 Vectors and random matrices. Thank you very much, and I'm very grateful for inviting me and letting me talk here. So these talks will be about delocalization, so let me first formulate what this phenomenon is. Let's take two modal cases. Before I discuss it, let me briefly describe the plan of these lectures. We will shortly talk about delocalization in general. And then we will concentrate on one particular manifestation of this phenomenon on no-gaps delocalization. And in the first lecture we'll see what can we do with the standard, the most robust method, the epsilon net argument, and what we cannot do with this argument. And then we are going to step a bit away from a random matrix theory and talk about random vectors and develop strong small probability estimates for random vectors. And these estimates will be instrumental in proving no-gaps delocalization, which I hopefully will do lecture three. And the final lecture will be about the application of different types of delocalizations to random graphs. So let me start with what delocalization is, and let's start with two toy examples. One is GOE, and the second one will be Geneva Ensemble. So the first one is Gaussian IID matrices, which are Hermitian, real IID matrices, which are Hermitian. And in the second one, let's consider complex Gaussian IID matrices with full independent on Hermitian. And we will assume that the real part of the matrix is independent of the imaginary part. So the distribution of the first ensemble is invariant under the orthogonal group. If we consider a unit eigenvector, then an eigenvector being a function of a matrix should share the same invariance properties. So the distribution of the eigenvector will be invariant under the action of the orthogonal group, which means that any eigenvector is uniformly distributed on the unit sphere, and the unit sphere in this case is real. And if we look at the complex non-Hermitian analog, the Geneva Ensemble, this distribution will be also invariant this time under the unitary group. And again, this causes the distribution of any eigenvector to be invariant on the unit sphere, and the unit sphere now is complex. We'll see that there is a significant difference between these examples later. But here we used invariance. If we have random matrices, say Hermitian random matrices, or just full independent case, but the entries are not normal, we cannot hope for any invariance. In particular, if the distribution is discrete, the eigenvectors will take values in a discrete set so no invariance is possible. But if we believe in a universality phenomenon, as n goes large, we have to approach the uniform distribution on the sphere. And this vague idea is the localization. So an eigenvector of a reasonable class of random matrices is distributed roughly uniformly on the sphere. And there are many different ways to quantify this vague idea. For example, one can take a few coordinates of an eigenvector, and as the size of the matrix approaches infinity, the joint distribution of these coordinates will tend to the distribution of the standard Gaussian vector. This was proved by Burgadin-Yau in the case of random Hermitian matrices. But we will take a different point of view. In this point of view, first, one takes the limit as an approach to infinity, which is a standard idea of random matrices. And second, one fixes the set of coordinates in advance. What we are going to do is we will consider matrices of fixed size. The size is huge but fixed. And we will strive for explicit probability estimate. We will want to get bounds with high probability, and the probability should be as high as we can get. The reason lies in applications. For example, random matrices arise in computer science in many different numerically linear algebra algorithms. And if you solve a linear system or a linear programming problem, you cannot let the size of the matrix tend to infinity. And you want to have explicit probabilistic guarantees that your algorithm will work. So we will first approach this question non asymptotically, which means that we fix a large n. And second, we will strive to get estimates which are not local, which do not hold for a fixed number of coordinates, but which reflect the global structure. And the most standard approach here is the L infinity, the localization. To describe it, let's first look at these toy examples. So if I take a random vector uniformly distributed over the sphere and it doesn't matter real or complex, it cannot have large coordinates. It should not be aligned with any coordinate axis. And actually the maximal coordinate should be rather small. It's easy to see if we have the uniformly distributed over the sphere, then this vector can be modeled. It has the same distribution as one over the standard Gaussian vector as a normalized standard Gaussian vector. And then this norm, this Euclidean norm is strongly concentrated about square root of n. And so finding the L infinity norm of the uniform vector over the sphere reduces to finding the maximum of an independent normal random variables, which is a probability one-on-one problem. So with high probability, we will have that the L infinity norm of a random vector uniformly distributed over the sphere is O capital of square root log n over square root of n. Okay. And then we would expect the same phenomena to hold for the matrices with independent entries. And this indeed was proved rather recently. Before I formulate the results, let me formulate the assumption. The assumption is that the entries of the matrix will have light tails. By light tails, I mean that the probability that absolute value of aij is greater than t is less or equal than, let's say, 2 exponential of negative t over k to the power alpha for some alpha greater than zero, and this holds for any t positive. So in this talk, k and c capitals is small, et cetera, we'll denote absolute constants. And if we have such light tails, the tails of exponential type, and if in addition, I assume say that the entries are centered and of unit variance, then in the Hermitian model, Erdesch, Schlein and Jaub proved that for any, that with high probability, the L infinity norm of any eigenvector in the bulk is bounded by c log. And to the power beta over square root of n. So we have the same phenomenon, the same delocalization as in the GOE case. And moreover, the probability can be adjusted, so the probability that this event occurs is greater or equal than 1 minus n to the negative t if I allow the constant c to depend on t. So I can choose any power type probability by allowing a greater constant here, and in particular this would allow to run Borel-Cantelli. Beta is a number depending on alpha, and in the most, probably the most important case when alpha equals 2, such variables are called sub-Gaussian, Erdesch, Schlein and Jaub got beta of 2 to be 9 halves, but very recently, Nguyen and Wu improved this to the optimal bound beta of 2 to be 1 half, which perfectly matches the GOE case. If you look at the Genebran ensemble, there is the same statement, sorry, for any t greater than 1, there is precisely the same statement. It was approved a couple of years ago jointly with Vershinin, and in this case, beta of 2 was by some random reason the same 9 over 2, although I don't claim that this bound is optimal in any way. Actually there are explicit known ways how to improve this 9 and 9 halves. So what the L infinity delocalization tells us is that if you consider the distribution of mass of an eigenvector, you can rule out peaks. So in particular, this is not an eigenvector with high probability. We do not allow large coordinates. But what we haven't ruled out is the chasms in the mass distribution. So here I consider coordinates from 1 to N, and the coordinates are the absolute values of VJ. We ruled out the large coordinates, and we haven't ruled out the small coordinates. But if we consider a random vector uniformly distributed over the sphere, again it doesn't matter, real or complex, it must have relatively large coordinates. We cannot assume that it is completely flat. There will be some deterioration. Again, we can analyze the model of the normalized Gaussian vector. But any set of, if I consider, any set of size epsilon N, it must carry a mass which would depend only on epsilon. And here the normalization will be epsilon N. The idea comes that we model the eigenvector over, and the matrix models an eigenfunction, and an eigenfunction cannot be zero on the set of large measure. The measure here is normalized counting measure on the set 1, etc. N. So the measure is epsilon, the size is epsilon N. This brings us to no-gaps delocalization, which is the statement that with high probability, any set I of coordinates carries a non-negligible mass. So if I take the eigenvector and consider only coordinates belonging to I, then this is greater or equal than some function. And this establishing this fact will be the main topic of the first three lectures. So before I proceed, let's fix the model. And we want to incorporate as many different models as possible. We want to incorporate both Hermitian and non-Hermitian matrices. So let's make assumptions which are maximally general. And first, let me make an assumption about independence. A ij is independent of other entries, except possibly a ji. So this encompasses Hermitian ensembles, full IID ensembles, skew Hermitian matrices, etc. The second assumption looks a little less natural, but let me formulate it and then I'll comment on it. So the real part of a ij is random. The imaginary part is deterministic. This looks a little artificial, but it is assumed to include two most important cases. First, the real random matrices, in which case I can take the imaginary part being zeros. And second, the complex random matrices with real and imaginary part independent. Then, if I condition on the imaginary parts, I'll reduce it to this model. And if I prove that there are no gaps in the localization, conditionally I can then remove the conditioning by taking expectation of the imaginary part. And then we'll have two results or two models. First, let me formulate a simpler one for the matrices with absolutely continuous entries of bounded density. Again, before I do it, let me introduce one event. Let's take M to be a positive number and introduce the event BAM. Is that the norm of A is less or equal than M square root of N. This will be a likely event if the matrix has uniformly bounded fourth moments, say, than the event BAM occurs with high probability. And so, first, let me formulate the first theorem, joint with Varshinin. So, I'll assume that A satisfies one and two, these assumptions on the right board. And also, assume density K is less or equal than some constant K and epsilon greater or equal than 8 over N any as positive probability, at least one minus CS to the power epsilon N minus the probability of the complement of this event BAM, any unit eigenvector, and for any set I of coordinates of size epsilon N, the norm, the mass carried by these coordinates is at least epsilon S to the sixth power. So, let's see what we have. First of all, let's take S to be a small constant, then this probability is exponential in epsilon N, and this is the optimal bound, and this is a small probability for any reasonable matrix ensemble. And then, if I take at least 8 coordinates, any 8 coordinates of an eigenvector, then these coordinates carry a mass which is polynomial in epsilon. No, for any M. One interesting feature of this result is how the tails of the matrix enter the picture. They almost now not appear here. I even do not assume that the entries have an expectation. The only way they enter it is that we want to say that the norm of the matrix is small with a reasonable probability. And this is the bounded density case, but let's consider a more general case where we want to drop this bounded density assumption, and we want to drop all the assumptions on the distribution of entries. Of course, we cannot drop all of them, otherwise the matrix would be deterministic, but let's reduce the assumptions to bear a minimum, and then there is another theorem. C would depend only on K. Again, joined with Vrashinin, so again I assume that A satisfies 1 and 2, and also assume that for any z complex, the probability that absolute value of A ij minus z is less than c1 is less than c2 where c2 is a constant less than 1. This is the minimal assumption I can put that the entries do not fall entirely into a small disc around C. Then, if I call this star, star holds, but in this full generality we are not able to get down to 8 coordinates. The number of coordinates will be relatively big, holds for any s relatively large, greater or equal than epsilon to the negative 7, 6 is n to the negative 1, 6 plus e to the negative c over square root of epsilon. And this condition...