Branching random walks and stretched exponential tails
Dyszewski, Piotr
Gantert, Nina
We will consider a branching process with a spacial component on the real line. After birth each individual performs an independent step according a stretched exponential (or Weibull) law. We will give a detailed description of the asymptotic behaviour of the position of the rightmost particle.
 Okay. Okay, and we are live. So, and also start the, you started the recording. Okay. So, I want to say today, welcome everyone to the fourth session of the week. So today we will have two lectures on both of being recorded. So if you don't wish your audio and video to appear in the recording, please keep your microphone. I'm going to switch stuff. And let me just also mentioned that next week we have rather full week of material with mini course from Ivan Corvin Monday, Tuesday, Wednesday, followed by several short lectures on related topics on Thursday and Friday. So there will be an email to the mailing list about that this afternoon. So with that, I'm going to hand it over to Nina Gantert to introduce the speakers over to you, Nina. Yes, thanks, Luigi. So today we will have two talks and the first speaker is Piotr the chef's kid. The second is Sam Johnston, you can see them both here. And Piotr, who is a postdoc in Munich right now and also in Wroclaw will start and his topic is, his topic is a branching random walk and stretched exponential tails. And the screen sharing works for me and I hope for you as well. Okay, Piotr. You can start. All right. Thank you, Nina. Yeah, so for the most of the Nina's lecture, she focused on on a branching random walk when where the displacements have fin tails. And what I want to do today is to show you how you can work with branching random walks whenever your displacements are heavy tails, have heavy tails. And what I mean by that is whenever the, you know, the exponential moments exist or not. So first off, let's just recall the setting. So I will, we will work on a real line. And we start with one particle at the origin. And after some time, this particle dies and gives birth to a random number of new particles. And we're going to assume that the reproduction law is the reproduction of the particles. It's governed by a supercritical Garten Watson process. So this in particular means that the mean of the reproduction law is is greater than one. All right. And we also want to assume that the displacements of the particles because after the particles are born, they are moved somewhere on the on the real line. They alter shifts are an IID copy of one given given random variable with so-called stretch exponential law. So this means that the tail of X decays as e to the minus t to the power r. And this and this twiddle means that the quotient of, you know, the both sides tends to one as t tends to infinity. So instead of, you know, giving you a full scope of what happens with heavy tail displacement, I'm just going to focus on the stretch exponential case and give you the whole treatment. But most of the phenomena's do replicate in with different tails. All right. So, you know, the death particles are removed from the system, right? And then they reproduce as previously. So they produce some random number of other particles which are then placed on the real line. And, you know, this system evolves for some time, according to this rule. And, you know, after some time, you do get some you do get a collection of particles on the on the real line. And the aim is to somehow understand how this this collection of particles evolves, right? And the first question you want to ask is the behavior of the of the extremes. So in our case, this is going to be the position of the rightmost particle. And this, you know, this case study of stretch exponential displacements basically boils down to explaining the math behind the result that I did obtain with Nina Gantt and Thomas Hufflesauer. All right. So Nina does Nina needs no introduction and Thomas is her former PhD student. And if my memory serves me right, he did graduate two years ago and now he works in the private sector. All right. So under some technical technical assumptions that I do want to do omit because I don't want to make my slides too complicated. We have the following. So assuming that the law of this generic random variable is, you know, such exponential and that this guy is centered and has variance one. And then it turns out that you have two different regimes of the behavior of your rightmost particle. So depending on whenever r is smaller or greater than the critical value of two thirds, your rightmost position either either fluctuate or you have almost true behavior. So in both cases, the leading term is the same. This is alpha times n to the power one over r. Right. But for smaller values of r, you know, the rightmost particle fluctuates in the neighborhood of this value with a limit being the shift of a Gumbel law. So here this w, this random shift is a martingale limit associated to the underlying Galton-Wolson process. And we will see exactly how this, how this guy pops up. And whenever r is, you know, big, you have more balanced behavior, which translates to almost a limit. All right. So let me just introduce the, you know, the setting that Mina did, did, did, did give in order to be self-contained. Let zn be the underlying Galton-Wolson process. All right. So maybe I, maybe I want to go back and say two things. So first of all, my aim here is to, you know, somehow explain where this exponent are coming from and where this different type of behavior is coming from. And most importantly, I do want to give you some high, some idea of how can you use the heaviness of your tails to your advantage when you study the branching random walks. All right. So as I said, let zn be the underlying Galton-Wolson process and assume just for the simplicity that this process never dies out. So assume that the probability that your descendant has no children is zero. And if we don't have this assumption, then everything that I'm going to tell you will still work. However, you will just have to condition on the set of survival of your population. All right. So then you can draw your Galton-Wolson tree, right? So then you assign labels to each of the individuals, so on the individuals. So let's say that, you know, the first particle has a label with, which is an empty word. And then particles from generation one has a label, which is a string of length one. Then particles from generation two has a label, which is a string of length two and so on and so forth. So once you have those, once you have your Galton-Wolson tree, then you take the displacements. So you take those, a family of IED random variables and assign them to the edges of your Galton-Wolson tree. So you think of, you know, of your labels, of your axis as the displacement of a given particle from the place of birth. All right. And then, you know, if we are interested in a position of a particle from generation n, then what you need to do is, you know, you just look at the label on your tree, look at the path from this particle up to the root, right? And you just sum all the displacements along this path, right? And then you get this quantity, which I'll denote by Vx, which represents the position of the particle with label X. And now, you know, your rightmost particle is just the maximum of those guys. All right. So one technical thing that we're going to need is the large deviations for random walks. So you've seen that in Nina's course, and she did use large deviations in the case when the displacements have fintails, when you have some exponential moments. And this is basically a common knowledge, right? In the case which I'm looking at here, that is the stretch exponential case. This is not so well known. So let's just look at what's happening. So the lack of these exponential moments morally, this translates to the so-called principle of one big jump. Namely, you know, if you look at a statistic, a reasonable statistic of your ensemble, then, you know, the probability that this statistic is big in some sense is asymptotically equivalent to saying that one of the guys in your ensemble is big. So for example, if you look at the probability that the sum of the two guys is big, this is asymptotically equivalent to saying that one of those guys is big. Now, we want to somehow try to push this phenomena and see if we can get some large deviations result out of that. All right. So let's take an SN to be the sum of those X-case, right, and take the maximum and denoted by XN star. And now we want to understand, you know, what the probability that this SN is greater than some threshold TN. And, you know, as it will turn out in the future, we don't need the, you know, the full scope of TN. But it will be sufficient for our needs to take T of the form C times N to the power one over R, right? So we want to somehow exploit this principle of one big jump. And naively, what you can do is say, okay, so the biggest displacement does something, right? And then the other displacement somehow contributes, right? So you can naively try to write this supremum and just see what happens. Yeah, so we know what is the behavior of this maximum, right? This is exactly this part. And then you check what's the behavior of the rest of those guys. And since, you know, you think of those guys as the typical displacement, right? You expect a Gaussian behavior from that. All right, so you have some expression in your exponent and you just simply optimize this guy, right? So you get this, it's exactly that. All right. Yeah, so now when you look at this probability of typical particles, what you can do is since you expect this SN to be close to Gaussian, you can write, you know, SN minus one divided by square root of N, and this is supposed to be greater than, you know, this constant that we somehow get times N to the power 3 over 2 minus 1 over R. So you can actually see from that that the value of R is being equal to 2 thirds. It's somehow critical for the behavior of the deviations. And this is exactly, you know, this critical parameter that we have in our result. All right, so you write down what's the behavior of the product, right? And, you know, this leading term from the maximum translates to the leading term in the exponent. And then those smaller guys somehow transfer to something smaller. The key fact here is that the leading term is the asymptotic of the deviations of SN really comes from the maximum. And then depending on the value of your parameter, then you can get something different or not. All right. So what can we do with that? Well, we can redo the first standard term of the asymptotic of the rightmost particle. Again, something that Nina did in the previous course, but still this will help me drive my point. So let's just do that. So the probability that MN is greater than, you know, C times N to 1 over R. And, you know, 1 over R is the correct scaling because, you know, I said so. So you look at the probability that one of the guys is greater than this threshold. And you can use the first moment method to, you know, bound this basically by expectation of the number of guys that are greater than this threshold. Right. And what you get is the expectation of ZN, which is roughly N to the N times your deviation. And this is something that we did compute just now. So what we get is, you know, asymptotically N to the N times E to minus C to the power R times N. So now we need to tailor this answer that this term compensates this N to the N. Right. And this leads to basically a prediction of the first other term being N to the power 1 over R times times alpha. And this alpha is exactly given here. All right. So what we can learn from that? Well, everything, you know, this first term really comes from this parameter, right? And we just said that, you know, this leading term, this asymptotic of the deviation comes from the biggest displacement. So we can try to argue that the first term, the asymptotic expansion of the rightmost particle is related to the biggest displacement in your collection. And this biggest displacement, right? This is just a maximum of IID guys. And the only twist is that this, the number of guys that you are considering is random. But since those two quantities are independent, this is still easy to handle. So you know that after a proper centering and scaling, you will see a nice limit, which is, again, a random shift of the Gumbel distribution. And this is something that's so easy that we can actually try to try to prove that. So, for instance, take some threshold XN that goes to infinity. And let's just try to compute the probability that this biggest displacement is below this threshold. Right. So what you do, well, you just condition on your tree. Right. So this leads you to writing that basically you are saying that you have a random number of guys that are supposed to be smaller than this threshold XN. And this random number of guys is just the number of the edges that you have up to time N. Right. And this is easily expressed in terms of your branching process. Right. And then, you know, looking at the expression under the expectation, it behaves as e to the minus the product of, you know, this YN and the tail of X. All right. So now, this, now the random shift comes into play, because we know that this YN behaves pretty nicely. That is, after scaling by N to the N, you get a martingale. And since this guy is non-negative, it will have almost a whole limit. Moreover, since I'm implicitly assuming that the Kerstens-Tigerm condition, this limit will be non-negative. So this Kerstens-Tigerm condition is one of those ugly assumptions that I just omitted in the second slide. All right. But now, you know, we know what the behavior of this YN, right? This is, again, something that grows in the same fashion with a random shift. All right. So now, we can take a very concrete threshold here and just start to compute things. So first off, we do only the simple bits. I mean, first off, we just take the asymptotic equivalent of YN. This is here. And we just plug in the definition of XN. So nothing happens. Then, we look at the exponent on the right-hand side, and we simply use the Taylor expansion. And when we do that, we notice that, you know, this exponent here is Taylor, such that it will cancel with a derivative that pops up from the expansion. Right? So you're left with a constant. All right. But now, we just need to recall what is the role of alpha. Alpha is taken such that this guy will cancel with n to the n. Right? So we are in business. We are in business, because if we call our constants, you know, constants accordingly, then we get a nice limit. And if we go back to what we have done, this basically shows that, you know, this biggest displacement after proper centering and scaling does converge to this nice random shift of the Goomba law. So as you can see, the analysis of the biggest displacement is pretty straightforward. So what can we do with that? Well, when we look at the first order asymptotics of the biggest displacement and the rightmost particle, they are the same. So it's not difficult to convince yourself that if you look at the positions of the particles, which didn't have a big jump along the branch, those guys are negligible somehow, right? And if you have that, then you know that you need only to concentrate on the particles that had a big jump, right? And how many are there? So all the particles up to time n, which we see in our system, the number of those is roughly m to the n. And then you look at, you know, those particles that did encounter a big jump, and those will be fewer, right? And it turns out that whenever r is small, you only see a polynomial number of these big jumps. And whenever r is big, you see something which is almost exponential, but not exactly exponential. And I do know that this line is somehow vague, because I don't tell you what do I mean by, you know, x having a big jump. So the takeaway message from this line is that for small values of r, we have small number of big jumps that contribute. And for big values of r, we have a lot of big jumps that can contribute. All right. And in the end, you know, those particles that did give a big jump form a small subset of the whole tree. So what can we do with that? If we look at the whole tree, the first step that you want to do is look at those particles that, you know, with the displacement comparable, you know, to the first order asymptotic. And color the edges green. Then, you know, you color the edges which are below in red. And then you look at the path that goes, you know, from the green edge to the root. And you will have an orange segment and a blue segment. And the way this orange color and the blue color works is that when you take into account other big jumps, what you will see is that, you know, you have a segment which is orange and which is disjoint from different paths. And the moment they start to meet each other, they will be painted blue. So the thing is that those orange guys are independent. Yes. Yes. Were you trying to have a question in there? Oops. Let me, there's a question in the chat. No. Yes. What's the question? I don't see a question. I think you can. Maybe it was sent privately, but. I don't see a question in the chat. Okay. I'm, I'm all right. So I just move on. Where was I? Yeah. So the orange guys are basically independent and those blue guys are the bits when you see some kind of dependence. And yeah. So the set of those admissible particles, I mean, those that you see at the very end, right, each of those positions of those guys actually can be the composed in the following fashion. So if you have this, when you, you know, you go along here, you see a blue path, you know, a path in the orange color, then is this one green big jump and then there's the red. All right. So what's the contribution of each color? If you really believe that those big jumps are a sparse subset of your London tree, this means that if you take two green edges, they will be far away from each other on the, on the tree. And this means that, you know, those, those paths will meet very high up. Hence those this blue bit, which will be relatively short. In conclusion, you know, this, this red part is, is negligible. So now let's focus on the, on the red part. And this is something that requires some computation, but since the maximum displacement grows faster than linear, right, this go grows like. And to the power one over R. When you, when you look where those displacement happened, I mean, in which generation, it's more likely that they will happen in a generation which is close to N. So this in terms tells you that those, you know, those red segments, they also be relatively short. So this, this red guy can also be neglected. All right, but this, this tells us that, you know, the, the positions of this admissible particles are asymptotically are just the, you know, the, the orange parts and those, those green jumps, right, and those guys are independent. So we are, we are quite happy. So now we need to compare those, those two. Right. So asymptotically, your right most particle is consisting of, of this big, big jumps, this an X, and the position of, you know, the typical particles. And you know that, you know, this big jump, well, it is big. And those typical particles. This, this is the behavior should be asymptotically normal. So you can do the same optimization game that we do the previously. And, and you would come to a conclusion that the contribution of those typical particles, it, it is also of the order and to the two minus one over R. So this is the exact same exponent that we did get from this optimization. On the other hand, we know what, what is the behavior of the biggest particle, right. So this is, you know, we had the leading term and we know what are the fluctuations. So now we just need to, to compare those two. Right. So if R is small, then, you know, whatever the contribution of the typical particle is particles is it's killed by the fluctuations of the, of the biggest particle. And, you know, this right most particle, it scales the same as the, as the biggest particle. And you get this nice, nice limit. Moreover, this one comment that I would like to make at this point, that is, we can actually show that the difference between the right most particle and the biggest displacement is negligible in compared to the. The order of the fluctuations. Right. So this M and this is something that consists of, you know, the maximum of your guys and this M and this is something that has, you know, the sums. So this, this line somehow really screams that one particle did all the, did all the job. What happens when R is big? Well, you need to go back to our picture, right. And what you do, you first look at the green displacements at the green edges, right. And this fixes the first order, right. And then you know that the contribution of the orange parts should somehow kill the fluctuations of those, those green guys. So what you can do is start looking at those orange paths and see what whatever is the is the biggest. And as I mentioned, you know, in this case, there are actually a lot of those, those orange, those green jumps. Hence, there will be a lot of those orange paths. So we have a very big number of those orange paths with which in turn gives you a more balanced behavior. And now you can basically play again the same optimization game and say that, all right, so this is the contribution of those, those orange guys is in fact of the other. To enter the two minus one over R, right. And if you do this, this optimization, right, you can squeeze out the, the limiting constant. All right. So what happens in the boundary case? Well, you notice that, you know, both of those exponents are equal one half, right. So now if the contribution of the typical, typical particles should be is of this other than you will conclude that they will actually fluctuate. All right. So what, but still, you can work around that right, you can, you know, take a family of independent gaussians. And if you look at the, the right most particle after centering and scaling, you will see, again, a maximum of the, you know, those admissible particles, right. So you see the proper centering of the, of the right most position and then those typical displacement would be, you know, asymptotically gaussian. But remember that, that we did the couple, the whole thing, right. So we can still basically work as a, as if we would, we would be working with a family of, of independent guys, right. So still, you know, centering this, this right most particle is the same as, you know, pardon the cumulative distribution function of the right most particle is the same as, you know, those big displacements with some random gaussian shifts. But still, this is something which is, which is manageable. So you do the exact same computations that we did for the, for the maximum, right. And this is that, you know, this gaussian shift transfers to this integral. And here, you know, this, this big displacement transfers to this, to this e to the minus e, minus e, pardon. And then, well, the gaussian and the variable transfers to this, to this gaussian term, right. But still, if you look at this integral as a function of, of t, this is still a constant times e to the minus minus t. So nothing, nothing really changes. I mean, those fluctuations, they do contribute, but the, the, the contribution can be hidden in a, in a constant. All right. So this, this is that, I mean, in, in the case when we're are is very small, right, the whole, the behavior of the, of the whole system is really governed by one big jump, right. And in this boundary case, this, this one big jump is supplemented by this, this gaussian, gaussian shift. However, this, this contribution can be, can be hidden in this, in this constant gamma. So in particular, this, this gamma, it's, it is not a, this is not a continuous function of, of our parameters, right. And in, if R is big, then, you know, we have a lot of the typical particles that somehow contribute, which gives you a more balanced, balanced behavior. Yeah. So remember that the, the way that we started our analysis is first by, by looking at the, at the biggest displacement. On one hand, it did give us, you know, all the, all the information we needed, or it allowed us to decouple the whole thing. And it, this, this was the, you know, the key step which we needed in our analysis. All right. So this was based on a joint paper with, with Fina and Thomas and it's available on archive. And let me just mention that, you know, this is first order asymptotic was established by, by Nina in the, in the 2000s. And if, and then large deviations for, you know, for the, for the script stretched exponential random variables, go back to the Nagaev in the late 60s. However, it's, it's not an easy thing to, to read. All right. So I'm out of slides. So I guess this is a good place to stop. Thank you. Thank you very much, Piotr. So maybe you can unmute people. And so other questions to Piotr as long as the screen is still shared. Some questions were already answered in the chat. It seems that there are no questions. You are muted currently. Yes. So we will stop. There are no questions right now. So then I think, are you stopped the recording for. Stop now the recording and start it again for some stock.
 Okay.  Okay, and we are live.  So, and also start the, you started the recording.  Okay.  So,  I want to say today, welcome everyone to the fourth session of the week.  So today we will have two lectures on both of being recorded. So if you don't wish your audio and video to appear in the recording, please keep your microphone.  I'm going to switch stuff.  And let me just also mentioned that next week we have rather full week of material with mini course from Ivan Corvin Monday, Tuesday, Wednesday, followed by several short lectures on related topics on Thursday and Friday.  So there will be an email to the mailing list about that this afternoon.  So with that, I'm going to hand it over to Nina Gantert to introduce the speakers over to you, Nina.  Yes, thanks, Luigi. So today we will have two talks and the first speaker is Piotr the chef's kid. The second is Sam Johnston, you can see them both here.  And Piotr, who is a postdoc in Munich right now and also in Wroclaw will start and his topic is, his topic is a branching random walk and stretched exponential tails.  And the screen sharing works for me and I hope for you as well.  Okay, Piotr.  You can start. All right. Thank you, Nina.  Yeah, so for the most of the Nina's lecture, she focused on on a branching random walk when where the displacements have fin tails.  And what I want to do today is to show you how you can work with branching random walks whenever your displacements are heavy tails, have heavy tails.  And what I mean by that is whenever the, you know, the exponential moments exist or not.  So first off, let's just recall the setting. So I will, we will work on a real line.  And we start with one particle at the origin.  And after some time, this particle dies and gives birth to a random number of new particles.  And we're going to assume that the reproduction law is the reproduction of the particles. It's governed by a supercritical Garten Watson process.  So this in particular means that the mean of the reproduction law is is greater than one.  All right. And we also want to assume that the displacements of the particles because after the particles are born, they are moved somewhere on the on the real line.  They alter shifts are an IID copy of one given given random variable with so-called stretch exponential law.  So this means that the tail of X decays as e to the minus t to the power r.  And this and this twiddle means that the quotient of, you know, the both sides tends to one as t tends to infinity.  So instead of, you know, giving you a full scope of what happens with heavy tail displacement, I'm just going to focus on the stretch exponential case and give you the whole treatment.  But most of the phenomena's do replicate in with different tails.  All right. So, you know, the death particles are removed from the system, right?  And then they reproduce as previously. So they produce some random number of other particles which are then placed on the real line.  And, you know, this system evolves for some time, according to this rule.  And, you know, after some time, you do get some you do get a collection of particles on the on the real line.  And the aim is to somehow understand how this this collection of particles evolves, right?  And the first question you want to ask is the behavior of the of the extremes.  So in our case, this is going to be the position of the rightmost particle.  And this, you know, this case study of stretch exponential displacements basically boils down to explaining the math behind the result that I did obtain with Nina Gantt and Thomas Hufflesauer.  All right. So Nina does Nina needs no introduction and Thomas is her former PhD student.  And if my memory serves me right, he did graduate two years ago and now he works in the private sector.  All right. So under some technical technical assumptions that I do want to do omit because I don't want to make my slides too complicated.  We have the following. So assuming that the law of this generic random variable is, you know, such exponential and that this guy is centered and has variance one.  And then it turns out that you have two different regimes of the behavior of your rightmost particle.  So depending on whenever r is smaller or greater than the critical value of two thirds, your rightmost position either either fluctuate or you have almost true behavior.  So in both cases, the leading term is the same.  This is alpha times n to the power one over r.  Right.  But for smaller values of r, you know, the rightmost particle fluctuates in the neighborhood of this value with a limit being the shift of a Gumbel law.  So here this w, this random shift is a martingale limit associated to the underlying Galton-Wolson process.  And we will see exactly how this, how this guy pops up.  And whenever r is, you know, big, you have more balanced behavior, which translates to almost a limit.  All right.  So let me just introduce the, you know, the setting that Mina did, did, did, did give in order to be self-contained.  Let zn be the underlying Galton-Wolson process.  All right.  So maybe I, maybe I want to go back and say two things.  So first of all, my aim here is to, you know, somehow explain where this exponent are coming from and where this different type of behavior is coming from.  And most importantly, I do want to give you some high, some idea of how can you use the heaviness of your tails to your advantage when you study the branching random walks.  All right.  So as I said, let zn be the underlying Galton-Wolson process and assume just for the simplicity that this process never dies out.  So assume that the probability that your descendant has no children is zero.  And if we don't have this assumption, then everything that I'm going to tell you will still work.  However, you will just have to condition on the set of survival of your population.  All right.  So then you can draw your Galton-Wolson tree, right?  So then you assign labels to each of the individuals, so on the individuals.  So let's say that, you know, the first particle has a label with, which is an empty word.  And then particles from generation one has a label, which is a string of length one.  Then particles from generation two has a label, which is a string of length two and so on and so forth.  So once you have those, once you have your Galton-Wolson tree, then you take the displacements.  So you take those, a family of IED random variables and assign them to the edges of your Galton-Wolson tree.  So you think of, you know, of your labels, of your axis as the displacement of a given particle from the place of birth.  All right.  And then, you know, if we are interested in a position of a particle from generation n,  then what you need to do is, you know, you just look at the label on your tree,  look at the path from this particle up to the root, right?  And you just sum all the displacements along this path, right?  And then you get this quantity, which I'll denote by Vx, which represents the position of the particle with label X.  And now, you know, your rightmost particle is just the maximum of those guys.  All right.  So one technical thing that we're going to need is the large deviations for random walks.  So you've seen that in Nina's course, and she did use large deviations in the case when the displacements have fintails,  when you have some exponential moments.  And this is basically a common knowledge, right?  In the case which I'm looking at here, that is the stretch exponential case.  This is not so well known.  So let's just look at what's happening.  So the lack of these exponential moments morally, this translates to the so-called principle of one big jump.  Namely, you know, if you look at a statistic, a reasonable statistic of your ensemble,  then, you know, the probability that this statistic is big in some sense is asymptotically equivalent to saying that one of the guys in your ensemble is big.  So for example, if you look at the probability that the sum of the two guys is big,  this is asymptotically equivalent to saying that one of those guys is big.  Now, we want to somehow try to push this phenomena and see if we can get some large deviations result out of that.  All right.  So let's take an SN to be the sum of those X-case, right, and take the maximum and denoted by XN star.  And now we want to understand, you know, what the probability that this SN is greater than some threshold TN.  And, you know, as it will turn out in the future, we don't need the, you know, the full scope of TN.  But it will be sufficient for our needs to take T of the form C times N to the power one over R, right?  So we want to somehow exploit this principle of one big jump.  And naively, what you can do is say, okay, so the biggest displacement does something, right?  And then the other displacement somehow contributes, right?  So you can naively try to write this supremum and just see what happens.  Yeah, so we know what is the behavior of this maximum, right?  This is exactly this part.  And then you check what's the behavior of the rest of those guys.  And since, you know, you think of those guys as the typical displacement, right?  You expect a Gaussian behavior from that.  All right, so you have some expression in your exponent and you just simply optimize this guy, right?  So you get this, it's exactly that.  All right.  Yeah, so now when you look at this probability of typical particles, what you can do is since you expect this SN to be close to Gaussian,  you can write, you know, SN minus one divided by square root of N, and this is supposed to be greater than, you know, this constant that we somehow get times N to the power 3 over 2 minus 1 over R.  So you can actually see from that that the value of R is being equal to 2 thirds.  It's somehow critical for the behavior of the deviations. And this is exactly, you know, this critical parameter that we have in our result.  All right, so you write down what's the behavior of the product, right?  And, you know, this leading term from the maximum translates to the leading term in the exponent.  And then those smaller guys somehow transfer to something smaller.  The key fact here is that the leading term is the asymptotic of the deviations of SN really comes from the maximum.  And then depending on the value of your parameter, then you can get something different or not.  All right. So what can we do with that?  Well, we can redo the first standard term of the asymptotic of the rightmost particle.  Again, something that Nina did in the previous course, but still this will help me drive my point.  So let's just do that.  So the probability that MN is greater than, you know, C times N to 1 over R.  And, you know, 1 over R is the correct scaling because, you know, I said so.  So you look at the probability that one of the guys is greater than this threshold.  And you can use the first moment method to, you know, bound this basically by expectation of the number of guys that are greater than this threshold.  Right. And what you get is the expectation of ZN, which is roughly N to the N times your deviation.  And this is something that we did compute just now.  So what we get is, you know, asymptotically N to the N times E to minus C to the power R times N.  So now we need to tailor this answer that this term compensates this N to the N.  Right.  And this leads to basically a prediction of the first other term being N to the power 1 over R times times alpha.  And this alpha is exactly given here.  All right. So what we can learn from that?  Well, everything, you know, this first term really comes from this parameter, right?  And we just said that, you know, this leading term, this asymptotic of the deviation comes from the biggest displacement.  So we can try to argue that the first term, the asymptotic expansion of the rightmost particle is related to the biggest displacement in your collection.  And this biggest displacement, right?  This is just a maximum of IID guys. And the only twist is that this, the number of guys that you are considering is random.  But since those two quantities are independent, this is still easy to handle.  So you know that after a proper centering and scaling, you will see a nice limit, which is, again, a random shift of the Gumbel distribution.  And this is something that's so easy that we can actually try to try to prove that.  So, for instance, take some threshold XN that goes to infinity.  And let's just try to compute the probability that this biggest displacement is below this threshold.  Right. So what you do, well, you just condition on your tree.  Right. So this leads you to writing that basically you are saying that you have a random number of guys that are supposed to be smaller than this threshold XN.  And this random number of guys is just the number of the edges that you have up to time N.  Right. And this is easily expressed in terms of your branching process.  Right. And then, you know, looking at the expression under the expectation, it behaves as e to the minus the product of, you know, this YN and the tail of X.  All right. So now, this, now the random shift comes into play, because we know that this YN behaves pretty nicely.  That is, after scaling by N to the N, you get a martingale.  And since this guy is non-negative, it will have almost a whole limit.  Moreover, since I'm implicitly assuming that the Kerstens-Tigerm condition, this limit will be non-negative.  So this Kerstens-Tigerm condition is one of those ugly assumptions that I just omitted in the second slide.  All right. But now, you know, we know what the behavior of this YN, right?  This is, again, something that grows in the same fashion with a random shift.  All right. So now, we can take a very concrete threshold here and just start to compute things.  So first off, we do only the simple bits. I mean, first off, we just take the asymptotic equivalent of YN.  This is here. And we just plug in the definition of XN. So nothing happens.  Then, we look at the exponent on the right-hand side, and we simply use the Taylor expansion.  And when we do that, we notice that, you know, this exponent here is Taylor,  such that it will cancel with a derivative that pops up from the expansion.  Right? So you're left with a constant. All right.  But now, we just need to recall what is the role of alpha.  Alpha is taken such that this guy will cancel with n to the n.  Right? So we are in business. We are in business, because if we call our constants, you know, constants accordingly,  then we get a nice limit. And if we go back to what we have done, this basically shows that, you know,  this biggest displacement after proper centering and scaling does converge to this nice random shift of the Goomba law.  So as you can see, the analysis of the biggest displacement is pretty straightforward.  So what can we do with that?  Well, when we look at the first order asymptotics of the biggest displacement and the rightmost particle, they are the same.  So it's not difficult to convince yourself that if you look at the positions of the particles, which didn't have a big jump along the branch,  those guys are negligible somehow, right?  And if you have that, then you know that you need only to concentrate on the particles that had a big jump, right?  And how many are there? So all the particles up to time n, which we see in our system, the number of those is roughly m to the n.  And then you look at, you know, those particles that did encounter a big jump, and those will be fewer, right?  And it turns out that whenever r is small, you only see a polynomial number of these big jumps.  And whenever r is big, you see something which is almost exponential, but not exactly exponential.  And I do know that this line is somehow vague, because I don't tell you what do I mean by, you know, x having a big jump.  So the takeaway message from this line is that for small values of r, we have small number of big jumps that contribute.  And for big values of r, we have a lot of big jumps that can contribute.  All right. And in the end, you know, those particles that did give a big jump form a small subset of the whole tree.  So what can we do with that?  If we look at the whole tree, the first step that you want to do is look at those particles that, you know, with the displacement comparable, you know, to the first order asymptotic.  And color the edges green.  Then, you know, you color the edges which are below in red. And then you look at the path that goes, you know, from the green edge to the root.  And you will have an orange segment and a blue segment.  And the way this orange color and the blue color works is that when you take into account other big jumps, what you will see is that, you know,  you have a segment which is orange and which is disjoint from different paths.  And the moment they start to meet each other, they will be painted blue.  So the thing is that those orange guys are independent.  Yes.  Yes.  Were you trying to have a question in there?  Oops. Let me, there's a question in the chat. No. Yes.  What's the question?  I don't see a question. I think you can.  Maybe it was sent privately, but.  I don't see a question in the chat.  Okay. I'm, I'm all right. So I just move on. Where was I?  Yeah. So the orange guys are basically independent and those blue guys are the bits when you see some kind of dependence.  And yeah. So the set of those admissible particles, I mean, those that you see at the very end, right, each of those positions of those guys actually can be the composed in the following fashion.  So if you have this, when you, you know, you go along here, you see a blue path, you know, a path in the orange color, then is this one green big jump and then there's the red.  All right. So what's the contribution of each color?  If you really believe that those big jumps are a sparse subset of your London tree, this means that if you take two green edges, they will be far away from each other on the, on the tree.  And this means that, you know, those, those paths will meet very high up.  Hence those this blue bit, which will be relatively short.  In conclusion, you know, this, this red part is, is negligible.  So now let's focus on the, on the red part.  And this is something that requires some computation, but since the maximum displacement grows faster than linear, right, this go grows like.  And to the power one over R.  When you, when you look where those displacement happened, I mean, in which generation, it's more likely that they will happen in a generation which is close to N.  So this in terms tells you that those, you know, those red segments, they also be relatively short.  So this, this red guy can also be neglected.  All right, but this, this tells us that, you know, the, the positions of this admissible particles are asymptotically are just the, you know, the, the orange parts and those, those green jumps, right, and those guys are independent.  So we are, we are quite happy.  So now we need to compare those, those two.  Right.  So asymptotically, your right most particle is consisting of, of this big, big jumps, this an X, and the position of, you know, the typical particles.  And you know that, you know, this big jump, well, it is big.  And those typical particles.  This, this is the behavior should be asymptotically normal.  So you can do the same optimization game that we do the previously.  And, and you would come to a conclusion that the contribution of those typical particles, it, it is also of the order and to the two minus one over R.  So this is the exact same exponent that we did get from this optimization.  On the other hand, we know what, what is the behavior of the biggest particle, right.  So this is, you know, we had the leading term and we know what are the fluctuations.  So now we just need to, to compare those two.  Right.  So if R is small, then, you know, whatever the contribution of the typical particle is particles is it's killed by the fluctuations of the, of the biggest particle.  And, you know, this right most particle, it scales the same as the, as the biggest particle.  And you get this nice, nice limit.  Moreover, this one comment that I would like to make at this point, that is, we can actually show that the difference between the right most particle and the biggest displacement is negligible in compared to the.  The order of the fluctuations.  Right.  So this M and this is something that consists of, you know, the maximum of your guys and this M and this is something that has, you know, the sums.  So this, this line somehow really screams that one particle did all the, did all the job.  What happens when R is big?  Well, you need to go back to our picture, right.  And what you do, you first look at the green displacements at the green edges, right.  And this fixes the first order, right.  And then you know that the contribution of the orange parts should somehow kill the fluctuations of those, those green guys.  So what you can do is start looking at those orange paths and see what whatever is the is the biggest.  And as I mentioned, you know, in this case, there are actually a lot of those, those orange, those green jumps.  Hence, there will be a lot of those orange paths.  So we have a very big number of those orange paths with which in turn gives you a more balanced behavior.  And now you can basically play again the same optimization game and say that, all right, so this is the contribution of those, those orange guys is in fact of the other.  To enter the two minus one over R, right.  And if you do this, this optimization, right, you can squeeze out the, the limiting constant.  All right.  So what happens in the boundary case?  Well, you notice that, you know, both of those exponents are equal one half, right.  So now if the contribution of the typical, typical particles should be is of this other than you will conclude that they will actually fluctuate.  All right.  So what, but still, you can work around that right, you can, you know, take a family of independent gaussians.  And if you look at the, the right most particle after centering and scaling, you will see, again, a maximum of the, you know, those admissible particles, right.  So you see the proper centering of the, of the right most position and then those typical displacement would be, you know, asymptotically gaussian.  But remember that, that we did the couple, the whole thing, right. So we can still basically work as a, as if we would, we would be working with a family of, of independent guys, right.  So still, you know, centering this, this right most particle is the same as, you know,  pardon the cumulative distribution function of the right most particle is the same as, you know, those big displacements with some random gaussian shifts.  But still, this is something which is, which is manageable. So you do the exact same computations that we did for the, for the maximum, right.  And this is that, you know, this gaussian shift transfers to this integral. And here, you know, this, this big displacement transfers to this, to this e to the minus e, minus e, pardon.  And then, well, the gaussian and the variable transfers to this, to this gaussian term, right.  But still, if you look at this integral as a function of, of t, this is still a constant times e to the minus minus t.  So nothing, nothing really changes. I mean, those fluctuations, they do contribute, but the, the, the contribution can be hidden in a, in a constant.  All right. So this, this is that, I mean, in, in the case when we're are is very small, right, the whole, the behavior of the, of the whole system is really governed by one big jump, right.  And in this boundary case, this, this one big jump is supplemented by this, this gaussian, gaussian shift.  However, this, this contribution can be, can be hidden in this, in this constant gamma. So in particular, this, this gamma, it's, it is not a,  this is not a continuous function of, of our parameters, right. And in, if R is big, then, you know, we have a lot of the typical particles that somehow contribute, which gives you a more balanced, balanced behavior.  Yeah. So remember that the, the way that we started our analysis is first by, by looking at the, at the biggest displacement.  On one hand, it did give us, you know, all the, all the information we needed, or it allowed us to decouple the whole thing. And it, this, this was the, you know, the key step which we needed in our analysis.  All right. So this was based on a joint paper with, with Fina and Thomas and it's available on archive.  And let me just mention that, you know, this is first order asymptotic was established by, by Nina in the, in the 2000s. And if, and then large deviations for, you know, for the, for the script stretched exponential random variables,  go back to the Nagaev in the late 60s. However, it's, it's not an easy thing to, to read.  All right. So I'm out of slides. So I guess this is a good place to stop. Thank you.  Thank you very much, Piotr. So maybe you can unmute people.  And so other questions to Piotr as long as the screen is still shared.  Some questions were already answered in the chat.  It seems that there are no questions.  You are muted currently.  Yes.  So we will stop.  There are no questions right now. So then I think, are you stopped the recording for.  Stop now the recording and start it again for some stock.
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Okay. Okay, and we are live. So, and also start the, you started the recording. Okay. So, I want to say today, welcome everyone to the fourth session of the week. So today we will have two lectures on both of being recorded. So if you don't wish your audio and video to appear in the recording, please keep your microphone. I'm going to switch stuff. And let me just also mentioned that next week we have rather full week of material with mini course from Ivan Corvin Monday, Tuesday, Wednesday, followed by several short lectures on related topics on Thursday and Friday. So there will be an email to the mailing list about that this afternoon. So with that, I'm going to hand it over to Nina Gantert to introduce the speakers over to you, Nina. Yes, thanks, Luigi. So today we will have two talks and the first speaker is Piotr the chef's kid. The second is Sam Johnston, you can see them both here. And Piotr, who is a postdoc in Munich right now and also in Wroclaw will start and his topic is, his topic is a branching random walk and stretched exponential tails. And the screen sharing works for me and I hope for you as well. Okay, Piotr. You can start. All right. Thank you, Nina. Yeah, so for the most of the Nina's lecture, she focused on on a branching random walk when where the displacements have fin tails. And what I want to do today is to show you how you can work with branching random walks whenever your displacements are heavy tails, have heavy tails. And what I mean by that is whenever the, you know, the exponential moments exist or not. So first off, let's just recall the setting. So I will, we will work on a real line. And we start with one particle at the origin. And after some time, this particle dies and gives birth to a random number of new particles. And we're going to assume that the reproduction law is the reproduction of the particles. It's governed by a supercritical Garten Watson process. So this in particular means that the mean of the reproduction law is is greater than one. All right. And we also want to assume that the displacements of the particles because after the particles are born, they are moved somewhere on the on the real line. They alter shifts are an IID copy of one given given random variable with so-called stretch exponential law. So this means that the tail of X decays as e to the minus t to the power r. And this and this twiddle means that the quotient of, you know, the both sides tends to one as t tends to infinity. So instead of, you know, giving you a full scope of what happens with heavy tail displacement, I'm just going to focus on the stretch exponential case and give you the whole treatment. But most of the phenomena's do replicate in with different tails. All right. So, you know, the death particles are removed from the system, right? And then they reproduce as previously. So they produce some random number of other particles which are then placed on the real line. And, you know, this system evolves for some time, according to this rule. And, you know, after some time, you do get some you do get a collection of particles on the on the real line. And the aim is to somehow understand how this this collection of particles evolves, right? And the first question you want to ask is the behavior of the of the extremes. So in our case, this is going to be the position of the rightmost particle. And this, you know, this case study of stretch exponential displacements basically boils down to explaining the math behind the result that I did obtain with Nina Gantt and Thomas Hufflesauer. All right. So Nina does Nina needs no introduction and Thomas is her former PhD student. And if my memory serves me right, he did graduate two years ago and now he works in the private sector. All right. So under some technical technical assumptions that I do want to do omit because I don't want to make my slides too complicated. We have the following. So assuming that the law of this generic random variable is, you know, such exponential and that this guy is centered and has variance one. And then it turns out that you have two different regimes of the behavior of your rightmost particle. So depending on whenever r is smaller or greater than the critical value of two thirds, your rightmost position either either fluctuate or you have almost true behavior. So in both cases, the leading term is the same. This is alpha times n to the power one over r. Right. But for smaller values of r, you know, the rightmost particle fluctuates in the neighborhood of this value with a limit being the shift of a Gumbel law. So here this w, this random shift is a martingale limit associated to the underlying Galton-Wolson process. And we will see exactly how this, how this guy pops up. And whenever r is, you know, big, you have more balanced behavior, which translates to almost a limit. All right. So let me just introduce the, you know, the setting that Mina did, did, did, did give in order to be self-contained. Let zn be the underlying Galton-Wolson process. All right. So maybe I, maybe I want to go back and say two things. So first of all, my aim here is to, you know, somehow explain where this exponent are coming from and where this different type of behavior is coming from. And most importantly, I do want to give you some high, some idea of how can you use the heaviness of your tails to your advantage when you study the branching random walks. All right. So as I said, let zn be the underlying Galton-Wolson process and assume just for the simplicity that this process never dies out. So assume that the probability that your descendant has no children is zero. And if we don't have this assumption, then everything that I'm going to tell you will still work. However, you will just have to condition on the set of survival of your population. All right. So then you can draw your Galton-Wolson tree, right? So then you assign labels to each of the individuals, so on the individuals. So let's say that, you know, the first particle has a label with, which is an empty word. And then particles from generation one has a label, which is a string of length one. Then particles from generation two has a label, which is a string of length two and so on and so forth. So once you have those, once you have your Galton-Wolson tree, then you take the displacements. So you take those, a family of IED random variables and assign them to the edges of your Galton-Wolson tree. So you think of, you know, of your labels, of your axis as the displacement of a given particle from the place of birth. All right. And then, you know, if we are interested in a position of a particle from generation n, then what you need to do is, you know, you just look at the label on your tree, look at the path from this particle up to the root, right? And you just sum all the displacements along this path, right? And then you get this quantity, which I'll denote by Vx, which represents the position of the particle with label X. And now, you know, your rightmost particle is just the maximum of those guys. All right. So one technical thing that we're going to need is the large deviations for random walks. So you've seen that in Nina's course, and she did use large deviations in the case when the displacements have fintails, when you have some exponential moments. And this is basically a common knowledge, right? In the case which I'm looking at here, that is the stretch exponential case. This is not so well known. So let's just look at what's happening. So the lack of these exponential moments morally, this translates to the so-called principle of one big jump. Namely, you know, if you look at a statistic, a reasonable statistic of your ensemble, then, you know, the probability that this statistic is big in some sense is asymptotically equivalent to saying that one of the guys in your ensemble is big. So for example, if you look at the probability that the sum of the two guys is big, this is asymptotically equivalent to saying that one of those guys is big. Now, we want to somehow try to push this phenomena and see if we can get some large deviations result out of that. All right. So let's take an SN to be the sum of those X-case, right, and take the maximum and denoted by XN star. And now we want to understand, you know, what the probability that this SN is greater than some threshold TN. And, you know, as it will turn out in the future, we don't need the, you know, the full scope of TN. But it will be sufficient for our needs to take T of the form C times N to the power one over R, right? So we want to somehow exploit this principle of one big jump. And naively, what you can do is say, okay, so the biggest displacement does something, right? And then the other displacement somehow contributes, right? So you can naively try to write this supremum and just see what happens. Yeah, so we know what is the behavior of this maximum, right? This is exactly this part. And then you check what's the behavior of the rest of those guys. And since, you know, you think of those guys as the typical displacement, right? You expect a Gaussian behavior from that. All right, so you have some expression in your exponent and you just simply optimize this guy, right? So you get this, it's exactly that. All right. Yeah, so now when you look at this probability of typical particles, what you can do is since you expect this SN to be close to Gaussian, you can write, you know, SN minus one divided by square root of N, and this is supposed to be greater than, you know, this constant that we somehow get times N to the power 3 over 2 minus 1 over R. So you can actually see from that that the value of R is being equal to 2 thirds. It's somehow critical for the behavior of the deviations. And this is exactly, you know, this critical parameter that we have in our result. All right, so you write down what's the behavior of the product, right? And, you know, this leading term from the maximum translates to the leading term in the exponent. And then those smaller guys somehow transfer to something smaller. The key fact here is that the leading term is the asymptotic of the deviations of SN really comes from the maximum. And then depending on the value of your parameter, then you can get something different or not. All right. So what can we do with that? Well, we can redo the first standard term of the asymptotic of the rightmost particle. Again, something that Nina did in the previous course, but still this will help me drive my point. So let's just do that. So the probability that MN is greater than, you know, C times N to 1 over R. And, you know, 1 over R is the correct scaling because, you know, I said so. So you look at the probability that one of the guys is greater than this threshold. And you can use the first moment method to, you know, bound this basically by expectation of the number of guys that are greater than this threshold. Right. And what you get is the expectation of ZN, which is roughly N to the N times your deviation. And this is something that we did compute just now. So what we get is, you know, asymptotically N to the N times E to minus C to the power R times N. So now we need to tailor this answer that this term compensates this N to the N. Right. And this leads to basically a prediction of the first other term being N to the power 1 over R times times alpha. And this alpha is exactly given here. All right. So what we can learn from that? Well, everything, you know, this first term really comes from this parameter, right? And we just said that, you know, this leading term, this asymptotic of the deviation comes from the biggest displacement. So we can try to argue that the first term, the asymptotic expansion of the rightmost particle is related to the biggest displacement in your collection. And this biggest displacement, right? This is just a maximum of IID guys. And the only twist is that this, the number of guys that you are considering is random. But since those two quantities are independent, this is still easy to handle. So you know that after a proper centering and scaling, you will see a nice limit, which is, again, a random shift of the Gumbel distribution. And this is something that's so easy that we can actually try to try to prove that. So, for instance, take some threshold XN that goes to infinity. And let's just try to compute the probability that this biggest displacement is below this threshold. Right. So what you do, well, you just condition on your tree. Right. So this leads you to writing that basically you are saying that you have a random number of guys that are supposed to be smaller than this threshold XN. And this random number of guys is just the number of the edges that you have up to time N. Right. And this is easily expressed in terms of your branching process. Right. And then, you know, looking at the expression under the expectation, it behaves as e to the minus the product of, you know, this YN and the tail of X. All right. So now, this, now the random shift comes into play, because we know that this YN behaves pretty nicely. That is, after scaling by N to the N, you get a martingale. And since this guy is non-negative, it will have almost a whole limit. Moreover, since I'm implicitly assuming that the Kerstens-Tigerm condition, this limit will be non-negative. So this Kerstens-Tigerm condition is one of those ugly assumptions that I just omitted in the second slide. All right. But now, you know, we know what the behavior of this YN, right? This is, again, something that grows in the same fashion with a random shift. All right. So now, we can take a very concrete threshold here and just start to compute things. So first off, we do only the simple bits. I mean, first off, we just take the asymptotic equivalent of YN. This is here. And we just plug in the definition of XN. So nothing happens. Then, we look at the exponent on the right-hand side, and we simply use the Taylor expansion. And when we do that, we notice that, you know, this exponent here is Taylor, such that it will cancel with a derivative that pops up from the expansion. Right? So you're left with a constant. All right. But now, we just need to recall what is the role of alpha. Alpha is taken such that this guy will cancel with n to the n. Right? So we are in business. We are in business, because if we call our constants, you know, constants accordingly, then we get a nice limit. And if we go back to what we have done, this basically shows that, you know, this biggest displacement after proper centering and scaling does converge to this nice random shift of the Goomba law. So as you can see, the analysis of the biggest displacement is pretty straightforward. So what can we do with that? Well, when we look at the first order asymptotics of the biggest displacement and the rightmost particle, they are the same. So it's not difficult to convince yourself that if you look at the positions of the particles, which didn't have a big jump along the branch, those guys are negligible somehow, right? And if you have that, then you know that you need only to concentrate on the particles that had a big jump, right? And how many are there? So all the particles up to time n, which we see in our system, the number of those is roughly m to the n. And then you look at, you know, those particles that did encounter a big jump, and those will be fewer, right? And it turns out that whenever r is small, you only see a polynomial number of these big jumps. And whenever r is big, you see something which is almost exponential, but not exactly exponential. And I do know that this line is somehow vague, because I don't tell you what do I mean by, you know, x having a big jump. So the takeaway message from this line is that for small values of r, we have small number of big jumps that contribute. And for big values of r, we have a lot of big jumps that can contribute. All right. And in the end, you know, those particles that did give a big jump form a small subset of the whole tree. So what can we do with that? If we look at the whole tree, the first step that you want to do is look at those particles that, you know, with the displacement comparable, you know, to the first order asymptotic. And color the edges green. Then, you know, you color the edges which are below in red. And then you look at the path that goes, you know, from the green edge to the root. And you will have an orange segment and a blue segment. And the way this orange color and the blue color works is that when you take into account other big jumps, what you will see is that, you know, you have a segment which is orange and which is disjoint from different paths. And the moment they start to meet each other, they will be painted blue. So the thing is that those orange guys are independent. Yes. Yes. Were you trying to have a question in there? Oops. Let me, there's a question in the chat. No. Yes. What's the question? I don't see a question. I think you can. Maybe it was sent privately, but. I don't see a question in the chat. Okay. I'm, I'm all right. So I just move on. Where was I? Yeah. So the orange guys are basically independent and those blue guys are the bits when you see some kind of dependence. And yeah. So the set of those admissible particles, I mean, those that you see at the very end, right, each of those positions of those guys actually can be the composed in the following fashion. So if you have this, when you, you know, you go along here, you see a blue path, you know, a path in the orange color, then is this one green big jump and then there's the red. All right. So what's the contribution of each color? If you really believe that those big jumps are a sparse subset of your London tree, this means that if you take two green edges, they will be far away from each other on the, on the tree. And this means that, you know, those, those paths will meet very high up. Hence those this blue bit, which will be relatively short. In conclusion, you know, this, this red part is, is negligible. So now let's focus on the, on the red part. And this is something that requires some computation, but since the maximum displacement grows faster than linear, right, this go grows like. And to the power one over R. When you, when you look where those displacement happened, I mean, in which generation, it's more likely that they will happen in a generation which is close to N. So this in terms tells you that those, you know, those red segments, they also be relatively short. So this, this red guy can also be neglected. All right, but this, this tells us that, you know, the, the positions of this admissible particles are asymptotically are just the, you know, the, the orange parts and those, those green jumps, right, and those guys are independent. So we are, we are quite happy. So now we need to compare those, those two. Right. So asymptotically, your right most particle is consisting of, of this big, big jumps, this an X, and the position of, you know, the typical particles. And you know that, you know, this big jump, well, it is big. And those typical particles. This, this is the behavior should be asymptotically normal. So you can do the same optimization game that we do the previously. And, and you would come to a conclusion that the contribution of those typical particles, it, it is also of the order and to the two minus one over R. So this is the exact same exponent that we did get from this optimization. On the other hand, we know what, what is the behavior of the biggest particle, right. So this is, you know, we had the leading term and we know what are the fluctuations. So now we just need to, to compare those two. Right. So if R is small, then, you know, whatever the contribution of the typical particle is particles is it's killed by the fluctuations of the, of the biggest particle. And, you know, this right most particle, it scales the same as the, as the biggest particle. And you get this nice, nice limit. Moreover, this one comment that I would like to make at this point, that is, we can actually show that the difference between the right most particle and the biggest displacement is negligible in compared to the. The order of the fluctuations. Right. So this M and this is something that consists of, you know, the maximum of your guys and this M and this is something that has, you know, the sums. So this, this line somehow really screams that one particle did all the, did all the job. What happens when R is big? Well, you need to go back to our picture, right. And what you do, you first look at the green displacements at the green edges, right. And this fixes the first order, right. And then you know that the contribution of the orange parts should somehow kill the fluctuations of those, those green guys. So what you can do is start looking at those orange paths and see what whatever is the is the biggest. And as I mentioned, you know, in this case, there are actually a lot of those, those orange, those green jumps. Hence, there will be a lot of those orange paths. So we have a very big number of those orange paths with which in turn gives you a more balanced behavior. And now you can basically play again the same optimization game and say that, all right, so this is the contribution of those, those orange guys is in fact of the other. To enter the two minus one over R, right. And if you do this, this optimization, right, you can squeeze out the, the limiting constant. All right. So what happens in the boundary case? Well, you notice that, you know, both of those exponents are equal one half, right. So now if the contribution of the typical, typical particles should be is of this other than you will conclude that they will actually fluctuate. All right. So what, but still, you can work around that right, you can, you know, take a family of independent gaussians. And if you look at the, the right most particle after centering and scaling, you will see, again, a maximum of the, you know, those admissible particles, right. So you see the proper centering of the, of the right most position and then those typical displacement would be, you know, asymptotically gaussian. But remember that, that we did the couple, the whole thing, right. So we can still basically work as a, as if we would, we would be working with a family of, of independent guys, right. So still, you know, centering this, this right most particle is the same as, you know, pardon the cumulative distribution function of the right most particle is the same as, you know, those big displacements with some random gaussian shifts. But still, this is something which is, which is manageable. So you do the exact same computations that we did for the, for the maximum, right. And this is that, you know, this gaussian shift transfers to this integral. And here, you know, this, this big displacement transfers to this, to this e to the minus e, minus e, pardon. And then, well, the gaussian and the variable transfers to this, to this gaussian term, right. But still, if you look at this integral as a function of, of t, this is still a constant times e to the minus minus t. So nothing, nothing really changes. I mean, those fluctuations, they do contribute, but the, the, the contribution can be hidden in a, in a constant. All right. So this, this is that, I mean, in, in the case when we're are is very small, right, the whole, the behavior of the, of the whole system is really governed by one big jump, right. And in this boundary case, this, this one big jump is supplemented by this, this gaussian, gaussian shift. However, this, this contribution can be, can be hidden in this, in this constant gamma. So in particular, this, this gamma, it's, it is not a, this is not a continuous function of, of our parameters, right. And in, if R is big, then, you know, we have a lot of the typical particles that somehow contribute, which gives you a more balanced, balanced behavior. Yeah. So remember that the, the way that we started our analysis is first by, by looking at the, at the biggest displacement. On one hand, it did give us, you know, all the, all the information we needed, or it allowed us to decouple the whole thing. And it, this, this was the, you know, the key step which we needed in our analysis. All right. So this was based on a joint paper with, with Fina and Thomas and it's available on archive. And let me just mention that, you know, this is first order asymptotic was established by, by Nina in the, in the 2000s. And if, and then large deviations for, you know, for the, for the script stretched exponential random variables, go back to the Nagaev in the late 60s. However, it's, it's not an easy thing to, to read. All right. So I'm out of slides. So I guess this is a good place to stop. Thank you. Thank you very much, Piotr. So maybe you can unmute people. And so other questions to Piotr as long as the screen is still shared. Some questions were already answered in the chat. It seems that there are no questions. You are muted currently. Yes. So we will stop. There are no questions right now. So then I think, are you stopped the recording for. Stop now the recording and start it again for some stock.
the on the real line. They alter shifts are an asymptotic expansion of the rightmost particle is your branching process. Right. And then, you and the position of, you know, the typical working with a family of, of independent guys, for the, for the script stretched exponential