 is a random variable and we would like to understand its statistics and in particular what happens in the long end limit. So the first thing that I showed you and I think that that's really one of the key point is that this cumulative distribution is just related to the survival probability of a random work. So I remind you that the survival probability is the following. So here is a picture of what it is. You take a random work that starts at y at initial time and then performs this random jump type of dynamics and you ask what's the probability that it stays positive up to step n. So it doesn't have the right, it cannot cross zero. So these two quantities are related via a simple transformation which I showed you yesterday. And furthermore, we had seen in the previous lecture that the survival probability satisfies a backward equation, basically a kind of Kolmogorov equation, and eventually this integral which is quite involved equation because it's an integral equation for which very few tools exist but it turns out that for the specific case, it's possible actually to solve it and this is solved via this so-called Pollux-Expezor formula, which I gave to you, which was a bit complicated but still I mean which contains essentially all the information that we needed. And from it, we derived two interesting and important results. The first one is I think extremely important and this is called, this is known under the name of Sparonder-Santheorem which gives you the survival probability when you start exactly at zero, which is indeed possible for random work. And basically from the Pollux-Expezor formula, you obtain, one obtain, sorry, this generating function and from it, we can invert this relation and obtain q naught of n in terms of this combinatorial factor. So that's quite remarkable because again it does not depend on anything meaning that you see that the distribution of eta does not enter this expression provided of course so this holds indeed for p of eta continuous and symmetric but still I mean it really covers a wide range of situations. There are actually I mean extensions of it to the case where p of eta is not symmetric. For instance, if it's centered around the finite value, if you have a linear drift, for instance, there are some extensions of it. If you're interested I could give you some reference on that but already this is quite nice. Probably at this stage it's already important also to mention that you see I mean this thing is really universal. Now of course it's universal only if you start exactly at zero and if you start at a different place meaning if not zero but say why, of course you lose completely the universality. That means that if I start now from why and if I ask what is this survival probability, it will of course depend quite strongly on the jump distribution and this was clearly stated in the Pollux-Expezor formula where you had on the right hand side you had really the explicit Fourier transform of the jump distribution that enters. So now that's a very nice result and we will see many applications of it actually. Today we will see one and later on when we will study the record statistics of this random box in the last two lectures you will see that this plays really a crucial role. So that was the first thing and the other thing that I showed you is how to obtain result for the first moment of the maximum and I showed you that from the Pollux-Expezor formula one can write explicitly the generating function of this average value. So in principle if I know this guy in principle I can compute mn simply by using Cauchy's formula. This might be a bit involved and also here it is involved because I have not written it because it's a bit cumbersome but phi is a relatively complicated formula but nevertheless we have something explicit. So I will not do the full analysis of how you now extract m of n from that formula because it's a little bit cumbersome and complicated. I mean it requires a little bit of you need to be quite comfortable with analysis but nevertheless I just want to give you at least the idea of how you should react when you have such a result and what can you extract from it. So I will not enter into the details but simply say a few general things on how to extract the large end behavior of mn once you know this right hand side. So this right hand side is known explicitly. It's some function. It's a bit complicated but it's explicit. You have an integral representation of it and now the question that you ask is extracting it from any n is probably complicated but what you would like to the question that you would like to answer is the behavior of mn for large end. So again I mean if you had I mean why are you are we interested in that. I mean it's clear that you see I mean if I suppose that I have a random work here where with gaussian suppose that these eta n's are gaussians then we know that typically if they were suppose that these these accents were completely independent then we know that if they if at least these accents suppose that they are gaussians the maximum would grow quite slowly within and it should go like square root of log n typically if you have gaussians. Okay now the question is what what are the effects of correlations on the growth of this of this of this quantity. Okay so that that's a quite natural question and that's really a very simple way to characterize the effects of correlations compared to the iad case. So how should I do that so what I want to understand okay so it's the behavior of mn for large end. So in general when you have to so that's really the something that that that one needs to know essentially one when you have such a formula here when you know the generating function of a series like that then if you want to know the large end behavior of this you need to understand how this series here behaves when s goes to 1. So typically I mean in most of the of the cases the radius of convergence of this of this series is finite and typically is 1 and what you want to understand is basically how this so let me call it say this function so that's the generating function of what what I what I wrote here and the idea is that the behavior so basically that the behavior of mn for large end that will be more precise in a minute but is controlled by the behavior of this quantity when s goes to 1. Okay so in other words typically what will happen is that again mn what we're speaking in most of these cases here mn will be will have a power low behavior when n is large and as a consequence what is happening is that you see I mean when s is larger than 1 this s to the n I mean diverges exponentially and eventually the series is not converging while on the other hand if s is smaller than 1 then you see that this goes to 0 exponentially with n and in that case the series is well defined so in most of the cases that that one has to deal with I'm not saying that I'm covering all the possible cases in the universe but in most of the cases that we have to deal with mn grows typically like a power law and that means that the radius of convergence of this series is 1 and if you want to understand how so that means that m tilde of s will be diverging when s goes to 1 and the way diverges is controlled by the large end behavior of mn okay it's a theorem of analysis it's essentially a one of the Tauberian theorem if you want and in particular let's be let me be a little bit more more precise and because that's the way we will use this this theorem here uh so in particular uh if m tilde of s is diverging say as a power law so let's write it in this way with beta strictly positive when s goes to 1 and by it has to be smaller than 1 of course such that this series is converging then in this case what you get what you will get is that m of sorry is that this moment here right this moment here will behave like n to the power beta minus 1 when n is large and n goes to infinity and the prefactor here is just even by a divided by gamma of beta okay so that's the so what is gamma of beta this is just this integral so that's just integral dx x to the power beta minus 1 exponential minus x okay that's just a reminder here and in particular you know I guess that gamma n plus 1 when n is an integer it's just factorial okay so that's a very nice very nice property okay so again you see that for s equal once strictly this series will be diverging obviously and the way diverges is controlled by this parameter here by the exponent there and eventually that means that you can do the reversed ways some somehow it's probably easier to understand but you can easily see that if you have a behavior like that inserted this way then it's really fairly simple to see that it will have this this behavior when s goes to 1 okay it requires some work and I would suggest here to take it as a theorem it's extremely useful it will be useful here and in the following we will use it quite quite quite frequently okay I guess you have already seen this kind of theorem of these results this is a standard result in analysis which is very useful here so that means that if you want to understand the behavior of mm then I need to analyze this right hand side here in the limit when s goes to 1 now I've not written here but I have not written it here but if you remember the formula for this phi was a little bit complicated so this requires a little bit of analysis to really extract the behavior of this quantity when s goes to 1 but that's doable and let me just you let me just give you the result and we'll see what okay we'll see we'll see what happens okay is that is that okay I just want to want you to buy this okay and then we will apply that many times I mean many times here we will apply it a little bit and later on we will apply it also quite quite a few I mean it's extremely useful and it's something that is okay I think it's useful to have seen it once in his life at least when you when you do theoretical physics so here what one one can show that this this quantity here so let's consider the case okay I will just discuss the case we are sigma is finite okay so now I need to distinguish between Levy works and and standard random work so I suppose that the jump distribution so that means that I will suppose that the the second moment of the jump distribution is well defined okay so that means that the random works converges to as I discussed it yesterday or the day before sorry the random works will converge to Brownian motion okay so it's not a it's not a it's not heavy tail so I have just standard random work and in that case so there is a sigma let me define sigma square this way so what you can show there in this in this in this case is that you can actually extract the behavior of this of this of this term and one can show that one can show what one can show that basically this generating function behaves as s goes to 1 behaves as sigma divided by square root of 2 divided by 1 minus s to the power 3 by 2 so that's the result of the analysis I had yeah initially I thought that I would present you this this computation I mean for those of you who wants to see it I could I could show you how it works so then you see that I can apply this nice theorem and immediately get that mn will have this behavior here so if you now so beta is three by is three by two here three half so that means that the exponent here will be three half minus one that means half okay so let's do it so that means that mn for longer so again I have beta so I will have so a here is just sigma divided by square root of two then I will get gamma gamma three by two and then I will indeed get yeah that's correct n to the power three half minus one which is just half okay now gamma three by two is just square root of five divided by two yeah you see I mean you have three by two basically these are related to some Gaussian integrals that's why you get this this product of pi so that's the result so again I mean this this results from this analysis and basically this results from that theorem okay so eventually you get you get this formula okay so let's get the coefficient properly so this is just now sigma so I just have sigma square root of two and over five so let's write it this way okay so that's our result when n is large so it's actually quite interesting because you see that now instead of having something that goes like square root of log n which you would have for a Gaussian collection of Gaussian random variables okay then what you get is something that goes much much faster and it goes actually like square root of n okay so that's the clear signature of the square root of n here is a very clear signature of the of the of correlations we would not get of course for the ad case in fact if you are even you can even show you can even compute the the correction here the correction is over the one this is something that you can compute explicitly there is a nice paper by my colleagues for more say Alan Conte and Satyama Jumdar who did this very nice computation here that's something that that that can be that can be done using this this formula okay so that's for the case of sigma sigma square finite okay now you can repeat the same although the analysis is even more complicated but you can repeat the same analysis in the case of levy flights okay so yes that's true yeah okay okay I was a little bit fast okay so I was just so why why do I say this the reason why I say this is that so it's just a remark okay let's consider the case which is this quick time and suppose that this eta n's let's consider this case suppose that these are iid gaussians so if there were gaussians basically then you would say okay I mean if I look at this extends basically would be would be iid random variables there would be also gaussians okay and if I look at the collection of iid random variables which are gaussians and if they are n of them then typically the maximum should go like square root of log n okay now here what you see is that the situation is quite different and it goes much faster it actually goes like square root of n so that that's what I meant okay so that's what okay that's that that that's what that's what I meant okay yeah yeah so now okay so so what I'm saying is that if you suppose that they are not correlated if you just look at these excise as independent random variables okay they are just gaussians okay and typically they will have if you look at the maximum we know that they are correlated okay but suppose that you have gaussians and forget about the correlations then the the growth of the maximum would be typically like square root of log n and here what you get is something much faster instead if you include the correlations then things are quite different yeah in this case so the point is that here you see I mean it's even more universal than than than anything that we encountered in in iid case because here it does not depend on the on the parent distribution of the of the eta n right so if they are exponentials gaussians uniformly distributed you will get this result okay now it turns out that the reason why it is universal is something relatively easy to understand it's just that in all the for all these random walks with a finite sigma we know that they will converge to Brownian motion and what you can show that's true but nevertheless we will see this in a minute so basically for Brownian motion n will be a t here but this result here you can actually get it from the Brownian motion limit okay that means that if you look at I will probably comment on this a bit later this leading behavior here in square root of n including the pre-factor here is actually given by the Brownian motion exactly so the scale that we get this square root of n peak this square root of n that pops up here is actually the square root of n of the diffusion absolutely I will maybe I want to discuss the Brownian motion limit so I hope it will be a bit okay this will clarify a little bit yes yeah okay so that's okay that's a good question so here if I if I if I need to okay I will we will see that because in a minute I will I will look at the full distribution so in this case actually so if I have in mind this a n and b n's that I had in iid case so in this case actually a n is zero and b n is square root of n I will I will I will see we will see that maybe in a minute that's that that's a good point indeed I would just I just want to to give you the result for the case of of levy flights because it might be interesting or important in some applications so suppose that now I have a p of ita which is like that with alpha strictly positive and in fact I want to take I want to talk about the the first moment so I need okay I need to to to restrict my myself to the case where alpha is in between one and two because if you go to smaller values of alpha in fact the first moment of m n is not even defined but in that case it is and what you find is that okay this is okay there is some constant here there will be some universal constant here alpha over five one minus alpha so what is this a here this is something that I already introduced before but if I look at the so in that case if I look at the the small k behavior of the Fourier transform we know that's that the central object then it will behave like okay I already commented on that when I okay so this a which is here eventually will depend on c and this is basically related to that so you need to to plug this the full distribution for p of ita look at the small k behavior it will have this behavior like this the a you need to read that and this is the a that you find here no no it's because I am a bit stopping okay thank you sorry okay so m n actually you see again so it goes faster than square root of n and you see also that this is it has the same scale as the levy flights right we know that if I have a levy levy random walk of n steps the typical scaling is n to the power one over alpha and that's this guy again is that clear please stop me if it's not so in both cases this is true right I mean for for for the diffusion we know that the typical scaling of the of xn will be over the square root of n and here it will be n to the power one over alpha and indeed the maximum has the same scaling as the other ones precisely which is not completely trivial in principle so now I want to to discuss maybe a little bit the to look at the full distribution if you want of the of mn okay so and and make some contact with with your question which is what are these a n's and b n's here okay so let's go to the full distribution of course I don't want to get it direct I mean for you can get this actually from the from the polax x pizza formula but this this is really a very hard work and I will not even quote the result but what I want to to discuss for you here is essentially the Brownian limit because there are things okay so it's the full distribution of mn when n is not so what happens is that if you look at this quantity here again we have seen already many times that it has this typical shape right if I plot it as a function of y so the typical value here so that's that's what I mean here is that the typical value there so it's something like this it goes like that when n is very large and the way the point at which this jump rises which is the typical value is actually mn okay so if you want mn here we really place the role of the typical value of the maximum and in fact we know that okay let's I will focus on the case where sigma is sigma is finite because okay so I don't I will not discuss the case of levy flights which is interesting but also more involved so this we have seen know that this is just okay it's not anymore on the blackboard but this is all that form sigma square root of n there is some prefactor of square root of 2 pi square root of 2 over pi if you want but let me include it otherwise you will be a bit confused so that actually suggests that this q of y n in the long end limit so this scaling here so one sees that the typical scale is over the square root of n and this suggests I'm not saying that it shows it but this suggests that somehow we have already seen it but this suggests that this q of y n takes a scaling form which is of that form okay so that will be something of that form okay so again the same argument that I did last time for the survival probability so q is a probability so it has dimension one and the natural scaling variable you see I mean is the scale square root of n okay so you see that that's so that's the cumulative distribution okay so this this this indeed has this form okay so that tells you that in other words what what I'm sort of saying here is that the typical scale of the that controls all the fluctuations of the maximum is of the order square root of n okay so that's the typical fluctuations of your maximum there will be a order square root of n and it's centered around zero so there is no a n here if you want so if you want to come back to the iid case this really amounts to have a n equal to zero and b n equal to square root of n okay because I'm saying that corresponds to if I want to make some contact and b n equal square root of n so I don't need there will not be a shift here f you will see is a very simple function in fact so so what I said before is that in this limit in the large n limit one can actually directly okay of course one way to obtain this f here is to go to the polax x pizza formula which you can do another way to do that is to observe that in the in the large n limit and when you look at the scale of order square root of n well the fluctuations of the random walk are governed by the Brownian motion okay so that means that this function here f will be given by the Brownian motion limit okay so that means that this f of y over square root of n uh it's okay uh this of course I should maybe I can be a little bit more precise if you want this of course holds in the limit when y is larger than one n is larger than one and you look at y over square root of n fixed okay so that's the that's the scaling limit right so let me okay let let me write it explicitly because it's important the scaling limit is really why large and large and you keep y divided by square root of n fixed and you know that if you do that all these observables of the random walk will be described by the Brownian motion okay so that means that this f function here f of z is given by the Brownian limit the Brownian limit you remember we have already studied it because I solved the equation for the survival probability directly in the Brownian motion limit and I found this error function okay so that tells you that this f of z is nothing else but an error function so in other words you can write directly uh you can pick up directly the result that we had last time right so that's this error function f of y divided by square root of 2n remember that okay so what I know is that y over divided by square root of y divided by sigma square root of n will converge to the Brownian motion and then that's that's immediately what I get okay I can redo all the things that we did last time but I think it's completely useful that is that clear okay so what I'm saying is that I am looking I want to understand what is happening to this random walk when there is a large number of steps now what I know is that in that limit when n is large we know that this random walk will converge to Brownian motion and it actually converges to Brownian motion if you rescale of course the coordinates by square root of n so in other words that means that in this kind of function you need to renormalize if you want all your distances by square root of n to get something which is well defined and has a good large n limit and eventually is converging to the to the Brownian motion uh to the Brownian motion results do you understand so in other words it's a bit like like like with like the remark that I was making last time when I did the didi id case somehow if you take this this formula here and if you blindly take the limit n goes to infinity you will not obtain anything really relevant except some theta function if you want to do something more precise what you need to do is really to do this to do this rescaling somehow introduce a non-trivial bn if you want now we know what this bn is because we know that there is convergence to this Brownian limit okay so so in particular if I had uh levy flights instead of having square root of n here I would expect something like n to the power one over alpha you mean for for bn well no actually because the okay so that's uh yes exactly yeah okay so that's that's the reason that's the final answer and in particular uh so that's the the the cdf then we can get the the pdf of the maximum it's better somehow the pdf is just dq dy and dq dy is just an error function it's just a Gaussian okay so that's just this result here sigma square n exponential of minus y square divided by 2 sigma square and now here one has to take care one thing is that this is only positive okay the maximum is necessarily positive so yeah y has to be positive yes well I mean yeah well the sign is that uh indeed that the the the the limiting distribution that you have is not a gamble is not a gamble is not is not fresh but that's a good point uh this this limiting distribution okay it's it's a bit trivial it's a Gaussian but it's a half Gaussian y has to be positive the reason for that is that it has to be positive because the random walk you remember starts at zero here and since it starts at zero the maximum cannot be smaller than zero so y has to be positive and indeed this is a half Gaussian so and that's that's a good point you need this distribution although being a Gaussian uh it's not it's not really uh trivial or it's not a simple fact to observe it here because we are looking at extreme statistics and one would not naively expect to see either the gamble or the fresh air of eyeball and the answer is that we observe something quite different okay so again this this half this error function here uh so that's a good point indeed which is what you mean you say because it's a Gaussian that's true so so the the the indeed the the the universality is also very I mean this is also extremely large here yes so it's a new it's another universality class uh for it's an important universality class I should say so it's different from iid universality class right which are again uh gamble fresh air or vibe okay this is not none of these one and that's that's obviously something different now maybe a comment on that I said that this result here actually comes from the Brownian limit so what something that you can check I leave you as a check this result here is precisely the result of the of the the Brownian limit so in other words if you compute uh okay maybe it's just as a remark here so I obtained this formula here before or at least I showed you that I can obtain it via this polaxx pizza formula now in fact you can also once you have done this analysis you can do it you can get it immediately by computing the mean value of this of this of this random variable okay so once you have this limit here essentially uh you can just get okay let's make me write it here the mn that I obtained here you can just obtain it this way from this integral dy y dq dy so again this is just the the mean value of of the Brownian can you read it so in other words I worked a lot with this polaxx I mean to get the result that I could have obtained in in two lines but of course I mean there are some good reasons for that in some cases you would like to need uh to have uh I mentioned that in some cases it's very important to get the sub leading corrections for instance there are many problems and there is a very nice problem uh in computer science uh which has to deal with the analysis of some algorithm where it's turned out to be quite important to get this this constant here there is a famous work by so that's an infant for this case of course you need really to uh to have this machinery and not simply the the Brownian the Brownian limit but nevertheless it's good to know the Brownian result uh it's already quite useful and in many cases this is even a good approximation to to the random work limit so we when as an explicit expression for Brownian uh in the Brownian limit now if you take levy flights there is no so simple expression instead that the one has some rather complicated integral representation of the distribution of the maximum but still I mean it's fairly explicit and things that you can get that you can get with the with this polaxx pizza and and extensions of it okay so uh I want to leave this uh yeah are there questions okay so I want to to just to to to have a last uh last remark or last application somehow of what we have been doing to compute another quantity which is related to uh to the extreme statistics so up to now I was looking at the value of the maximum now if you look at the the the process uh of as a Brownian motion the very quite natural quantity is to ask when does this maximum happen okay so if you look at your random work uh on the say up to step n so you start at zero and you will have these kind of things and you would like to know where when this this maximum happens okay so that's that's the question that you want to ask you and that's the time the time at which the maximum is reached so I want to show you this because it looks like a bit like a complicated question of course it's quite important I mean right I mean if you have in mind that you are looking at some I don't know if you think that it's the the price of some assets or whatever and it's of some options it would be quite important to know at which time this this happens yeah the mean value while it's like square root of n so the maximum goes like square root of n so n is the number of steps okay so we have seen that mn goes like square root of n is that fine so it depends on time somehow right when n is larger it gets larger and now you ask okay given that that I have this I look at this random work on on the on the interval zero n and I want to know when it happens at which time steps does it happen is exactly does it answer your question so yeah again this this this is obviously a quite quite important quantity now again maybe a remark here I'm looking at the case where again just to fix the notation but p of eta is is symmetric that's one thing but it's continuous and since it is continuous there is no degeneracy right so that means that the I mean with probability one this maximum is reached at a single point okay so almost surely one should say almost surely the maximum is reached at a single point okay so there are not two points which are the same values okay simply p of eta is continuous so now you want you want to do this and to do this thing so you want to compute the distribution of the time which I will denote m if you want so question is how do I do I do I obtain that so the idea is to do so you will do some construction that that is very useful when you look at this this random walks you just separate it on two sides the first side which is before the the maximum and the other one which is after the maximum okay so let's just look at these two parts okay so I just divide this two independent parts okay so I just divide it in these two two segments okay so I have basically here the first zero m and m and minus one okay so these two segments of course because the the the process is Markovian these two segments here are just independent why do I say n minus one is probably not correct it's n yes true and m and and then so basically I will I want to write the the the probability so this is the pdf of of of so I want to compute this p of m m okay so that's the pdf that the probability so it's a true probability because m is discrete here okay so m is a discrete random variable so it's the probability uh that the max is reached okay so I will again write the things in this way so to have the maximum in m so basically I need the first to propagate from say zero to m reaching the value mn but staying below this line okay so on this part here you see I mean I stick to I need to just because this guy is a maximum so all the points have to stay below this value mn here okay so I have a first part here and this guy can be anything so you see I mean you can just view this part here which is of course independent of this guy well you can do again what we did before right so the the probability of this path here is the same as so let me reverse the thing so if I just take this this this part here I will first do one thing which is that you can also write it in this way right so I just move the origin here so I have this point here and then I'm doing something like this and then so I'm doing two things okay I'm changing the the direction of the y-axis and also the x-axis okay so that gives something like that okay so you are just reversing this and reversing this one is that fine now this will say mn is the value so let's call it the value of the maximum so that's basically so the probability to arrive here and touch this point for the first time arriving at y is basically the same as starting from zero staying positive and arriving at y we have done this already this kind of manipulation okay now what is nice is that when you do this of course the value of y can be anything okay because I'm looking at the value of of m but the the value itself of the maximum can be anything so I will eventually integrate over all the possible values of y and when you integrate over the values of y well you see that this the probability of this part here is nothing else but the survival probability starting at zero okay so the the weight of this path here this first part is just q of zero m survival probability starting at zero okay so I just did this very simple manipulation reversed the x and reverse time which I have the right to do here because the the the random walk is completely symmetric regarding these two transformations so this is one one piece here one piece so I have q so this is the first part and now I have to investigate the weight of this other path here but the weight of this other path is also quite simple right because you see here on this on this part again so I have a random walk of n minus m time steps okay I have n minus m time steps here I start at this point and I should never recourse it during the n minus m simply because this guy is the maximum okay so there cannot be any other points higher than this one on that side so that means that on that part I start so I can again do this similar transformation as I did exactly the same so I well the same I just need to in this case adjust to basically shift the origin at that point and make an reverse reverse space and that will again be the survival probability starting from zero but now with n minus m time steps is that okay so that means that the the probability of the second part is just q of zero times n minus m do you like it yes it's not starting at one so it's starting at zero right because you you arrive here so you you reach at this point so in m you really need to touch this point and is that fine yes okay why yeah why was this point here okay so you start why is the value of the maximum suppose that m n is equal to y is the maximum and I start at zero so if I look at this part here what I'm doing is that I first I shift the origin so the origin now is in y so that means that this guy would be in minus y if it's zero then it would be minus y then I reversed the y axis yeah okay I see so now what what I said is that in fact you have to integrate over all the possible values of y because you can reach the maximum at some value y but actually y can be anything so you have to integrate over all these possible values is that okay so the maximum of course so here I I just draw a specific configuration where the maximum is y and is reached at time m but eventually to compute what I am after I need to integrate over all the possible values of y right I mean y can be 10 can be 100 can be 0.3 so in other words you would have another configuration right which would be this one for instance oh sorry this one doesn't doesn't doesn't work but say something like that and then here I would need something whatever okay so then I would have another values of y so I need to integrate over all these possible values okay so what is equivalent here is when you really integrate this over y okay so the integral is that okay okay so now we have these results but we actually have much more because we know that this guy is just given by the spawners and theorem where it's fairly explicit not only explicit but completely universal it does not depend on anything so that means that this probability to reach I mean the to compute if you want the distribution of the the maximum the time at which the maximum is which now you can just use power understand so that tells you that this will be just one over two to the power two m two m choose m then I will have so this is n minus m so two to the power n minus m and you will have n minus m n minus m okay you can just rewrite it eventually as it's a beautiful beautiful formula and again which is completely universal does not depend on p of m on the p of eta you can ask what's what's happening in the large and limit and if you look at okay I will not probably should I no I don't I don't I will not do it in details but you can have you can then use stealing formulas to look at the large and limit of this formula and then what you what you find is that okay so if you now look at the limit n goes to infinity m goes to infinity keeping the ratio x m over n fixed okay so that's the most natural limit that you can do when you look at the at the time and the large time limit right so in the time direction m has to scale like n then what you find is that this guy actually goes to some function f of x which is m over n and this function f maybe you have seen it is very very famous famous function okay so this is just this one over pi square root of x one minus x and this is called the arc sine law and this was discovered by levy in the 50s this is called the arc sine law levy's law levy's arc sine law no 50s much much much before that actually so is it is it clear or I was a bit fast at the end yes yeah the same symmetric and continuous so here I'm working with that right so that's true also for levy flights okay so you don't I mean it's universal means yeah well I want to emphasize is the fact that this is valid for sigma finite or sigma infinite okay so if you plot this function it's actually quite interesting but if you plot this function you will see that well of course it's positive and it has these kind of things yeah so it is diverging in 0 and 1 so that means that with high probability you will reach the maximum either at 0 or at 1 and what does it mean well it means that a typical trajectory so it actually means something rather deep for the Brownian motion is that in fact when you look at the so what I mean why are they so that means that you it's much more likely to reach the maximum close to 0 or close to 1 okay and now this has to do with the stiffness if you want of the Brownian motion if you want to see to see it as a polymer is that a typical trajectory in fact is drifting in the sense that of a random arc will be either something like this right so you would just do something like that okay and then that means that the maximum will be reached very close to the origin when you start it or it will drift on the other way okay so that's actually these are actually the typical trajectories when you think about a random arc in fact most of the trajectories in fact look like that okay so that means that here okay i'm just m is typically over the zero and m is typically over the end yeah yeah it's still almost surely but okay these are these are two different kind of four four objects right i mean there is one probably to come back to the origin which indeed will happen at very large time this is this this indeed will happen but typically if you look at a given time and that's actually what what will happen eventually this kind of guy will somehow probably come back to the origin but it will come back i mean it can come back but then if it come back then again it will just drift away yeah and it's it's it's also to do with the fact that this distribution between the zeros actually are have a very power low tail i mean so it's one over t to the power of three by two so these are actually extremely extremely flat i mean extremely fat tails right so typically that will be extremely large okay so this is very nice result i think you see i mean how we can get this this result using the what we have what we have shown before for the for the i mean we could use actually this parandas and formula yeah i didn't write it but of course i used i used this parandas and formula when i used that q or not q0 of n is is simply not is it okay so that more or less closes what i wanted to say about the extreme statistics of random walks and random motions i guess we will come back to this random walks a bit later at the end of the lectures when we will look at the records and i will show you how this in fact more or less the same tools survival probabilities and and also these parandas and formulas are quite useful to study the records but before going to the records i just wanted to show you another example of extreme value statistics for strongly correlated variables and okay so i thought it was it could be a good idea to show you a little bit about what kind of interesting questions one has or there are in the context of random matrix theory so as i said i will not enter too much into the details of rmt i will give you some background just to understand what is going on and i will discuss some applications of essentially the largest eigenvalues of random matrices okay is that fine no questions about about this okay so let's move on to that to that subject okay so let's let's go to uh so that's another again i mean the the the logic the rationale behind the the organization of the lectures here is that we have seen the extremes of id quite in detail then i showed you the extreme value questions for strongly correlated case one example was this random walk and i showed you how essentially this is intimately related to this survival probabilities and first passage problems and now i want to discuss another examples there is basically another example of extremes strongly correlated variables and that will be the case of random matrices and in particular the kind of questions that i have in mind is to show you that okay when you look at these random matrices it turns out that in many cases the the properties of the largest eigenvalues or the largest or the or the eigenvalues of the with the largest modulus are actually quite important random matrices and that means i will discuss largest eigenvalue statistics somehow so i will actually start with an example which i found quite nice which i hope will give you the a bit the motivation as to why what one should the one should look at this okay so let's start with some introduction and motivations so the example actually is taken from a very nice very nice paper from the 70s by robert may that's a nice paper from was published in nature in 1972 it has a huge number of citations several thousands so may was actually an ecologist a quite famous one and he's still alive i think and so he was studying this kind of he had some model he was studying some ecology i mean model related to ecology and what he was looking at the the the following system so he was considering an ecological system where you have say and different species let's at the moment let's suppose that they are without interactions so okay let's let's first announce it this way so you have some end species which which which are living in some given ecological environment and they are described and they are equilibrium whatever it means dynamic equilibrium at some density roi star okay so the the it's a very simple model where you have these end species and he's saying okay let's look at the case where they are just at static i mean dynamic equilibrium which which is such that at equilibrium they have a density roi star and what it means is that and and he wants to to study the case where without interactions okay so we suppose that suppose for for a while that that these species are just non-interacting and is let's assume that in the absence of interactions these species are just in an equilibrium which is stable okay so what does it mean well it means that essentially if you if you perturb your system so if you define say this kind of variable like psi of t which would be the density at time t minus the density at equilibrium then essentially if you write the equation of motion for this for this i well then they will slowly so if you slightly perturb it away from their equilibrium value then they will gently relax to their equilibrium equilibrium state okay so that's that's the idea okay so and i suppose for simplicity that the time scale here over which they all these species relax to their initial value is the same for all of them and it's one okay so i just set it equal to one so it's a very simple and boring problem now what may ask he wanted to ask now suppose that i switch on the interactions between these these species under which condition will this equilibrium this global equilibrium will remain stable so what he proposed to do is just to look at what happens what happens when if you switch on interaction okay so you switch on interaction between these these two these different species under the following case i mean in this form okay that's that's the way he introduced this model so you have a matrix that couples now all these different species okay and this is of course a dynamics that you have for all i between one and n okay so this represents the interactions it is this this is somehow the the simplest interaction that you can imagine between these different species so alpha is basically the the strengths of interactions okay and then you have some matrix that models or characterizes the the the interactions between between these guys okay so now the question that he asked is was the following is is basically what is the probability so he has this question so suppose that i have now the system is quite complex okay that means that n is very large and the true interactions between the different species are extremely diverse and complicated to characterize in details so let's assume that gi j is a random matrix okay let's assume that these gi j are just random random numbers that take into account the values the diversity of the interactions okay so for such a complex system gi j can be is random variable is random take Gaussian random numbers if you want and to simplify a little bit the analysis he assumed also that the interactions are symmetric so xi interacts on xj in the same way as xj interacts on xi one can discuss this kind of approximation i'm not pretending to make a very precise model of of ecology and i'm just trying to show you that how this extreme value questions naturally arises in some simple model okay so you assume that gi j is real because these are interactions and they are just symmetric and then he asked the question is okay if i do that so without interactions you see the system was stable and the question that he asked is if i when i when i tune the interaction what is the probability that the system remains stable that's a quite reasonable question right and a natural that the system remains stable so how do i study this problem so again this was this this question was by me many people after after him actually studied similar type of questions so how do i i analyze this well it's not that complicated i mean i just need to to analyze the the linear stability of the system which is by the way already linear so linear stability analysis tells me that i need to know something about the eigenvalues of this guy okay so i will diagonalize this this matrix okay let's that's really a bit more precise here so how do how do i analyze this so i have a linear stability criterion and that tells me essentially that i can rewrite if you want this so x now is a vector which encodes this x1 x2 xn so i just use a matrix notation if you want and i get that this is alpha j minus the identity on n times x not fine so i just rewrote this linear system like this and now let me introduce the eigenvalues of this matrix j okay so i will introduce as lambda 1 lambda 2 lambda n the eigenvalue of of j okay so j is a symmetric matrix real symmetric matrix so all these lambda i's are real okay yeah yeah no it's yeah yeah you will have to remember that xi is actually the distance with respect to the to the to the rho i star okay so it's actually it's yeah should remember that xi is the distance yeah it's rho i minus rho i star okay so it's rho i of t okay no that's fine so that means that in the absence of interactions again everyone will gently relax to their uh if i slightly perturb the system then the system will gently relax to its equilibrium equilibrium states okay so here you see i mean they are all real okay i chose i mean they are like this because again j is a real symmetric matrix okay so its eigenvalues are are all real so now what does this the stability criterion reads i mean it's pretty simple right so the system is stable if and only if the eigenvalues of this matrix are all negative right that means if alpha of lambda i minus 1 is strictly negative and this has to be true for all i and this is just such that lambda i is smaller than 1 over alpha for all i so that means that in other words i want that if i look at the largest eigenvalues of this of this set i need to have so if i define lambda max so here comes my extreme observable lambda i now this condition is just equivalent to say that lambda max is smaller than 1 over alpha so now i have a partial answer to this question but this probability i could recast it on a question related to the eigenvalues of some random matrix which is that this probability is actually so let me call this probability p stable of course it will depend on alpha when alpha equals zero this probability is just one and p stable is just the probability that lambda max is smaller than 1 over alpha so you see that this question here this question of stability boils down to say something about the statistics of the largest eigenvalue of this random matrices okay may didn't know so much about about random matrices but he was a very clever guy and so you see i mean this is already the indication that random matrix theory can help to understand this question now what it may observe i mean he was doing some numerics in fact yes okay so you agree with that this is this question that that you're asking okay so okay sorry dx okay so if i suppose that i have the eigenvalues of j so i will essentially diagonalize this so i will write this this equation essentially in the eigen basis of this random matrix okay and that means that now for each of the eigenvectors somehow i will have an equation which essentially tells me that if alpha lambda i minus one is positive then so you see i mean the solution to this to this to this linear system is just an exponential okay so i need that all the the exponential are decaying because if i have a positive some positive eigenvalue here then that means that there will be some eigen mode that actually diverges exponentially is that okay yeah g is symmetric yeah so that's why i said yeah yeah of course here j is symmetric so indeed i mean i can diagonalize it and i know that all the eigenvalues will be real so there would not be any oscillating parts so they are all real and that means that the system is stable if and only if all the the all the the eigen modes will actually decay exponentially okay if there is one mode that does not satisfy that then that means that i will have one of the eigen mode that actually diverge diverges exponentially and of course the system will not be stable is that clear okay so so what what what may observe is that so he was just plotting i mean he was just doing some numerics and one may observe this something which is quite nice so and there are some assumptions okay that i will so what what it may really do may it shows this j ij not in a in a in a random way so if you want to have a nice large end limit you need that this so it shows the j the j ij so we will see this a little bit later as gaussian random variables and that's the simplest model that you can think of which were centered so j ij zero and then if you want to have a well defined large end limit we will see that later on but you need to have the variance you need to choose the variance as to be of order one over n it's clear that if here i mean you see easily i mean there is there is a number the very large number of term here so if you want to have you want to have this over the one okay and the central limit theorem tells you that basically this will be over the square root of n times the variance of that so square root of n times one over square root of n so that means that you choose this j square over the one over n you want to have this simply over the one otherwise of course it doesn't make sense yeah that that's that's another way that that that would be another way to see it so i could actually put a one over square root of n here such that okay one over square root of n and then g ij would be would be okay and then by central limit theorem one over square root of n times the sum of iid random variables is indeed finite that's that's the idea so that's the natural limit that you have that that you have to take and what you observe is that if you basically if you you plot this distribution as a function of alpha i'll say as a function of one over alpha which i will call w in the following so what he said is that well you see i mean again if alpha is very small that means one over alpha very large you expect that the system will be with a high probability will be stable okay because that means that if alpha is stable if alpha zero strictly then we know that the system is stable okay so that means that this probability of course is bounded by one so if you are very large then you would expect that it is simply one and on the other hand if alpha is very big well you would expect or you could expect that if the interactions are extremely strong and given that they are completely random then it's with high probability they will completely destroy the stability of the system so when alpha is very large that means one over alpha goes to zero you expect this probability to be essentially zero then what he was observing actually in his simulations is that there is actually a quite sharp transition he said he was looking at he was saying that there is a critical value wc such that in the large n limit you really observe something like this there is a true transition in this model now of course so this is the large n limit when n is finite of course you would expect some finite size effects and these finite size effects will smooth smoothen out a little bit this this transition that would give instead something like that okay so that will be for finite and say 10 okay so you have your transition this wc we will see i mean we will be able to compute it later on wc uh it's not it should be actually one over square root of two you will see why in a minute i mean maybe in a minute so so you have two region okay so you have a region so what what is quite quite remarkable you see is that there is really a transition such that if alpha is say larger than wc then your system is unstable with probability one or unstable or stable with probability zero so you have a phase which is unstable and here you have another phase which is actually a stable phase and this transition actually has been observed in many other many other systems this transition is called sometimes called the vignard may transition so may for obvious reason because he introduced the model he saw it now vignard because this turns out to be vignard was one of the pioneering researcher in the field of random matrices and you will see that this this transition is really related to uh to to uh to random matrix theory and in particular to the vignard semester okay this i don't know i mean uh i don't know i mean you can at least do it in numerics i mean it's easy to do now in a real system i don't know i mean i guess there are many many different systems to encourage this type of this type of interactions i don't know i let you imagine what kind of setup okay there are depending on the system that you have i mean you can do it sometimes i mean by increasing temperature or decreasing temperature or i don't know you can i mean if you have different species for instance uh people i mean people i don't know there are for instance in some cases they force the the the i don't know the species they have to to look for food for instance and that increases the the interactions between them because they want to cooperate there are many different ways to do that but again i mean keep in mind that this is a toy model right i'm not pretending to describe any concrete experiments i really want to show you that these questions actually arise in simple models where uh you can try to say something quantitative okay yes why the graph is like that no that's normal actually i mean it is you mean well i mean what do you mean counter-intuitive i mean what do you mean exactly yes yeah so no i think it goes in the no this is intuitive i suppose yes yes okay so one of one of the alpha increases okay so this probability is one the probability that lambda max is less that infinity is one and so that's what i show here okay one of the alpha is that right i think this is intuitive okay what is not intuitive is that there is a threshold but apart from that the two limits is that clear okay fine okay so there are many questions actually that that one can ask and we'll try we will try to answer some of the questions that that that are there i mean one question which i think at least for statistical physicist is is is interesting is that is there any okay it's a dynamical system here and i'm looking at some probability associated to that one actual question is to to know whether there is a thermodynamic interpretation of this of this phase transition does it have a thermodynamic interpretation and if it's so what is the order of this transition so that's one question that we will answer here and it turns out that this transition is quite peculiar in nature we know many systems that exhibits one i mean first order or second order of phase transition it turns out that this transition is a third order phase transition and so we will see how it how it shows up and how it can really be studied in detail using the RMT techniques techniques of random matrix theory again i will not enter into the details but i will you will really see that the analysis of this of this transition is deeply related to to extreme value questions that that i will now try to to study and in particular so now at some point now i just want to give you a little bit of background on on RMT at least the minimal things that you would okay that you're supposed to know to really address this question so let me just define now let me be a little bit more specific and and just go to some backgrounds on RMT right so very basic so the basic model that we are looking at will be the model that may be studying right so we have RMT i will usually use this acronym so the basic model that that that i want to consider here is really the well i can actually go to this to this same notation as as this guy so i have g so which is a matrix and it is typically of that form right so i have a matrix j let's just have it like this so it will j 1 1 j 1 2 j 1 n it has to be symmetric so it's here is just j 1 2 again and here i would get j 1 n the same as this one and here i would get this j of n n okay so this is my matrix it has to be symmetric so it's already there so g i j is equal to g of j i very nice and and they are real i'm constructing i'm constructing sorry the the sort of probability i need to give some probability measure on this on this matrix ensembles so i will do it in a rather simple way and i will do it in the such a way that i choose all these random numbers as being gaussians and independently one from the others okay so that means that the p of j will be up to some pre-factor will be simply a product of gaussians okay so i will write it in this way so it will be a product over i and j of exponential minus j i j i j square and actually i need to have the variance to be to be like like like that i said so i would choose it to be one minus n over two j i j square okay so this n here is just because of that on the variance to be about a one over n otherwise it's a it's a global rescaling i mean which is an important but i prefer to have it this way now you see that this guy you can still write it in a different way i mean the product of gauss so this is a natural measure that you that that you are right i mean you just take gaussian random numbers and you put them in your matrix and then you will look at the the eigenvalues of it now let's just rewrite this a little bit this product of exponential i can just write it as the exponential of the sum so that will be exponential of minus n over two sum of ij j i j square okay you like it so now let me manipulate it a little bit i want to rewrite this in a nicer way so now i want to use the fact that j i j is equal to j g i okay so that means that the p of j that i have i can again write it like this so it's exponential of minus n over two and i would have i can write it this way right i have g ij jgi okay i didn't do anything i just use the fact that the matrix is symmetric and now this you see you can just write it like this so it's sum over i all the sums are going from one to one i just try don't write them explicitly because i'm too lazy but yeah exactly so this is just j square ii and this is just the trace of j square so we like that why so that means that this is just one of our z n exponential of minus n over two trace of j square so we like this because now it has a nice invariant invariance property and i didn't have time to really make a full to give you a full historical background of rmt but rmt was initially introduced in the context of nuclear physics yeah okay so here you see i mean i have a product of two i can see as the product of two matrix a ij when you have this use sum over j this is by definition this is ab ik and you apply this formula to a equal b equal j and i equal to k yes so the reason why i want to have you will see that if you okay there are two reasons the first one is that you will see in a minute or maybe tomorrow that if you look if you if you take this the measure like that then typically if you i will be interested in the eigenvalues of this matrix and what happens when you choose this this this this scaling here that means if i choose the variance to be of order one over n it turns out that the typical value of the of the eigenvalues are order one in the limit when n goes to infinity okay so if you choose your things like that if you take n very large ten thousand if you look at the eigenvalues then all the eigenvalues typically will be of order one and they will converge to something finite when n goes to infinity if i don't do that so if i i guess you would prefer to take something of order one okay then if you do that what will happen is that your eigenvalues are over the square root of n so it's just that it's just a simple rescaling and that's somehow a little bit more comfortable to work with quantities which eventually are over the one now in maize model in maize model that that the reason is that again you want to have an interaction form which is also typically over the one when n goes to infinity otherwise the strengths of the interaction would be too much too much too strong and the interactions will would always always win in that case okay so it's just maybe you will see that yeah probably today i will not have time to show that but the eigenvalues are over the one here that yeah yeah yeah that's true yeah yeah yeah yeah that that's true i could i could just so absolutely yeah i could just what you are saying is that in other words here i just want to it's it's sort of more convenient to write it in this way because like this like this i can see this this i mean nicely i can see this this trace of j square but you are right i mean somehow i would not need to sum over it over it yeah okay so that that means that in that case if you do that then you you have you the point is that in that case if you do what you want to do then you need to take random variables which are not exactly the same so the the diagonal values will have a variance one over n and the and the non-diagonal ones will have a variance which is double or half okay so this is just a matter of choice i mean but but that's true i want i want to have something which i eventually read like this depending on how you define here your random numbers then you will need to show to choose them either completely identical or you will have to single i mean to to differentiate between the diagonal and the non-diagonal terms yeah yeah you're right yeah okay it's it's a bit right okay consider it as a bit technical that's why i didn't enter into this point but but you are definitely right okay so why is it interesting i mean because as i said i mean people in the in the early days of random matrix theory people were modeling heavy nuclei so in nuclear physics they were modeling the Hamiltonian of this of these nuclei by these random matrix theories saying that basically instead of considering a complicated Schrodinger Hamiltonian why not consider just a simple random matrix right i mean because the interactions between all the components of a heavy nuclei is so complicated so let's try to assume that this is just a random and then at that time there are of course driven by some a symmetry reason and they really easily i mean quickly realize that you cannot choose any types of random matrix theory they need to be invariant under some kind of transformation and then here this is one of this model which is invariant under rotation indeed if you if you define a new matrix j prime which is obtained from j by simple change of basis so you just do this this conjugation where all is orthogonal matrix okay then it's easy to see that you can compute easily the trace of j prime square okay so trace of j prime square is just trace of so let's do it so it's all g odiger all g odiger now odiger always just identity so this is just the trace of all j square odiger and then the trace is cyclic so i can just put this odiger here and odiger always you then his identity so this is obviously equal to trace of j square right so that means that if you take two matrices which are equivalent up to this autogonal rotation if you want this autogonal transformation then basically they have the same weight okay so this defines some given weights to each class of equivalence if you want in this in this in this thing okay so so that means that these two guys eventually what it means is that this is p of j prime is equal to p of j okay and that's actually quite important i mean that was quite important in the early days of rnt and this is one of the most the the ensemble that i am describing here is the most well known is certainly one of the well yeah most emblematic ensemble and this is called under the name of the garshan garshan you have understood why because jr garshan autogonal now i think you have understand also because the probability measure is invariant and the orthogonal transformations ensemble okay so this is go exactly yeah exactly then i will just end up with this with this with the following thing is that now because of this uh invariance property it turns out that if you look at this random random matrix so one way to characterize that this random matrix is to give it or give you the or describe the eigen its eigen values on the one hand and its eigen vectors on the other hand now because of this because of this property here it turns out that this for this or such Gaussian random matrices the eigen values and the eigen vectors are completely independent and this is true for any n i mean this is not an asymptotic result this is true for any value of n and so that makes sense then to study only the the eigen values of such random matrices and so if i define lambda one lambda n the n eigen values so these are random variables right because you see that if you take one matrix of this ensemble and then another one and if you look at the eigen values obviously there will be different okay now the remarkable property is that for such models it is actually possible to write explicitly the joint law of these random eigen values okay and that was that was done essentially by by vigno himself and it would like that so there is a some pre-factor here there is a first term which is a bit trivial somehow which really comes from the fact that these are Gaussian simple simple Gaussian random variables so this lambda i square really comes from this trace of j square and now there is a non-trivial part which comes from the fact somehow from the the integration over the the unit the orthogonal group so you when you perform the integral over the the the matrix also the eigen vectors and there is a non-trivial term that comes out of that so that's actually quite nice because again this is just now the equivalent of I mean this is simple this would be simple independent Gaussians so that would be a simple case of iid as we have discussed in the earlier lectures but now you have a non-trivial interaction term okay so this can be as a Jacobian this can be viewed as a Jacobian this is a Vandermonde determinant and this is really the interaction so that tells you that this eigen values are actually extremely strongly correlated there are various ways to understand that but the most quantitative ways to do that now yeah yeah yeah I know it's it's really I mean it's I would need half an hour to derive it yeah the trace will be that but then of course you have to you have a measure in front you have a Jacobian right you have a Jacobian that initially you are essentially here well basically you can just see it as a Jacobian right this is mathematically this this comes this comes out as a Jacobian right lambda i are the eigen values of j of j yes okay yeah so just to answer your question again I mean you indeed initially this p of j you see I mean so you have to to essentially to hear that the measure is in is in as a function of the the matrix elements and you want to reparameterize this measure in terms of the eigen values and eigen vectors so there will be a Jacobian between within this transformation and that's what is non trivial and and this produces this this this term okay now it turns out that there are many other interesting models that people have studied which are sort of a variant of these guys so maybe I will just just mention it here briefly before we before I stop you can actually introduce here parameter beta and beta is strictly positive and you have a nice a nice collections of models and we have seen beta equal one so beta equal one is the GOE yes exactly yeah exactly so now I come to the description I just described up to now the the beta equal one ensemble and there is another model which is corresponds to beta equal two and which is known under the name of gaussian unitary ensemble so this is a model which you you see here we have dealt with matrices which were real symmetric so you could also work with Hermitian matrices Hermitian matrices would also have real eigen values so this is this ensemble the GUE gaussian unitary ensemble because they would be the measure is invariant under this kind of transformation but where oh actually becomes belongs to the unitary group so it's uu dagger and there is another group okay which probably is always a bit frightening and I will not mention too much this is the beta equal four and which is called the gaussian symplectic ensemble so now here we have real symmetric here we have Hermitian so we put gaussian numbers but now you could also put quaternions if you like it if you really like it and and then you will end up in this fourth model beta equal four number recently people realized that there were also construction very nice construction for any beta this was done by Edelman and and Dumitriou in the 2000 where you can actually have some three diagonal ensembles for any beta I will not comment too much on that but so later on and in the next lecture I will show you how one can understand this probability measure in terms of statistical mechanics of interacting particles and namely I will show you that there is a nice Coulomb Coulomb type of model behind this this guy here and then I will derive some basic basic stuff and essentially then arriving at some properties of the largest eigenvalues which we are after okay okay thank you very much