 OK, oh yeah, it sounds much better. Sound OK. Ça va vous m'entendez? Je ne peux pas trop le mettre plus bas là. Je vais le mettre sur le... OK, cool. I'll wait for the mic. All right, so let's continue with the... Here is Kami Mal, welcome Kami. Thank you. He's a CNLS researcher in the University of Bordeaux. He's a specialist of free probability. So we go more from the let's say to the actual mathematics side. And welcome and we're looking forward to know about free probability. Thank you very much and thank you for the invitation. So I will present to you a subject which is at the intersection of free probability and random matrix theory. Free probability can be developed a part of random matrix theory and I will introduce you the general context to put the setting. So the kind of problem you should... where you should consider free probability to solve a random matrix problem is the following. Imagine you are given a matrix that is written as a polynomial in several matrices. A matrix A1N, ANN. For me, big N is the size of the matrices. They are all of size N by N. So this size is going to tend to infinity. Okay. So you assume that you have a matrix of this form where P is a fixed function. And let's say a non commutative is a fixed non commutative polynomial of rational function if you want to generalize this. So what is a non commutative polynomial? You just make a product of matrices and you know that the order in the product matters. This is why we say non commutative and you do linear combination of this. Okay. And the kind of assumption we want to make on the AA is not really specifying a model but having a generic position. And addressing the question of what does that mean being generic position is quite rich and there are different levels that we are going to consider. But what is important for us is that the matrices A, L, N these matrices, we assume they are independent. This is really what we want to do. Independent random matrices. So this is several, sometimes called the several matrices problem because several matrices are involved. What we want to understand is a spectrum of matrix matrix. When the size goes to infinity. So maybe you are used. This is common for you, but if not let me remind you the notion of empirical Eigen value distribution of a matrix HN. So this is a probability measure. I will denote it L index HN. And so it is the sum of direct mass with weight 1 over N on the Eigen values. So delta is the direct mass and lambda I of HN is the Nth Eigen value of HN. In all the talk, I will assume that the matrices, I want to compute the spectrum Hermitian matrices because otherwise it's a different problem, much more complicated. So we just consider this. Moreover, this is a random probability measure if HN is a random matrix. I would rather consider the expectation of this guy. And if you're interested in the random guy, you just build on this construction and try to find concentration. And just what is the meaning of that? If you evaluate a function, it's just the expectation of one other end, the sum of F applied on the Eigen values. OK. Here it is. So what will be the strategy? The strategy is given by the theory of free probability by Voie-Coulet school. We start constructing this setting, the abstract setting in the 80s and apply it 10 years later in random matrices. Just to say that the theory exists before. And the idea is to see to consider a notion of limit for the matrices when the size goes to infinity. And to consider a random matrix, not like a collection of a huge number of random variables, but just as a single random variable of a different nature than what you do classically in probability. A random variable which is non-commutative. OK. This is just the construction of the spirit. But it comes with a notion of non-commutative probability. The concept of non-commutative probability is an analog of independence in a very deep sense. So I don't have a lot of space in this tiny black box. So let me just organize this like this. So everyone is, I suppose, familiar with the notion of classical probability where we have the concept of law of random variable which is important. Voie-Coulet school created the concept that you know here have analogs in this place. And analogs not in an obvious way which have their own mathematics. And sometimes there are some things that you, a problem where it is difficult to compute on that side but much easier in this side. This is what we discover with this Eigenvalue problem. But what is important to note here is that in this context you know you have classical independence the expectation of the product is the product of expectation. Here we have a notion that we called free-ness or free-independence. The definition is much more complicated that the expectation of the product is the product of expectation. But there is a definition. This notion is as strong as this one is found in probability. There is a notion of Gaussian random variable as you know, or Gaussian processes. Let me say random variable. Here there is an analog that we will discover during this first lecture today with classical GU or Vignard matrices. In the limit when the side goes to infinity there is an abstract object which replace the Gaussian process and which describes the Eigenvalues of these matrices when we consider such a generic problem. This is what we call a semi-circular system or circular system for non Hermitian matrices. So we will discover this. What can we mention also? Something we Marc mentionned this this morning. You have the convolution of random variable which explains the distribution of the sum of independent random variable. If you have a major mu on a major mu you know what is the convolution of this and you know what is described. In this context you have the free convolution which is a different rule but which plays an a log role. We will denote this operation differently because it's a different operation. It is a boxed convolution an operation between two measures that give you one measure such that if you have two matrices two random matrices which are in the context of this topic, let's say a rotational invariant and you are interested in the distribution of the sum of these two matrices in a generic position and in the limit when the size goes to infinity this is the free convolution that will give you this answer. This is just a notion for us right now, how to compute this and how to compute this when we are not just in this context will be the subject of this lecture also. Okay, we have some notions of subordination property this morning this is what we are going to explore. Okay, but this work this context of freeness under kind of robust on a kind of strong assumptions this context is no longer applicable it happens from kind of classical matrices that you can construct easily but if you consider the distribution of the sum it is not given by the free convolution and this is the kind of ensembles that I want to focus on during this week. So this is applicable when for the matrices under consideration the eigenvector vector basis the random matrices are asymptotically uniform. This is heuristic and this is not a precise definition what does that mean being asymptotically uniform I'm not going to precise this but your random matrices if you look at the eigenvectors they are very they should be to apply this theory very close to what happens if you sample it uniformly independently of the eigenvalue. I will introduce some natural example of matrix model for which this is not true and so we cannot apply this and to solve this question I will talk about a third generation of probability theory I call it traffic probability and this guy comes with another notion of independence and this notion of independence is much richer than what we do in these two contexts because the notion of law of distribution is much richer we are not just considering moments we are considering something that we introduce that take much more information on the matrices and if you take much more information of the matrices you are able to describe much more a good variety of phenomena what happen is that this notion of independence actually encode both freedom and classical independence because it is a richer notion and it also encode other notion of non commutative independence that appear in the zero ok and we will see how to compute things like this the interest of what has been discovered since about 20 years in these free probability techniques is that we just we don't have just a conceptual framework we have analytic tools with this the subordination property and all the numerical methods that you may use in random matrix theory have a great theory of analysis in this non commutative setting ok so this is what we call free harmonic analysis free harmonic analysis maybe is a world afraid it's strange but think about the stiges transform the air transform on what you heard about this is just a world to sum up this ok so ten years ago I developed this theory to deal with this kind of weird ensembles and I was asking can we develop the analog of free harmonic analysis and it's not just a question for fun just because the theory that we had 5 or 6 years was just combinatorial in the sense that we had a theory where we were able to compute the spectrum in more cases but computing means having moments of the empirical distribution and if you know the moments of the distribution you don't know a lot of its qualitative properties you don't know what is the behavior of if there is a density or something like this and with the team that joined me on this question we address this problem of finding analytic tools and what we discover and I will introduce this in the second lecture is that if we take this theory it has something that I can mention is the analog of conditional independence ok if you have a function of x and z a function of y and z x z x y and z are independent these two functions are not independent because of the z but they are independent over z conditionally on this variable ok this has an analog invented 40 years ago by Vocule school which has a different name it's called amalgamation over an algebra amalgamation is just because it's coming from group theory where it has a sense ok just say it's the analog of conditional independence it exists in this area and what we discover is that traffic independence implies a certain notion of conditional independence in this non commutative way and the result of that is that we can use the analytic tools the free harmonic analysis in this setting and we get algorithms that allow to compute eigenvalue distribution in this situation ok so this is an overview this is maybe a lot of information at the moment so we will discover this in detail today I will talk about some aspects which are classical in the series of free probability we will see how to compute what is called the non commutative distribution of several normaticies for this it will be the opportunity actually to introduce the traffic tools in the background but I won't put in the front series the second lecture I will focus a bit more on the problem that we solve with these tools and go to this notion of freeness over the diagonal which is this analog of conditional independence we use to solve this question ok these two topics will be very combinatorial I will be dealing with moments under using these combinatorial techniques of traffic probability in the last lecture I will introduce more analytical problem how to deal with this still just transform for these ensembles and I will focus on these outliers problems when we have finite trunk perturbation so if the program is clear and everyone is happy to start let's go and if you have any question you have the opportunity to ask it so what is a non commutative polynomial is the question that typically you take a b plus b a something like this you can put a complex coefficient ok so it's a linear combination of words in the matrices I mean the word a b is not the word b a the other matter let's start with some basic notions in M specifically what I want to do right now is to give you a short zoology of random matrices zoology M is a short cut for random matrices for us today ok so what kind of random matrices do you know you know these covariance matrices with i, i, d and 3 we are about this so I won't go back but I want to define the definition of a vigno matrix because we will compare with different so if you miss this definition it will be a problem a vigno matrix what is it I will call it x so it's a real complex matrix we assume that the entries of the form x a j over square root of n implicitly we have a noise which is quite small we divide it by a very small amplitude ok so we have a small fluctuation ok we assume that matrix is a mission it is written like this we assume that the upper diagonal entries are i, i, d independent of the diagonal entries which are i, i, d as well of course if your matrix is complex you have different distribution in general and we assume that let's say the entries are centred it is not very important but let's say that and we have a finite variance and I will assume for us for using these techniques that we have finite moments of any order all orders but the really important assumption is that this is finite for k equals 2 otherwise with truncation argument you will manage to deal with that ok so you may know the Wigner theorem and we will give a proof of that which tells us that the empirical eigenvalue distribution of a Wigner matrix xn converges to what is called the semicircular distribution but if you have a complex matrix you have complex entries there and real entries there and you don't really care about the detail of the diagonal entries actually what is important is an easy variance of the diagonal entries for this theorem you have universality this result does not depend on the exact distribution we put on the entries ok, the semicircular distribution if the variance is 2 as a support minus 2, 2 your eigenvalue will be most of the eigenvalue will be around this interval and if you draw a finite by large size matrix on histogram it will be very close to the semicircular ok so the second matrix that I want to consider looks very similar to this one but the behavior is different we call it a Bernoulli matrix we consider a real symmetric matrix where the xn aj iid but we don't assume that they are normalized like this with a random variable whose distribution is independent on n and such that we assume that these guys are Bernoulli random variables of very small parameter let's fix an integer p and take a Bernoulli random variable parameter p over n which means that with probability p over n, which is very small this guy is 1 otherwise it is 0 ok so this is a rousse matrix if you want you center the matrix by removing all entries on normalize the variance but it is good like this this guy so let maybe I should use another symbol like y the empirical and value distribution of this guy has a limit which is very different from the semicircular distribution it converge to a guy that I will call L of y just a symbol I prefer to the limit of matrices but no free probability will give a meaning for this way which depends on this parameter p ok and this measure is very singular it's not we don't have an expression there is no density in general and knowing the properties of this distribution is complicated having an expression for moments is quite easy but knowing really what this distribution is complicated and in particular if you take a generic random matrix or a deterministic matrix and you consider this matrix plus such a matrix you won't have the same phenomenon as what you do with a vignard matrix a deterministic plus a vignard matrix in general under the good assumption we'll have an eigenvalue distribution which converge to the free convolution of the two eigenvalue distributions but for this kind of matrices it's true but we'll see that freeness gives a good solution not only the solution but also the algorithm on the numerical methods to compute concretely the spectrum yes why the universality results like of Tao and Wu does not apply to Bernoulli matrices because it's not a vignard matrix in this situation you have a field of smaller random variable where you have some entries most of entries are 0 and some entries are 1 you know it's actually there is a model which is similar to this one which are vignard like matrices with heavy tailed entries that is you remove the assumption that the second moment is finite you have a Couchi distribution you must normalise such a matrix in a different way and then you will see the same phenomenon there is no universality you will depend on the heavy tailed distribution and if you look at the sum of such a matrix with a deterministic matrix it will be the convolution over the diagonal that gives a solution there is a question somewhere yeah the question is if we replace the Bernoulli bar Hanmarreur is a symmetric Bernoulli but it's just if I'm not wrong it's just shifting the matrix yes you can do that what is important is to have a matrix which is not uniform in the noise you have some entries which are very big compared to the others and here big is just being 1 compared to 0 but for heavy tailed entries that is clear that you have some entries which are huge but then Rademacher I think it would be like a vignard matrix at that point if you divide your Hanmarreur by square root of n Rademacher would have all the entries ah c'est symmetric plus or minus 1 and then you divide by square root of n ok you will probably it depends on kind of if you think of the symmetric Hanmarreur you will need to normalise it like this but you can cook something which is a bit weird but you have to have something which is really not symmetric so it's maybe not what Bernoulli Rademacher where you have many zeroes the entries are minus 1 you must cook something which is very unbalanced ok ok we will talk about Hanmarreur until wise product but it will be another subject absolutely and more generally so there is other kind of matrices which are heavy tailed other kinds of graph we start adjacency matrices of sparse graphs ok if you like this kind of graphs and look at the eigenvalue of polynomials of the adjacency matrices or something else you are in this kind of bad ensembles or weird ensembles for which we will need freeness over the diagonal or not just the plane notion by your equation ok ok so this is just to give so the sample of matrices so matrices of graph was in my list I want to conceptualize something a bit more general but about these two notions as you know there is as you maybe know there is one vigno matrix which is very specific which is a GU matrix which is very important in the theory and it is important because it is a unitary invariant matrix this morning we said rotationalia invariant this is the same notion but here in the complex case we have when we have a matrix A n it is unitary invariant if it is equal in distribution to the matrix U A n U star and this for any U which is a unitary matrix again a matrix which sees property will be a nice matrix that will consider with a theory of Euclides and the matrix that we consider with the theory they have a weaker assumption which is the following that we call permutation invariant matrices permutation I mean invariant by conjugation by any permutation matrices that's a matrix that is invariant a load definition is quite simple when we have this one but this is assumed for all permutation matrix ok so if you just know this assumption you don't know a priori if very cool school theory will work it may happen but not necessarily but you can be confident that a freeness over the diagonal and traffic independence should say something interesting about this ensemble ok and if you take, you consider the matrices which were on this blackboard the Ardoche phonograph, the Bernoulli matrix they satisfy this assumption ok this assumption is not really satisfied for Wigner matrices but we don't really need matrices to strictly be unitary invariant we just need that assumption to please the eigenvectors are more or less uniform and actually this is what happened for Wigner matrices they are not exactly unitary invariant but we don't care it's most of the same ok and just to mention that this assumption is much more cold than this in the sense that if you consider t vectors there is much more chance that it is invariant when you exchange the vectors than invariant when you apply a nortagonal matrix that will mix all the entries so for the point of view of modellisation it is clear that it will be much easier to justify this kind of assumption than this one of course if this one works you can start safely assuming this and it's more difficult to be in this situation with this unitary invariant yes sure of course so what is the permutation matrix it's a matrix so you have sigma permutation of one n so for me one n I will denote it with a bracket like this so it's just a b-jection of this assumption ok the permutation v is the matrix associated to sigma if vaj is indicator then I think that the definition will be this one but you choose a convention I think that this is the one I use ok so it's just a way to encode the permutation in the matrix this will result in the matrix where in each row on colon you have one entries which is 0 for instance this block tells us that you exchange one and two no it says that these two guys are fixed point and these guys say you exchange three and four ok so just a way to encode the permutation to have a more basic way to see that if you are given a matrix an what is this matrix it's a matrix which describes the same operator as an but you just exchange the order of the vector of the basis when you express the operator ok this is a different way to see that other question ok let's go we have 10 minutes to give a little bit of free probability then in the second part of the lecture we will consider the distribution of ignore matrix ok, so now that we have some characters to tell a story let's consider a good framework so what is an encumutative ? random variable donc je vais essayer d'évoiler une lecture très abstracte, mais je vais vous donner quelques ingrédients que nous utilisons en practice. Le formalisme est assez simple comparé à la probabilité classique. Une probabilité non commutative est le data of a couple, a phi, a is a space and phi is the expectation. The space is an algebra, just the field where you can do products. And we assume that it is a non commutative algebra, typically an algebra generated by random matrices, matrix algebra. And phi is a linear map, a linear formative, it is a linear map from a to the field of complex numbers. It tells you what is the expectation of a, if a is a non commutative or non variable. In classical probability you define sigma field, you define some very difficult actions, here it is much more accurate. There are a little bit of assumptions that we can assume that I will not want to tell so much, but we also assume that these guys are endowed with an application star, which is from a to a, which is a non-tilinear involution, such as the complex conjugate of matrices, anti-linear involution, such that when you take the star of a product, you exchange the order of the product as for the complex transpose of matrices. Okay? Why we want a star? Because we want a notion of positivity. I will not explain all the details about. But let's consider examples of these kinds of ensembles of spaces which are related to random matrices. So first before random matrices are random variables. Example one, you have omega, Fp, a classical probability space. How to see that the classical probability space is a non commutative probability space. That way, we consider as an algebra, the algebra of boondid, non commutative, non boondid commutative random variable, define omega with value and complex number. And this symbol just means that the random variable, I assume they have moments of all orders. All right? And for five, I will consider the expectation. It's a linear form and it's always well-defined on this algebra. And the star is just the complex conjugate. Okay, just to say that this setting of non commutative probability is an extension of the notion of probability space of complex random variable. There is of course a restriction with this. We want to develop all the probability setting. So the second example is the example of matrices or random matrices. Let's take for A, an algebra generated by some matrices, by certain matrices. You can take the full matrix algebra, but you can restrict to matrices you are given and you want to study. Okay? And five, I will put n as for the size of matrices. And n by n matrices. And this expectation that we will consider in this non commutative setting is one of our answer traits. Okay? Why it is an expectation? We'll see just in a moment. But it plays a role of an expectation. And if you're considering random matrices or certain n by n random matrices, you can combine these two settings and just you will put an expectation here. And this will be our setting. I did not mention, but this guy is complex conjugate, complex transpose. I don't know how you call it. Right? Okay, we have defined the objects, but as in classical probability, this is not what matters. What matters is the notion of distribution. In probability, this is really the notion we should focus on how you realize random variable is not really important. So let's talk about the notion of non commutative distribution. Okay, definition. Given a family of non commutative random variable, so I will denote this family with this bold symbol, just mean that it is a collection of guys and x j. Okay? So we consider this guy in our setting. So this is a collection of an element of a. This is what we call non commutative random variables. And we call the distribution, we call it the star distribution to distinguish from the usual notion of distribution. And because it involves a star, that's just a trick. The star distribution of a is, so it is a linear map, which gives you all the complex number, you can compute from a, given this setting. So if you are given a family of variable like this, because you are in algebra, what you can do is making products, some, and multiplying by complex numbers. That is considering a non commutative polynomial in the variable. In the variable and in the adjoint, because we have this star. Okay? So this gives you variables. And then you can apply phi. And it gives you complex numbers. And we can sum up like this. This map, we will note it phi and x a, because it's given by phi. It takes p. It is a non commutative polynomial. I should, okay, I should introduce some symbols. The set of non commutative polynomials in some family x will be denoted like this with this bracket. And here it's x and x star because we will have a family and the variable with the star. Just x is just a family of undeterminates. They are symbols, which are indexed by the same ensemble. So this is just a word in symbols, where there is an obvious way to replace this symbol by the variable under consideration. Okay? And you associate phi of this polynomial. So maybe I have chosen a way very obscure to define it. But an example of this is just taking phi of a1, a2 plus a2 star, a1 to the third, which is a third power, a2. Okay? And if you evaluate your linear form in a non commutative polynomial, you get a complex number. And this is what we call a non commutative moment or star moment called, sure, just a notation for the space of non commutative polynomials in some variables. So here I just denoted the family of variables xj in xj's and xj's star. We should have a void to use this notation. It is quite confusing. We will not use it anymore later. Okay? It's just the notion of non commutative distribution is the data of non commutative moments or star moments. For the moment, it's kind of abstract. But it's evaluating for us. And we start with this, the second part of the lecture, the expectation of the normalized trace of any polynomial in the matrix. So this is the end of the first part of the lecture. We have a 15 minutes break. We illustrate more concretely what is this definition and what we are going to compute with that. Thank you. Okay? So let's resume this. So we finish the session with this kind of abstract definition. And we're going to illustrate in the two examples that we have considered before. So you know what is the notion of distribution in classical probability. The distribution is a map where for every let's say boondi, the measurable function, you associate the expectation of f of x. x is the collection of a variable. And this is the data of these numbers which characterize the distribution. You also know that you have other ways to characterize the distribution, which may be very convenient, such as the characteristic function. If it is allowed on CT or accumulative functions for one variable and so on. Okay? But here, this is one notion. What do we have here? We have something which is almost this, but a bit more restrictive. Here, so just, right? It's a classical notion of distribution. The notion of distribution on where f is a measurable boondi. Okay? Versus, the non-commutative notion, the restriction, it is, it is more algebraic. The non-commutative distribution or star distribution will be the collection where for each polynomial on, in this context, it is not relevant to consider a non-commutative polynomial. The order doesn't matter when you evaluate to complex random variables. So it's just a classical polynomial that you denote maybe in C, x, x bar. Okay? A polynomial in the variable on their conjugate. And you associate the expectation of the polynomial in these variables. What, what's the name for this? This is called the moments. Okay? So, there are the moments, the joint moments, if you have several variables of the variables. Okay? So, does the moment is really the distribution of variable? Not always. If you have a very weird distribution, maybe it has no moment. Maybe it is not characterized by its moment. If you take the log normal distribution, the moment are defined, but there are different distribution with the same moment. But they are very weird situation. If you have a Gaussian variable or a sub-Gaussian variable, a boondi don variable, it is entirely characterized by its moments. So, just to sum up, there is the main part of a classical probability, where if you know moments, it's the same as knowing the distribution. And there are some random variables which are not, where this is not true. And this is maybe a problem if you want to see free probability as an extension of classical probability, but that's a life. We just restrict to this algebraic notion. Okay? But we, we don't really care about this. We care about random matrices. So, what we must know is, what is the notion of non-commutative distribution for matrices? And why do we care? Okay? Given the family of matrices, xn, let's call them xn, xn, xn, those questions? Okay. The non-commutative distribution is the data of the expectation of one over n. The trace of, if you take one matrix, l1, let me remove the dependence on n on the matrices, just a question of clarifying the notation, times a second matrix, times another matrix, and you take either the matrix or it's a joint that I will encode with this index here. And this is so for all l1, so for all big l.