 Hello, welcome back, before introducing the next lecture, Gregory, who already introduced himself, so I'd like to announce the first change in the program. So tomorrow, the second lecture is by Maurizio Fagotti, and he asked us whether this could be postponed to Thursday, and because tomorrow there is this seminar by Emmanuel Bloch at CISA, and this Bozeman lecture, so you can read more on the website of CISA, it's going to be at the same time, at 11 a.m. So he would like to attend, so we thought maybe we could have this shift so that, and also for those of you who want to attend, maybe we can think about, well, there is a shuttle bus going from CISA, from my CTP to CISA, I don't know about the time, but maybe we can try to see whether something can be arranged. So, but in any case, I will tell everything to Erika, and she will update the website, so you just look at the website, and everything will be clear, and so now it's the time for Gregory for his first lecture, and so I'll just let him start. Okay, thank you very much, good afternoon to everybody. So in this afternoon lecture, I will start to talk to you about relatively broad subjects, which has to do with the statistics of extreme events, say of extremes in general, this covers various questions, and somehow the title of these lectures contain this precision, which is that I would like to show you how we can indeed deal with statistics of extremes in correlated systems. So I guess that many of you, some of you at least, have probably heard about these such questions, and it's pretty clear, in fact, that extreme value statistics and more generally statistics of rare events actually play a quite important role in various areas of science, and this includes many, many topics. Probably we can try to elaborate a list of these topics where rare events actually play a quite important role in science. Certainly, if you think about it, the first area that would come to your mind is probably the right subject of environmental sciences, which includes many extreme events, such as, say, for instance, earthquakes, war graves, many other questions related to climate science. And in these areas, it's pretty clear that having a good idea or good estimates of the statistics of rare events is quite important because, of course, although these events are rare, they may have disastrous consequences, obviously. And so that's one subject where these kind of questions have been studied quite a bit, and there is actually a huge literature in that context. Another white subject where these kind of questions are quite natural is the questions of, I mean, the areas or the realm of finance where it's clear that one usually has to watch quite carefully when such big events, when such cracks or crash, say, financial crashes that may happen, which may have disastrous consequences, at least in that field. Sometimes, people also talk about avalanches in this case, which are quite closely related to this kind of questions, and it's clear that in this kind of context, one would also like to have quite good control on when and how high would be such big events. Now, of course, here this is a school on complex systems in statistical physics, and that will be probably, and that was actually my main motivation when I started to study these questions. It turns out that the statistics of rare events play also a very important role in the context of complex and disorder systems, and that was realized around the early 90s where people working on disorder systems, in particular in spring glasses, realized that these kind of questions are actually quite relevant, and I would like to give you some concrete examples to give you a kind of ideas why we should care about this problem. But before that, maybe I will just, because I think that this is a little bit nostrical to avoid to breaking it, but so let me just elaborate a little bit more on this aspect. So, for instance, in the context of complex systems, we usually like to have this kind of cartoon where you look at the energy landscape of your system. So, basically, in this simple cartoon here, you plot the configurations of your system, which is usually a high-dimensional space, but to make the picture easier to draw, let me just do it, imagine that it's one-dimensional, and here, basically, you would like to think about the energy of such a configuration, either the energy or free energy. And typically, I mean, in many of these complex or disordered systems, in a wide sense, at the moment, I don't speak about any specific models, but you would imagine that you have a quite rugged energy landscape like this, typically. And if you look at the low-temperature physics of these kind of systems, then obviously, this low-temperature physics will be dominated by the lowest energy states. So, typically, at low-temperature, your system will lie in one of these minima of the energy landscape. And typically, so if you look again at the low-T thermodynamics, for instance, low-T thermodynamics of such complex systems, then typically, you would be dominated by or given by this minima here, typically given by the minimum. And so that means, in particular, that if you look at the fluctuations of the free energy, for instance, of such a system, then it will be intimately connected to the distribution of these low-lying states. So that's clearly, obviously, an important quantity to compute, and you would like to say something about the minimum of these guys. So that's for the statics. I was talking about thermodynamics. Now, if you look at the dynamics, it's also clear that if you look at what happens about the dynamics, well, in the large-time limit, you know that the system will be dominated by the largest barrier. So typically, suppose that you are here, but you are still far from the, so you are close in energy from the minimum, but still to get to it, you will need to cross this very high barrier. And in other words, that means that the relaxation times in these systems will be dominates if you think that you have a kind of a renews-type dynamics. So if I just denoted by BL here, which is the barrier in your system, then typically the relaxation time TL would scale like exponential of BL divided by KBT. And that means that if you look at the distribution of the relaxation times in your system, then the distribution of TL, this relaxation time, there will be related to the distribution of the barrier. So that's quite obvious here to study so relaxation dominated by the max of BL, the max, the largest barrier. Okay. So you see two examples here. Of course, it's a bit qualitative, but you already see that in both cases, if you now look at the low T dynamics, you will be basically given by the max what I've written. So you have already the idea that these questions are actually extremely important in this context. And that's why people started again in the mid 90s, started to study these questions with the tools and the questions also of statistical synthesis. Yeah. Right. So this actually is called the Arrhenius law. So that's typically, so if you really study this escape time problem, so you really look at this kind of bystable potential, imagine that you have a two wells, then what you can show is that the time that you need to cross this barrier is indeed given by this exponential time. So that's called activated dynamics, Arrhenius law. Now this, okay, if here, that's good question. So here I chose to represent the energy landscape. So in principle, the energy does not depend on the temperature. Okay. And then, of course, the dynamics itself will depend on temperature. And in particular, you see here that the temperature is here. And this is correct at low temperature, so on. Okay. But that's temperature is here. But in principle, at least if you look at the energy landscape, it's kind of, it's static. Okay. It depends on the kind of dynamics that you give to your system, but usually the kind of dynamics, relaxational dynamics that we study because it's relevant from an experimental point of view indeed drives you towards these low lying states. Okay. So this is, okay, maybe two words are attached to that. So this is called activated dynamics. And this specific form is called under the name of Arrhenius law. Right here. Okay. So that was to set up somehow the stage of the motivation somehow of as to why one should care about these questions. So in the following of the lectures, of course, we will see some more concrete examples where the extreme value statistics appears in more concrete example. But that's roughly the main motivations. Now, the question is, what do we know about these guys? Okay. So what do we know about these kind of statistics? Okay. So it turns out that, oh, sorry. Yes, that's true. Here. Okay. I, okay. This is, this is, this is a name. This is Arrhenius law. Arrhenius was a chemist. So question now is what do we know about these guys? Okay. So it turns out, and I will say more during the lectures about it, but it turns out that roughly speaking, the statistics of rare events in the general sense is completely understood. Oh, it's well understood, I should say. It's nicely more precise. It's well understood in one case. And this case corresponds to the situation where all these, suppose that all these energies here are just independent and identically distributed random variables. Then a lot is known about the statistics of these guys. And similarly in that case. So that means that the statistics of rare events is very well understood for identically, so for independent identically distributed random variables. So what do I mean by that? So I will usually use this shortcut IID for independent and identically distributed random variables. Okay. So one example, rare events actually contains a lot of various questions. Some of them will be touched upon here during these lectures, but for instance, if you really look at the statistics of the maximum, so you just give you a set of random variables, say x1, x2, xn. So that could be the energies that are here indexed by the by the configuration or the barriers here indexed by L or the temperature at a given time I on the day I whatever. So these are supposed that these are IID random variables. So you imagine that you have, you look at one realization of these n random variables and then you look at the maximum among them. That's basically what I was playing with before. So you look at this guy. Okay, so xmax here is one of these rare events. Okay, so that's what I call here rare events. Now, for instance, one rare event is the maximum, okay, or extreme event if you want. So if you look at the maximum of these xi's, then in this case, in the case where these guys are just independent and identically distributed, everything is known about the maximum. That means we know very well the statistics of xmax is completely, completely is known. Okay, so I will assume that many of you probably have not heard too much about this aspect and I will start in a few minutes. I will start by reviewing the results that are known from that. I know that some of you have already heard about it, but okay, I hope that it will be good also for you to refresh a bit your memory about it. But one has to start with something and I chose to start with the simplest case. But the good thing to know is that in this case everything is well known and you will see that it's still quite rich, known but quite rich. Now, the point is that even though this is already quite useful in various contexts, it turns out that in statistical physics and in particular in the context of complex systems, it turns out that this variables xi's are usually strongly correlated. So let me just give you a few examples where this is indeed the case. Okay, so this is nice. There are other examples of rare events where all these things are well known for iids. This is also the case, for instance, of the record statistics, which I will talk a bit about later on. Yeah. Yeah, exactly. Yeah, so that exactly, that's what I mean. So this is one example. Another one would be the, for instance, the record statistics, which are also rare in some sense because they are singular if you want, and as you said, above some average. But just to say that in general, these kind of questions are very well understood for iid random variables. Of course, then you need really to sit and to say and to ask really what is the precise question. But what I mean is that usually, once you have asked these precise questions for iid, there will be a precise answer. Okay. So that's good to know that already. But nevertheless, in many cases, these random variables that we have to deal with in concrete situations are not iid. And therefore, one needs other, other tools. And that will be somehow one of the, the, the, the theme of these, of these lectures is basically how to handle these extreme statistics in the case of strongly correlated systems. So let me just give you a few examples of basic examples. Just let me write this explicitly. However, in many applications, concrete applications, as you will see, I mean concrete. Many problems, at least in many applications, the x is i not iid. So one first example, which I already mentioned before, one example is the example in finance. I mean, in finance, there is a nice model that people like to study, which is the Brownian, I mean, random work model. So typically, if you look at the stock option, the price of a stock option, say, at step n or step t, if you want s of t, then a good model that people like to, to study is this exponential of Brownian motion. Okay. So b of t, here is a random work or Brownian motion. I will come back to this a bit more, a bit more detail later on if we've needed. But just to set it as an example. And so typically, you will have this kind of things, right? It's an exponential because the price cannot be negative somehow, okay? And then, okay, you will have something like that. But usually, the idea, so for instance, you would like to know when, what important question is when, when do you reach the maximum here? That would be quite, quite important. So this is s of t as a function of t. Now, this case, in this case, you can convince yourself that the prices, s of t, are strongly correlated. And they are strongly correlated because the Brownian motion itself is a strongly correlated path in the sense that if you look at the correlation between bt and bt prime, at two different times, then this will be proportional to the minimum of t and t prime. And so there is no decaying correlation in this case, right? So you see that there is, there are very strong memory effects in this kind of things. And so that means that if you look now at the, for instance, if you look at s max, which will be the max of our given interval, say between zero and t of s t, then obviously because of these properties, these s t there are strongly correlated. There is no decaying correlations there. And so you are facing here a problem where this s t is there are strongly correlated. Or maybe it's already two down here. Is that clear? Okay, so that's, that's one example. And obviously in this case, what you will know, what we will learn from this IID framework is just useless. Is that okay? So that's one example from finance. Of course, these questions related to Brownian motions also are quite, quite important in various areas of physics because it has many applications. And I will actually, later on, I will cover in detail the extreme statistics for such Brownian motion. You will see how far we can go. Now let me just give you yet another example, another famous example I should say, from statistical mechanics of these ordered systems where you could easily see that the correlations are strong. This is the famous example of the directed polymer in random media. So that's another example where the IID case is completely relevant and where you need, where you need some additional tools, which is that, this is really in physics in this case. And this is the problem of the directed polymer in random media. So, so of course I will not treat the whole, the whole problem in full details because it's, it's quite, quite hard one. But what is it about? So you, let's consider a square, a square grid, a square potential, say like this. Okay, so this is my square. I will start to, to do some random work if you want, which are directed and I will construct, and this random work will construct a polymer. So how do you do that? So you start, you start from the top here and then at each time step you just go randomly to go to the right or to the left. And you will have something like that, right? Typically. And you will end up to some point here. So that's, sometimes people like, I mean, we like to, to think about this direction as time. So it's directed, okay? So it just, it can always only go bottom wise. Now, so this is one polymer here. Now I want to assign an energy to this polymer. So how do I do that? Well, what I will say is that on each, on each side here, so I have a side ij. On each side, I have some energy, epsilon ij, which is a random variable. Okay, so ij, okay, I mean, you can have, that would be the direction i if you want. That would be the direction j if you want. Now, so this is one configuration of a polymer. So it's a configuration C. And now I assign some energy to this polymer. And this energy is simply the sum, the sum, sorry, of the energies of the points that you have met along the, along your path here. Okay? So that's basically the sum of this energy plus this one, plus this one, this one, that one, this one, this one, and finally this one. Okay? So basically the energy of a configuration, E of C is just the sum over all the, the couple ij, which belongs to the path of epsilon ij. Okay, I mean, it's a big complicated way of writing something really trivial. Okay? So that's just the sum of these, of these energies. And of course, I mean, one important question in this, in this problem, and that many problem actually can be rephrased in terms of this directed polymer. The main question that you may ask here is, is basically an optimization question. The question is, what is the minimal energy that you can, that I can find for a given realization of the configuration? That means that I would like to, to look at the mean, which will be the minimum over all the, the minimum or the maximum, depending on, on your problem. But okay, that's typically the kind of question that you would like to analyze here. Okay, so you would like to find the optimal path if you want. Okay, so that, that's the path that costs the less energy. Now it's fairly simple here to see that these random variables here are actually quite strongly correlated. Indeed, suppose that, so I've, I've drawn here one of this, of this, of this path. Now let me look at another path. Then I could have this kind of path here. I could have this kind of guy. Now you see that on this path, it's pretty clear that it will have a non, non zero overlap with the, the other guy here. So I will have a C prime here. And in other words, that means that in other words, E of C and E of C prime will be actually quite strongly correlated simply because there is quite significant number of sites which are the same for the two paths. Okay, so obviously in this case, this E of C turns out to be extremely correlated. Okay, and that's, okay, you have here a graphical way of seeing it. And this directed formula model is actually very important. New tools to study this problem. So the values E of C gets from the point of view. So that means that the IID case of obviously, okay, is it better? So let me do it more carefully. Okay, let's try. So again, that means that the IID case cannot hold here and does not hold and is completely useless. Now it turns out that the full problem turns out to be related to another extreme value questions, but in a different context. It turns out that probably I'm not sure I will have time to really cover this, but let me just tell you about it. It turns out that the fluctuations, so if you ask about the distribution of emin, for instance, so how is it possible to say something about the fluctuations of emin? Well, it turns out that the fluctuations of emin are given by, are the same in fact, are given by the statistics or by the fluctuations of another extreme, which turns out to be the largest eigenvalue of random matrices, largest eigenvalue of Hermitian random matrices. So what is it? What is it? Well, you take a matrix and buy in matrix and then you fulfill the elements with Gaussian independent random variables, both real and complex, such that your matrix is Hermitian. Now you look at the eigenvalues of this matrix. It's Hermitian, so they are all real. And if you look at, and you say, okay, I will look at the largest of them, let's call it lambda max. Now it turns out that lambda max has the same distribution as this guy. Okay, that's a relatively recent result from, from the 2000s now, it's getting a little bit old, but and this is related to maybe something that you have heard about, which is the traceridum distribution. So if I have time, I might say a few words, I mean maybe more than a few words about random matrices and extreme value questions in random matrices, but you already see that it has very nice applications to start make models. Yes, I've seen you. Yeah, that's true. Yeah, yeah, yeah, yeah, absolutely. Yeah, yeah. I mean, of course, you can consider all the cases, but what I'm saying here holds actually for IID. Yes, you mean which configurations? Well, I mean, the path you see, I mean, okay, it's themselves, they are all a bit correlated, right? Because geometrically, they, they, they, especially at the beginning, they will have a rather strong overlap. So they are not, they are not IID in that sense. Okay, to talk about IID, you would need to define probability measure on the path, which you can do. But no, they are not IID. And what is clear is that what I want, what I want to emphasize here is that these scalar variables here are not, are not associated to the path are not, are obviously not IID for the reason that I mentioned. Okay, because if you look at E of C and E of C prime, then you will have many of these epsilon i's will enter into the sum here. Yes. Yes, because, because of that, yes. But here, of course, I'm in the geometry of the graph itself makes in use of some correlations on the C. Yes. That's true. Yes. In that case, so I impose, yeah. In that case, I impose them to have, say, capital N, capital N steps, basically. Yes, that's true. So they will all end up at a fixed set. That's if, if I think it as a time direction, all the random walks here have a fixed length. Yeah, fixed number of steps. Okay. There are some, some cases. That means, so here you see, I mean, I've not specified too much things. So, but I have actually here, I said these epsilon i's are random variables. They are IID. Now, it turns out that for some specific choice of the distribution of this epsilon ij, for instance, if you choose them to be exponential, positive and exponential, and if you don't look at the minimum, but say in that case at the maximum, then in that case, there is, okay, there is an explicit connection with random matrices that you can construct with some model of random matrices which are called Wishart matrices. In that case, you can write explicitly the distribution of the minimal energy here as the largest eigenvalue of some, but that's a very specific model. And what is believed and now, which is more and more, I mean, for, for which there are more and more indications and sometimes with those proofs, that this is actually quite universal. That means that it does not depend too much on these epsilon ij's. But the one to one correspondence, first, is quite mathematical in the sense that there is no very good physical argument. And the other one is restricted, second is restricted to a very specific choice of epsilon ij's. So, yeah, so indeed, so in any case, yeah, so this in general, of course, I mean, this universality will be true, will hold in the limit where the polymers get very long. That means that here, and on that side, you need to take large matrices. Essentially, the size of the matrix here, which is n, is roughly the number of steps that you have here. Are there questions? So, if not, that basically closes this introductory part where I hope I could more or less set up the problem and also give you some motivations as to why we should care about these questions. So now, let me just give you the outline or at least the, yeah, the current outline in my mind of the following lectures and we will see how it goes. And of course, this outline is not frozen. We could adapt it. I don't know how far I can go, but basically how, so that's how I plan to go on. So basically in the first part, I want to do the extremes of IID random variables and that I will cover in quite detail. I want to go to these extremes, but for one, I want to cover in detail one example of extremes for strongly correlated sets for strongly correlated examples. And okay, I chose to, well, in fact, there are not so many systems for which we can say something about strongly correlated systems, but I chose to talk mainly on random walks, random motions. So random walk is discrete in time. Brownian motion is continuous in time. And I would also discuss the relevant case of levee flights. So given that, so I want to discuss the extreme statistics of these guys, but since, okay, I thought that maybe at least I wanted to be sure that everyone is clear about what these objects are, I plan to have something here in between where I will do some overview of minimal requirements about that will be just a reminder. I mean, I will just recall this random walk, et cetera. But that will be a reminder, okay? Either a reminder for those for those of you who knows, know this by heart already. And otherwise, for those of you who are less easy with that, that will be the occasion just to learn the, the minimum. Okay, I want, I want these lectures to be as self-contained as possible. So that's why I chose to do that. Okay, so then, so that will be for the extremes. And then I wanted to, I want to discuss another kind of rare events or rare events type of questions, which is the case of records. So I will discuss basically record statistics for IID, and you will see that it has already a very nice structure. So that will be one. And then I will, of course, because we will see that I will have developed in principle all the tools to study then the record statistics for random walks, random walks, heavy flights, and all this family. Okay, and then depending on whether I have time or not, I would like to discuss various other examples of strongly correlated systems. And I had in mind to discuss a bit random matrices. Okay, so that will be, again, you see if I write here, it's already too, too, too down. It's, it's too down. So the last, okay, that will be more or less these kind of things. I mean, of course, I will probably not discuss this tricep rhythm, these results in detail, but at least give you, I would like, I mean, if I have time, that would be basically other instance, I mean, other examples of strongly correlated systems for which you can say something about the extreme statistics or rare event statistics. So, yeah, I thought about random matrices because this is pretty universal in the sense that random matrix is used a bit with everywhere in many, many areas. And given that your topics seem to be quite broad, quite heterogeneous also, probably random matrices and random walks are sufficiently universal so that everyone can have some interest on. I could also discuss other, other, other examples. One would be, could be for instance, in the context of fluctuating interfaces, stochastic interfaces like the Kpz, the Karbar-Parisian equations, but okay, this would be maybe a bit more technical or to focus at least. So that, that, that's the plan. Okay. Is that fine? Okay. So let's start with this, with the first, with this, with this first. Okay. Okay. So that's not the outline anymore. That's, okay. One, for instance, which would call it one if we want, before it was zero. Okay. So I will really stick to this case. And here is the, the, the, the problem we already saw hit. So I have xn, x1, x2, xn, which are n random variables. That means that, okay. So I will use this, this letter to denote the probability. So the probability that xi is less than y is typically given by this. So it's just some notation if you want, bx, p of x. And so all the, all these, you see that it's, it's independent of i here. So they are identical and they are independent. So p of x, this is what I will call the parent distribution. It's a density. And that, okay. So, and I will stick to the case, in fact, where these random variables are continuous. Okay. So they have a density. There is no delta, delta component. So this is, this excludes, sorry, for instance, the random variables plus one, minus one. I mean, all what I'm saying, I mean, can be easily transposed to this case. And the reason is the following. The reason why I'm, why I'm choosing this is the following is that I will actually look at this random variable and I want to avoid some possible degeneracy. Okay. So that means that if they have a continuous distribution, it's clear that with probability zero, there will be, there will, there will be two random, random variables which are the same. And that means that instead if you take plus one or minus one or whatever, discrete random variables, you see that you might have some degeneracy from x max. This gives rise to some additional combinatorial problems, which I just want to avoid here to start. Okay. Fine. So they are identical here. And now I want to discuss this and obviously I could also discuss this guy, right? This mean that's another, I will focus on this guy because, okay, because I prefer to think in terms of this one, but of course everything can just go through and be transposed to x min. So that, but I will be mainly focused. Now, what do I mean by independent? Well, I mean the following that if you look at the joint law of, so let me define a joint PDF of x1, x2, xn, which is roughly the probability to observe x1, x2, and xn within dx1, dx2, dxn, then this joint probability is just a product. Okay. So that means that this joint density will be just this product here. So that will be p of x1 times p of x2 times p of xn. Is that clear? I mean, so this, the fact that here the right hand side is independent of i means that they are identical. Now here they are independent because of this factorization property. Okay. So the probability to observe jointly this x1, x2, xn is just the product of observing x1 times the product of observing x2, et cetera. Okay. So that's good to think about. That's basically the independent side. Here this is more identical. So again, we would like to observe, I mean, to say something about this. Now, so this is the maximum. Now you define the maximum, you define the minimum, but usually more generally one can define something which is slightly more general here. And one can define what we usually call order statistics. What do we usually mean by order statistics? You mean that you order these values. So you just rank your values. So you rank say m1n, which is larger than m2n, which is larger than say mnn, such that this one is actually xmax. This one is actually xmin. Okay. So you have a collection of ordered sequence. And if you look at the middle here, then you will have the case maximum. Okay. So in the middle here, you would have a guy which will be mkn. And this mkn is the case maximum. And it's sometimes called the case order statistics. So that's another kind of generalization of this. Okay. It's more complete if you want. And that's basically the question. Okay. So mk of n is what we call the order statistics, the case maximum. So in general, so this is a bit the formal definition of the problem that we have. So we have this joint distribution. We have these random variables. And the question that we ask is basically, so that's our main question, we want to compute the cumulative distribution of the case maximum. And in particular, you would like to say something about what happens in the limit of larger. Of course, this is where you expect to have interesting things from a practical point of view. n is typically the number of degrees of freedom that you have in your system. So taking the large n limit is like the thermodynamic limit, if you want. Okay. So that's the problem. So I would like to show you essentially how one can completely understand this, fully understand this problem in this case. And I will do it in two steps. I will first present you a kind of heuristic argument to get some information about the typical scale of this x max. And then I will show some concrete computations that allows you to compute it explicitly this quantity. Or at least today, I will certainly focus on k equal one, so the maximum. And I want to show you that for k equal one, one has a closed expression in this IID case. And then one can do the large and asymptotic analysis. Okay. Yes. So typically, if I had in mind, if you had in mind this polymer problem that I had, then typically what is interesting is when your polymer is quite long. Okay. And if the polymer is quite long, then you will have a lot of different configurations, typically two to the power n in that case, but typically an exponential number of configurations exponentially in the size of the system. And that's usually where you expect to describe some thermodynamic properties of a system. Okay. And so that means that that's really the limit that is interesting. I'm not saying that having explicit results for finite n is uninteresting. I mean, this is obviously, in some cases, this can be quite interesting. What we will see is that in the large n limit, in this thermodynamic limit, it turns out that this limiting distribution is to a large extent independent of the power n distribution. Okay. That's a bit equivalent to what we observe when you look at the sum of random variables. Then you have a central limit theorem that tells you that under some technical assumptions, you will have Gaussian fluctuations of this sum independently of the initial random variables. It turns out that this is also, there are equivalent results, not with the Gaussians, with some other functions that we will uncover here, but there is also universal behaviors that will emerge there. And that will emerge in the large n limit. So let's try with some heuristics. Okay. So it's okay. No, I will just, I will just look on the first maximum, say X max. Okay. And so I just want to show you some basic facts and formula. And then we'll see. So the first basic fact is what I want to call the typical value. So I want to estimate, so you give me a set of random variables, n of them. And I would like to estimate the typical value of X max. So let me just to set up the problem properly. Let me make some assumption on p of x to start with. So I will suppose that p of x has a fine, I mean, is upper bounded. I mean, the support of p of x is upper bounded in the sense that if I look, if I look at p of x as a function of x, then the distribution of p of x is actually vanishes beyond a certain value, say X star. Okay. So X star might be infinite, but you will see that to think about it, it's sort of nice to, at least for me, it's sort of easier to think about it when X star is finite. So I don't know, maybe p of x could be just a uniform distribution between minus one and plus one if you want. Okay. Now, the first fact, which is, I think, quite intuitive. So imagine that, imagine again that you have, you just take some numbers between minus one and plus one. You just generate them randomly. You take a huge sample of them, 10 millions, and you look at the largest one. And obviously the largest one will be very close to one or close to the upper bound. Okay. And in fact, it will be even closer to X star if you increase n. So in other words, the first thing that should be clear to you is that m1 n will go with probability one will go to X star as n goes to infinity. Does it sound reasonable to you? I will not prove this, but this can be announced as a theorem. But that's a fact. And I think it's quite reasonable. Now, of course, if X star is infinite, suppose that you are taking Gaussian random variables, that simply tells you that the maximum is, okay, it goes to infinity as n goes to infinity, but it does not tell you how it goes with that. Does it go like a power law? Does it go exponentially? Does it go whatever? We'll see that. So that's basically the second thing that you would like. So I think that your question is, how close are you from X star when n goes to infinity, essentially? Okay. So that's the second question, but more precise question that you want to answer is basically how close are you from X star? And for that, I will, okay, denote mu n as the typical value of X max. So is it possible to estimate it? And in particular, to estimate how mu n is close from X star. So there is a very nice and rather simple argument to estimate that, which was, one can, this can be also proved rigorously, but let's try to get this heuristic argument, which is quite nice, which goes as follows. So imagine that you look at the realization of this, of this Xi. So there is X star here, and you just order them. Okay. So I don't know. For instance, here, I will have X, sorry, I will have X 127, here 1235, here I will have X 4. I don't know what I will have. And somehow here I will have X max. I will have another guy here, which would be X7. Now, the question is, how do I, what is the tip, how would I estimate the typical value of mu n? Okay. So if you give me such a sample, and if you need to have an estimate of X max, then basically what I claim is that a good estimate of the maximum, say mu n, is basically around here. And this mu n, which is the typical value of X max, is basically such that you have a finite amount of values, typically one, between mu n and X star. Okay. So that's how I estimate this mu n. Let me just write it explicitly. So what I, what I, what I, what I, what I claim is that mu n is such that the number of Xi's in the interval mu n X star is typically one. So it's over the one as n goes to infinity. That, that's what it means. Okay. So that's, that's the idea. So indeed, if you take, if you say, okay, no, mu n is somewhere there, and it's such that there is a huge number of values between mu n and the maximum value X star, then it's pretty clear that it would not be a good estimate of, of the maximum, right? Because there are many, too many of them, which are in between. And that means that this does not give you, this does not give you any information about the actual value of the maximum. On the other hand, it's also clear that if you are much greater than X star here, or at least much greater than X max here, such that you don't have any values in this interval mu n X star, then typically it's also, you are very off, you are really off from the typical value. But the correct, the correct way to estimate this is typically by saying that, right? This is not an arbitrary definition. Okay. What is arbitrary is this one here. Okay. But one should understand the thing by saying that this is over the n to the power of zero. Okay. So that's when n goes to infinity, this has to be over the one. Okay. So the fact that this is one here is indeed arbitrary. But the fact that, what is not arbitrary is the fact that it does not depend on n. Okay. So when n goes to infinity, that's what gives you the correct estimate. I will set it to one for convenience. And it turns out that the theorem that you can prove, show you that this is one exactly. But that gives you an estimate of mu n, right? Because how do I estimate this? So I can, so this is an estimate for mu n. So I can just now compute the average value of the number of X size, which are in this interval. Okay. So that's, if I want to compute, say, the average number of X size, which are in mu n X star. Well, this I can easily compute, right? Because what is the probability that one of X size is in this interval here? So for one variable, this is mu n X star dx p of x. Okay. So that's, if you give me one of the X size, that's the probability that one of the X size lies within this interval. And now you have capital N random variables like that, x1, x2, xn. So the total average number is just n times x. And this tells me that this should be over the one. Okay. So that's what fixes the value of X star. Okay. So mu n is basically this quantity here. So what I, what mu n is, is, is the typical scale. So that means that I want to estimate how X max will go as n goes to infinity. Okay. So I want to set up to estimate this value. You will see that this later on, it has some more precise, it coincides with some other precise observables associated to these other statistics. But at the moment, I just want to have an estimate of the scale of X max. Okay. So, and, and, and again, the way I, I, I'm setting, I'm setting that is just by saying that this mu n here is just such that you see, I mean, it's roughly X max. So we X max is basically such that there is only a single random variable in that, in that, in that range here. Yeah, exactly. Yeah. So this, so what I'm saying is that this mu n here is, is exactly that, right? I mean, mu n, mu n is not one of the random variables, right? Mu n is just a scale that I fix from, from, okay. I just look at this, at this, at, at one realization of the X size. And I just want to, to, to have a, a reasonable estimate of, of this value. Okay. But mu n is, is none of these values. Okay. It's just, I'm saying that this is just an estimation of the value of X max. And that should be such that the number of eigenvalues within this is specifically one, one, because this one should be X max itself. Okay. And later on, you will see that in many cases mu n actually coincides with the average value of X max. We will see that. Yes. Right. So X star is the, yeah. So X max is bounded by X star. Okay. Because all the, suppose that, let's take this, this, this, this simplest case here, right? So let's think that you have this kind of guy, the distribution. Okay. So I, I, so p of X is this, this quantity. So X, this is X star. Okay. So this is the, the, yeah, exactly. This is the, the edge of the support. Okay. So X max now is the following. So imagine that you are drawing at, at random. You are drawing some random numbers between minus one and plus one. Okay. Then it's clear that none of them will be exactly one. None of them will be exactly one. But what is happening is that if you take ten, ten random variables, okay, there will, you will have some which are quite close to one. If you take one million, I mean you will see that the largest one obviously will be quite close to X star. And as n goes to infinity, it will be very close, as close as you wish from X star. But for finite n, there will be some fluctuations. Okay. And that's what I'm trying now to estimate. Is that clear? Other questions? So now it's nice because we have a, a very nice formula when one can evaluate it for value of distribution. You will see, this, this is not the, at the moment you cannot get it. I will come to that in a minute. I should maybe hurry up. Okay. So now one can, let's, let's, let's try to evaluate it for a few, few cases such, such that we get some, some estimate or that we see what, what it gives. So let's just evaluate this formula on some concrete examples. Okay. So let's, let's look at this model, which is some other, the simplest that I just mentioned with you. Yeah. So I mean I'm even doing something even, even simpler, but even simpler. Let's look at some examples. So for instance, I will do three examples. So let's, let's take p of X is equal to one, if X is in between zero, one and zero, if it's not. Okay. So that's the same door distribution, right? So that's zero, one. So X star is equal to one here. Okay. And so now let's, let's just evaluate this integral. So it's just, it's n times the integral from mu n to X star, which is one times the X, p of X, which is one. Okay. P of X is just one. And that should be equal to one. Okay. And so you immediately get that here, mu n. So you get, so if I just integrate this, okay? So you get one over one minus mu n is just equal to one. And that gives you the typical scale of mu n. Mu n is just one minus one over n. Okay. So you see again that if I look at the typical value, it's probably better to think like this. I mean, basically, one minus mu n is equal to, is over the one over n. So that's basically related to the question that you're asking, that you were also asking Mark a little bit before. This is the rate, if you want. So you have your random variables. There is one here. And if you look at X max here, well, what you get is that typically X max. So this is now, okay, X max, if you want. It's the typical distance is one over n. Okay. That's what it means. Yes. So indeed, I mean, so for instance, here it's quite simple. But if you have something that vanishes, for instance, like one minus X to the power alpha, then this will behave like one over n to the power alpha plus one. Yeah. This crucially depends on what happens at the edge. But you see also that it only depends on that. So now you don't care. I mean, p of X here could be anything, right? I just need, you see it on this formula. You really only, what controls the large n behavior is really the vicinity of X star. You don't care too much about what happens for X negative, for instance, or for X equals zero. You don't care. You just need to know how it behaves close to X star. And that will control the typical scale. And not only the typical scale as we will see, but the full distribution. Okay. That is mu n. So that's the typical scale. Yeah. Thank you. Let's look at another example, which is quite nice to study, which is the exponential case. Exponential is for extreme value statistics. It's quite nice because, oh, so let's just investigate this formula. So let's just get another example, right? Where I will get p of X is just exponential of minus X when X is positive and zero otherwise. So this is a case where X star is infinite. But this formula also holds when X star. Maybe it's nice to point it out or to write it explicitly that also holds for X star equal to plus infinity. Okay. I mean, you can just repeat the argument. There's nothing. I mean, everything will go through. So here what you get, if you apply this formula, you get that you will get this, this, this quantity, right? Now, this is just exponential of X. This is equal to one. And then you see that it gives you an exponential of minus mu n is equal to one. And that immediately gives you that mu n is log n. Yes, is infinite. Well, because if you think a little bit about it, I mean, you can just, just repeat everything that I said with the next star, which is infinite. Yeah. So you want to, yes. Well, you just say that you want, you want, you estimate mu n such that there is just a single variable between mu n and plus infinity. This is a well defined question. And it admits a well-defined answer, I believe. No, no, you are not happy with it. Or it does not, it's not, I mean, you can do the limiting procedure even. I mean, you can define that. Okay. So that's one case. You could actually, I can leave it as an exercise that I will not do it. But if you have a Gaussian, so you see that it goes very slowly actually. I mean, if you take exponentials, I mean, the maximum is very, is just, log is almost a number. I mean, it goes to infinity, of course, but right. So if you take p of x, which is one over sigma square root of 2 pi exponential minus x square divided by 2 sigma square, I leave you as an exercise to show that, that the maximum is even, grows actually even, even more slowly. It actually evolves like square root of log n. It's very slow. And in fact, the coefficient you can actually compute it is just sigma square root of 2. That means, so that's, you see, I mean, in this case, the maximum is something which is extremely, extremely small. And now let's look at a heavy tail distribution. Let's look, for instance, the case of Cauchy. So I take one over pi divided by times one, and x is real in this case. So this is the Cauchy law. It has a power low tail. So it's pretty different from the Gaussian, pretty different from the exponential. And also, of course, pretty different from the other cases where you have a finite support. And then you can just repeat what we did, right? That means that in this case, you have n. So in principle, I have x dx over pi one over x square. This should be equal to one. We have to solve this equation, can integrate it, but you know that mu n here is very large. So I can just approximate here one over one plus x square by one over x square. So I would get the pi here. And then you see that you get one over x square equal to one. Now this is just one over mu n. And that gives you that mu n is just n over pi, n by pi. So in that case, you see that things are quite different, grows quite fast. And in this case, mu n grows linearly with n. Right? So you really see that, as already mentioned, the growth of mu n as a function of n depends quite a bit on the distribution of p of x, right? We have seen all sorts of behavior, the case where x star was finite. And when we have fluctuations over the one over n at the edge, then we have seen this log n behavior for the exponential and an infinite support, this growth of log n. And now we get yet something else for the power law. But what is nice is that we have this formula. And this formula actually encodes already a lot of information. And it's valid for any p of x. Okay? So that was kind of, yes. Well, actually, you can get even, okay. So you can get the, you can get even faster. So you can imagine that you have the fastest that you can imagine, okay, fastest. At least faster than that. One over x to the power one plus alpha. So if you have, okay, so that's, in this case, you will have one over x to the alpha. One, so, sorry. Okay, let's do it. So it's just a remark. Yeah, yeah, sure. So that's why you need to, so p of x can be of this one. One plus alpha with alpha positive, okay. And then you immediately see that then you will get higher powers as it goes. Okay, so we can look at it. So you will have n, and here you will need to have basically, yeah, that's just one over mu n to the power alpha, which is over the one. And that gives you a mu n, which is n to the power one over alpha. Okay, but alpha can be very small. And so, yeah, yeah, you can have anything you want. But still algebraic. I mean, it's hard to get something exponential or whatever. Though it's not, for IID it's quite hard. I don't know any cases. Okay, so that's for the mean, so the typical value, which as we will see is essentially the mean value. Now we would like to have something more. We would like to have the full distribution of the maximum. Okay, so that's where that's where we are now. Okay, so we really want to have a complete, an exact expression for the cumulative distribution of the maximum. Okay, so that's an important thing here is that to compute these statistics here, and in fact, in many cases in probability, instead of computing the PDF, that means that the density, it's usually easier to compute the cumulative distribution. Okay, so that's the second. So this is again in this part of about facts and formula. And here I want to show you that it's an exact formula. So for what I call the cumulative distribution function. So this CDF. So what is that? This is, okay, the definition. This is what I will denote by F1N of M. One, because this is the first maximum. And because this is the total amount, the total number of random variables and M, which is the variable that I want to focus. And this is just the probability that X max is smaller than M. So that's actually very convenient to compute, because if you think a little bit about it, what is this distribution? What are the events that contribute to this probabilities? Well, this is simply the probability that X1 is smaller than M. X2 is smaller than M. XN is smaller than M. So essentially the probability that the maximum is smaller than M is just the same as the probability that all of them are smaller than M. But now these guys, these random variables are independent. So these joint probabilities here that you are computing, they are just products. Okay, they just factorize, because you remember that the joint PDF of the X size is just a product measure of the P of X size. So this is just the probability that X1 is less than M times the probability that X2 is less than M dot, dot, dot times the probability that XN is less than M. Okay, so here I use the fact that they are independent. But now we know that they are all identical, they are all the same. So actually this probability that X1 is less than M is the same as the probability that X2 is less than M. So that means that this is just essentially the probability, so this is just the product from I equal 1 to N of this probability of each of them. This is each of them. I started basically with that. This is nothing else but the probability from minus infinity to M P of X dx. Okay, so here I just use the fact that they are indeed identical. And in other words, this is just this guy to the power N. Okay, so eventually you see that, and that's really the reason why the IID case is extremely simple here, is that we have a quite explicit formula, an explicit formula in fact in this case to compute this object here. Is that clear? So that's quite important because you have an exact formula to start with. That's first usually a very important fact. And in addition, it turns out that here this explicit formula is also quite simple. Nevertheless, if you don't know anything, if you have never saw this, if you've never saw this problem before, analyzing the larger limit of that is not completely trivial and it requires some computation. So let's try now to see how it goes through. I mean, how we can get the asymptotic analysis of that. I will of course not do all the details. Actually these computations are now already, it started in the 20s in fact. The first results were obtained in the, I mean, 25, between 1925, 1930. And essentially the big theorem was eventually established by a Russian, by Nedenko in the 40s, 42 or 43. So I will not repeat all this here because the analysis is also pretty heavy. But I just want to give you the flavor of what happens in the larger limit. But the first thing that one has to say is that if I look at this distribution, let's have a look at how does it look like, this function. So this function here, if you plot it, you see it's a cumulative distribution, okay? So if I look at this guy as a function of, as a function of m, so roughly speaking what happens, you see I mean when m is very small, then basically the probability that x max is smaller than a very small number will be zero. So it will go from zero when m is very small. And eventually when m is very large, it will go to one, okay? So this s from n is essentially, roughly speaking, it's a step function. Now when n goes to infinity, if you don't do anything, if you don't do anything, suppose that x star is finite, what will happen is that in the limit when n goes to infinity, this function will be just a simple step function at x star. There's nothing. So x star, remember that it's just the edge of the support of prx. So in the limit when n goes to infinity, of course this function is not really interesting, it's just a theta function. So what it means really is that theta of, so it's one when m, yeah. So what it means is that you need to do some rescaling to obtain a non-trivial limit. And what it means is that to obtain a non-trivial limit, well you, we need actually to center and rescale this variable m. And so that means that, that means the following. I will write it and then I will comment. One needs to find a n and b n such that, is it still readable if I write here or just here? And then such that what? Such that this guy, so let's write it in a proper way. Limit when n goes to infinity of 1 m of n a m plus b n z. So I'm writing m as a n plus b n z such that this goes when n goes to infinity to some well-defined function, some function g of z. Okay so let's, I need to be a little bit more precise. So first I convinced you that the cumulative distribution is not very interesting, but something maybe which is more, I mean, which is more interesting to plot is the pdf. So that means that the derivative with respect to m of this quantity, so that's really the pdf. So if I, so what I'm saying here roughly, I mean essentially is that if you compute this, if you look at this guy as a function of m, what do you expect? That's basically written here. What you expect is the following. You expect that, so you remember for instance, have in mind that the case of the exponential case, we were saying that the maximum, typical value of the maximum goes like log n. So that means that if you look at the distribution of the maximum, it will be roughly speaking centered about mu n for the exponential. It will be something like log n, but something which is proportional to mu n or similar to mu n. We will say exactly what it is, but, and then it will have this shape, right? The precise shape we don't know, but essentially it will be picked around some value. And this mu n, this is exactly what I call here a n to be precise. So that's precisely what a n is, okay? So I am looking at the distribution of f1n, but close to a n, okay? So that's this value here, which turns out to be indeed similar to mu n. And then it will have also some width. And the width basically is roughly, is this bn. That's what bn is. The similar kind of things actually happens when you look at the central limit theorem. When you look at the, let me just make a comment to make the parallel between the two cases. So that's quite important. I will just let it here. Just a remark, I mean, just to tell you that what we are looking at, what we are looking for here is quite, I mean, should not sound too crazy at least. So if you think about the sum of iad. So that means that I look at Sn, which is basically the sum of this random variable. And suppose that Xi has a well first defined moment, which is mu. And suppose that it has also variance, which is also well defined. Then you know that the law of large numbers tells you that Sn, so when n is large, Sn is basically n times mu. So that's the law of large numbers. So in that case here, in our case, this is the equivalent of An. That's the typical value. If you look at the PDF of the sum here, it will have some, it will be centered around the mean value, which is n mu in this case. And then we know that there are some fluctuations around this mean value. And the fluctuations are over the square root of n, which is my bn here. There is a sigma here, which comes from that. And then there is a random variable here, which I call chi index j. Probably you know of this notation, n01. n01 between the Gaussian. So it's a Gaussian random variable with unit variance and centered around zero. This is the equivalent of that. So n mu here is the equivalent of An for us. So that's this An here. And this bn here is just this guy. So that's basically what I'm looking after. So in that case, of course, if you do that for the sum for Sn, what you will find is as g of z is itself is a Gaussian random variable. Now, the question I am asking is what are these distributions for the IID case? What are the equivalent or what is or what are the equivalent of the Gaussian normal distribution? So that's random variable. So the question is really whether I can find these numbers, An and bn, and what is then the, in that case, what is the limiting distribution g of that? OK, so that's our goal. Is that clear? I mean, if not, I should repeat it because I think it's important, you know. Yeah, OK, so the idea is that, so An, these are just numbers, and this is my m. OK, so this is initially my m. So z is the random variable associated to this one. OK, so I just said, OK, in other words, yeah, OK, maybe I should write it this way. What does it mean? Another way to interpret this is that I write, so basically what is this g of z? I am now writing the maximum, sorry, x max. I choose to write it as the following. I write it as An plus bn times the random variable chi, and the cumulative distribution of chi is precisely the g of z that I wrote there. I mean, that's the probabilistic content of this formula. OK, so I write x max. I'm saying that this is An plus bn times some random variable. This random variable does not depend on An. That's the crucial point. And I want to know the distribution of this chi. Again, if instead of x max, I would have the sum of random variables, then that would simply be a Gaussian random variable with unit variance. Now in our case, the question is, what is this g of z? Is that OK? So the pdf or g prime of z will be this limiting function, right? I think I still have some few minutes, right? OK, so at least I would like to give you more, I will not do the full analysis, but I will tell you what the results are. So that's, again, the nice thing, of course, is that we have an explicit formula for f1 on n, right? This is just the formula that I showed you before in terms of the problem distribution p of x. And now one can do the full analysis. So in general, the full analysis is a bit lengthy to do. I will not do in detail, but instead what I propose you is to give you the results and maybe give you, treat some simple cases and to convince you that the general results are quite reasonable to believe. So let me tell you the result of the full analysis. So the first part was this facts and formula, and now the second part is what I call limiting behavior. It turns out that in general, a n and b n are a little bit complicated to compute, while a n in many cases is just mu n that we have seen before. b n is something else. But the striking facts, but in general, so a n and b n depends on p of x. Now, the striking and quite nice results is that the g of z here cannot be anything. So there are essentially three universality classes. So that means that g of z can only take three different forms that I will just give now. So there are only three distinct universality classes, which are indexed by this number row, which can take the three value, no, okay, I will take one, two, three. These three distinct universality classes, they only depend on the behavior of the parent distribution close to the edge. I will write it and I will give examples. So they depend only, depending on the behavior of p of x. So I remind you that p of x is the parent distribution of the d x i's, of p of x near x star. So near x star, if x star is finite, you have the behavior at infinity, x star is infinity. So that already tells you that depending on you have, whether you have a finite support or not, you will get different universality class. So let's begin with the first class, row equal one. This is probably the most famous, and this is called the Gamble universality class. And this corresponds to the case where x star is, okay, where p of x, okay, p of x decays faster than any parallel when x goes to infinity. Okay, so that means that p of x decays faster than any parallel. So in most of the applications, x star is finite, but it can also be that it is, it has a finite support, and this would corresponds to a p of x which is quite singular, vanishing like exponential of one, like exponential of one over x minus x star. But okay, usually x star is infinity. I will just leave it like this. Okay, so you have something typically the exponential that we have seen seen before, or the Gaussian exponential, one of variable Gaussians, et cetera. So that's the first class, and in that case, one has a specific limiting distribution. Maybe I can already say it. So in this first case, I will come back to this probably in the next lecture, but just... Yeah, exactly. Precisely, yes. So in all these cases, you don't know to know, I mean, no need to know anything about, okay, at least to compute the limiting distribution. The limiting distribution does not depend on the details of the p of x. So again, it can be exponential Gaussian. And in this case, the g of g, the function g of z is actually g1 of z is actually this double exponential that you have seen. And this is defined on the total, on the full real line. Right? So that's the first universality class. I guess I will probably come back to this a little bit more with some concrete examples, but that's nice to see this universality class. Now there is a second universality class which precisely corresponds to the fact when, to the case where you have a power law, dk. Okay? So in this case, you need to have x star, which is infinite. And then p of x decays algebraically. Okay? So you have p of x, which is say x to the power minus eta, or how did I choose it? Yeah. Okay? So suppose that it's like that. So it's x to the power minus 1, minus alpha. So you can have some constant here when x goes to infinity. Okay? So you can, that means that you have an algebraic dk, but alpha can be big. I mean, alpha can be, it's not necessarily a fat tail. I mean, alpha need not to be small. Alpha can be a hundred if you want. And in this case, you have another class, which is called the Frechier class. And you have another limiting form. And g2 of z in that case is just exponential of, yeah, we have this form. So it's basically exponential minus z exponential minus alpha for z positive and is 0 otherwise. Right? So that's quite different from this guy. And there is a third class, which is the class that I initially discussed before. For instance, the case where you have uniform distribution between 0 and 1. And this is called the Vibral distribution. So that's the third case. I guess I should better write it here. I just want to have it, let it here. And so that the first case is the third case, rho equal to 3. And this is the case where x star is finite. And where now you have what we said before, but we already discussed a bit before. So you have rho p of x, sorry, which decays now x star with some power low. Yes? That's true. Yes. Well, okay, this is most of the cases. Because I didn't want to, okay, there is, so most of the case that we encounter is the case where p of x decays where x star is infinite and decays faster. Okay. Okay. There is some details around it for x star finite, which I don't want to enter, but exactly. Yeah. In all these cases, I write x max. I write it as a n plus b n z. And, okay. And the distribution of z is this guy. The cumulative distribution of z is this guy. So what I will do probably tomorrow is that I will analyze some specific cases in all the three universality classes such that we understand really what's happening here. Yeah, I just want to give the results. And then eventually if, okay, so p of x now will decay as x star minus x to some power. Let's call it new, for instance. So let's call it, no, let's call it alpha again. Alpha minus one. Okay. So you have these kind of things, but it can also be diverging, but with an integrable singularity. And then you have yet another class, which is called now the viable class. I'm sure you have heard about these names here. Of course, so it's, okay, it has this form, which is one, okay, let's go with for z, is one for z positive. And for z negative, it has some expression that I'm writing here. Exponential minus mod z to the power alpha when z is negative. I hope it's readable. Can you read everything which is written there? These three functions here, the gumbel, the Frecher, and the viable functions here, they are the equivalent of the Gaussian distributions for the sum of IID random variables. So they are as universal if you want as the Gaussian. And that's actually why they pop up a bit everywhere. I mean, they, many problems you will, you will find this function appearing. And the reason is because they are attached to some universality classes. And that means that they do not depend too much on the details of p of x. Only the large x, only the large x behavior of p of x happens. Matters, sorry. Now, I didn't say so much about the coefficients a n and b n. We will come back to this tomorrow. In many cases, a n is more or less like a mu n. I didn't say anything about b n, but there are, there exists some formula to compute it. So I guess that that's somehow what I would like to do tomorrow probably is to, again, I will not prove these things. I mean, you can do it as a physicist would do. I mean, it's a bit of work. But I would like instead to show you some examples where one can rather easily obtain these distributions and then, so that you are quite confident that these results are reasonable. And then I will discuss briefly what these a n's, b n's are. And I will then end up by generalizing these computations. So here I've discussed the first maximum. But as I said, one can actually look at the distribution of the case maximum. That means that instead of looking at the first guy, you can look at the 10th guy or the guy number 20 and ask the same question. Is it possible to derive a limiting distribution? The answer is yes. There are some very nice formulas for that. And I just would like to show you rather briefly because the computations are quite simple and give you the, finally, the asymptotic results in the larger limit for the order statistics. And that will close what I would like to tell you about the IID case. And then as I said, I will go to, I would like then to discuss basically how these results do extend in the case of random walks, which is a nice example of a strongly correlated system. And for this, I would like to remind you some basic properties known or not so well known properties about random walks. And then we will see how it goes on to extract the extreme statistics for that. And then, as I said, we will go to the records and see how we can handle all these questions, both for IID and for random walks again. Okay, thank you.