granted those students start with that you almost take it for granted that semicircular is a natural object that comes out immediately and yes it is the semicircular is a natural object for Wigner matrices you see very quickly that there are many many other mature random matrix ensembles whose global density is not at all the semicircular law and actually Wigner knew it very well if you go back to the physical example which I will discuss on Wednesday if you go back to the physical examples then there is absolutely no reason to expect that the global density is the semicircular for example if you take random shading or operator it looks completely different but nevertheless the local statistics the microscopic statistics for example the statistics of the gaps between neighboring Gaigan values is very robust and it's very universal ok so now here is the the sine kernel in the simplest case I just formulate it in a precise way especially the necessary ingredients for that so you take the probability density that's the function of all the Gaigan values so this is a function of n variables which is supposed to describe how the joint distribution of the Gaigan values lambda one through lambda n look like it's a complicated function it has n variables and in principle it can look big mess and n is a big number so eventually I would like to start the angles to infinity so essentially there is no hope to understand this function of n variables as n goes to infinity and usually you don't do that also in statistical physics if you think about Gaigan values as a statistical physical process statistical physical system like in the low gases you do you never studied the endpoint function as a whole it's too complicated instead of that you study the k-point correlation function so the k-point marginals which are defined in the following way you just simply take the endpoint function the total distribution the total density function of all Gaigan values you symmetrize it sometimes I denoted the Gaigan values as an increasing order and I will keep that convention except for that definition so here it's easier to talk about densities if you are to symmetrize the Gaigan values you take this function of n variables and integrate out all but k-variables if you are interested in the so called k-point correlation function then you get a function of k-variables here so x1 through xk and this is just the the marginal of this big function of the k-point marginals now you are always thinking in such a way that n is a big number n will go to infinity and k is fixed so k is 1 or 2 or 3 so for example if k equal 1 then it's exactly a density we have seen before as a semicircular but now we are interested in more local properties of this correlation function so we rescale the correlation function accordingly and we rescale it in such a way let me draw a picture so we rescale it in such a way that it is a big semicircular very large scale and then you have fixed an energy somewhere that's this energy we fixed the energy in the bulk bulk in this whole business means that that's a point where the density is positive so for example if it's a semicircular density then you take e between minus 2 and 2 so you take an energy in this bulk and now you want to understand the eigenvalues so you assemble the eigenvalues in a distance 1 over n so now you want to zoom it out so you take you magnify this picture in such a way that you see the eigenvalues having typically a distance of order 1 because you want to see that the local eigenvalue process converges to something to converge the point process as n goes to infinity in order to do that so you have to first zoom it out to get everything to order on scale so now this is a complicated formula but this form exactly does that so here take the k-point correlation function but you want to understand eigenvalues near the energy e on a scale 1 over n it's a technical way of doing it that you take the original k-point correlation function and you evaluate it at points which are essentially at e where this fixed energy and then you go away from this energy by an amount of 1 over n so you go away in the first argument you go to e plus x1 over n second argument e plus x2 over n and so on this little row n here is just a constant it's just guaranteed that after this rescaling it guarantees that the gaps, the distance between neighboring eigenvalues is not just order 1 but on typical expectation it's exactly 1 that's why you rescale it by the density this is how you get the so-called rescaled correlation function so don't get scared about this formula this is a very natural rescaling and then the statement is about this function so the local correlation statistics about gu for gu e says the following that the k-point correlation functions converge as n goes to infinity k is fixed, e is fixed n goes to infinity it converges to a fixed function of course the deterministic and universal function which is very nice, which can be written very nicely this is the famous Dyson sign kernel so remember it's a function of k variables k is fixed, k is say 3 and then you need a function of k variables and you can get it by a k by k determinants and the matrix elements of this k by k determinants is simply computed it's simply explicitly written as the s of x i minus x j i and j runs from 1 to 3 in that case and this s function is the famous Dyson sign kernel so explicitly given by sin pi x over pi x so you get a very, very explicit a beautiful formula and it's not just a formula beautiful you get it as a determinant it's not expected now if you look at it why should a correlation function be written as a determinant and in this way you see all the correlation functions k can vary of course k is fixed but as n goes to infinity but the formula is given in a very nice uniform way for every k and it's always a determinant of k by k determinants it's a beautiful formula okay so this was computed this is a highly non-trivial calculation even for the Gaussian case of course you are used to from probability theory that Gaussian is a trivial regime and in some sense this is the easiest case but this table is quite hard and it needed God and Dyson and Meta doing very detailed analysis asymptotic analysis of hermit polynomials and further tricks to come up with this result it's a beautiful result already shows a type of even the Gaussian setup already shows a type of universality because you see the left-hand side is independent of E principally depends on E the right-hand side does not depend on E so this already shows that no matter where you are in the spectrum you are at that point E equal to minus one half or at that point E equal to one the local statistics is always the same but of course the true universality is a much stronger statement but the real universality is what's called the Wigner and Dyson Meta universality which says that it's independent the same statistics is independent of the fact that God and Dyson metacalculation was Gaussian