 So thank you very much. So the work I'm presenting today is joint work with my two PhD supervisors, Emmanuel Charzère and Damory Lambert. And together we work in the LPMA in Paris. And we also have an interdisciplinary team which is called SMILE, for Stochastic Models for the Inference of Life Evolution, where we work together with biologists. And the work I'm going to present today was inspired by the following experiment, which was carried by Enrique de Otonio at UNS. So he studies senorabditis elegans, which are small transparent worms. And we start with a population of 180 individuals, which are sampled from distinct populations. So here, each line represents the chromosome of one individual, and they are all distinct, so that's why I paint them in different colors. And he let the population evolve for some generations. And after some amount of time, the chromosomes of the individuals look like this. They are mosaics of colors, because each fragment of chromosome has been inherited from one of the ancestors in generation zero. And after 140 generations, he genotyped all the individuals in the population, so here each horizontal line, this is real data, each horizontal line corresponds to one individual, and it's the chromosome, and each color represents one ancestor. And so we want to study this mosaic of colors, because it may be informative about, for example, if some colors are more present than others, this may mean that some individuals in the generation zero was fitter than the others and had more progeny. So to study this mosaic of colors, we are going to use a model. So the question I'm going to ask, so if this is one chromosome, I call a segment a maximal set of connected points which share the same color, so for example, this is one segment, and they call a cluster a set of points sharing the same color, for example, the blue cluster is here. And the kind of questions I want to ask is what is the size of a typical segment and what is the length or the diameter of a typical cluster and how many segments of clusters are there in a given interval. So the model I use is an absolute right fissure model with recombination, so I assume I have a population of constant size n, and each individual carries one chromosome of size r. And my population follows a right fissure dynamics, which means that I have a discrete time dynamics, and at each generation t, each individual chooses two parents in the previous generation, and then two things can happen. So with a probability 1 minus rho, the individual copies exactly the chromosome of one of its parents. This is what happens here. And with the probability rho here, the individual will inherit a chromosome that is a mixture between the two parental chromosomes. So the two parental chromosomes are cut at what they call a crossing over point, whose position is randomly located along the chromosome, and he will inherit half of the genetic material of the father and half of the mother. So this is the picture of what happens in the population, which is cut with n individuals, which are pairwise distinct. Each one has a chromosome painted in a unique color. In generation 2, individuals may have two colors, or only one, and after a lot of generations, my chromosomes look like mosaic of colors. And what I know about this model is that as I have a fixed population site and my population is finite, my mother will reach fixation, which means that after a finite number of generations, all the chromosomes in my population will look the same, because it's the only observing state for my micro chain, and they want to characterize this mosaic of colors. So I will consider... I will define a process at p and r, which is the partitioning process, which corresponds to the color partition of the system and the equilibrium for a population of size n with chromosomes of size r. And then I will make my population size tend to infinity, and my probability... I will make my probability of recombination of rho dependent on n and r, and I assume that as n goes to infinity, n times rho converges to r. So I make my probability of recombination of rho dependent on the population size. So then what I say is that as n tends to infinity, my process converges to a process pi of r in low, which I call the r-partitioning process. And my question is, what can I say about this process when r is large? Which means for a large chromatome. So I will justify this limit that I know that r, for humans, is quite large. For example, for a human chromosome, the size r is 5 times 10 to the power 4. So the size of the chromosome is denormalized by the effective population size. So the first result I want to prove is the following theorem. So I will look at the color clusters that covers the origin of the chromosome. So here it would be the one that I call it in red. And I call L of r the length of this cluster. And I renormalize L of r by log of r. And they say that as r tends to infinity, this quantity converges in low to an exponential of per meter 1. So the idea to prove this theorem is that I'm going to use a process that is dual to my original process. In fact, what I will do is to use the ancestral recombination graph. So I will choose two sites in my chromosome. So if this was my chromosome at the origin, I choose two sites. It's x and y. And I will follow my population backwards in time. So here each line represents the ancestral line for one of these sites, which corresponds to the individual in the past population that is the ancestor for that given gene. So here I choose one individual at random in my population and I will follow which parent was carrying locus x and locus y. And I will see if this is my initial population to which color do they arrive. So if they arrive to the same individual here, if both lineages arrive to the same individual here, that means that in the partition, in the present population, these two points will be of the same color. And if as here they reach different individuals in population at time zero, that would mean that one of my locus will be colored in green and the other in blue. So here for two locus, I will follow two ancestral lines. So they start together and the two lines will split into two with a probability that is L over N, where L is the distance between x and y. And both ancestral lineages will coalesce with a probability that is 1 over N, which corresponds to the probability that in this generation, this individual and this individual have a common ancestor. So what I claim is that this process called the ancestral recombination graph is due to my initial process. Why? Because if here you have my ancestral recombination graph represented for my whole chromosome and I follow backwards in time the lineages that carry each of my genes. It will look like this, where red dots correspond to fragmentation events and the blue dots correspond to coalescent events. And what happens is that here I have my partitioning process that is represented. The probability that the two points are in the same cluster corresponds to the probability that two points reach the same ancestor in my ancestral recombination graph. So what I need is to define my ancestral recombination graph. I show you how it behaves for two particles, but I need to define it for N particles. So for N particles, we have a coalescent fragmentation process, which is a process valued in the set of partitions of 0N with the following transition rates. So each group of lineages coalesces at rate 1 and they have fragmentation at this rate. So if I have some particles that are located here, the probability that a fragmentation event occurs here depends on the distance between these two points. And I show that there's a duality relation, which means that the probability for my partitioning process and sites in my chromosome belong to the same cluster in the color partition corresponds to the probability that my ancestral lineages are together for my ancestral recombination graph. So then to prove the theorem I wanted to prove, what they will use is a method of moments. So I will use a Carleman condition, which says that it is enough to prove that for all N, I have the expectation of LR at the power N, renormalized by log R at the power N is this, which is the moment for the exponential law. And what they do is that I will rewrite this expectation like this. So the idea is that I want to look at the points that are in the same color partition as 0. So I want to integrate between 0 and R, the indicator function that a point is of the same color at the point in 0. And as this is at the power N, I can just rewrite it like this. So it corresponds to the probability that N particles and sites in the chromosome are of the same color as site 0. And using my duality relation, this is the probability that N lines in my ancestral recombination graph are together. So then I need to study this coalescence fragmentation process. So the thing is that the ARG is known to be computationally intractable. And so this is impossible to compute exactly. But I use the fact that as ARG goes to infinity, my fragmentation probability is much higher than my coalescence probability. So if I start with lines that are separated, for example, I will show you an example with three. If I start with three lines that are separate, this will coalesce at rate one into this, and this will be fragmented with the probability that it's proportional to R, etc., and to reach the state that I want, which is this one. I need to coalesce again. And still this will recombine at this rate. So what I show is that this state is the most likely configuration and has a probability that goes to one as R goes to infinity. And configurations that are at one coalescence event distance from this configuration have a probability that is of order one over R. And this one has a probability that is of order one over R squared. And using this approximation, I can replace it here. And they found what I wanted. So I found that this converges to factorial n. So the perspective, I have other results that allow me to characterize the law of a cluster that is conditioned to be at any point x of the chromosome, and also about the number of clusters in my chromosome. And what we want to do with this is to work on a neutrality test based on amplotypes. So that would mean that once I can totally characterize this mosaic of colors, I can compare my prediction to the real data. And if the real data fit my prediction, that would mean that the evolution was neutral, as described in my model. But if for some part of the chromosome this hypothesis is violated, I believe one loci is selected. I expect that then I would have a color cluster that are larger or smaller than predicted. And so that could be like a test for selection, which doesn't need to have mutation, that can work for populations that have a short time. And that's all. So thank you very much for your attention. So thanks Veronica. So Anhi? No, not this time. I don't understand. Is the first cluster, is it special? It's not that it's special. I could do this for any x here. So I think that it's somehow special, because this point has only right neighbors. So the thing is that to compute this, I need to choose one point and then I integrate over its neighbors. But I didn't show it here, but I can take any point here and just I will, what I show if it's, that if I take the cluster that is in any point x, then the length of the cluster in this segment is also an exponential and in the left segment it's also an exponential and at the limit for large r they become independent. So it follows a gamma law. Okay, so the number of clusters is of order n over log n. So sorry, r over log r and this Poisson process with constant intensity? It's not that easy, because the thing is that the number of clusters is the constant c times r over log r and the constant is not one, it's like 1.3. And this was found using simulations in 1977 or something like that by Ruf and Hein. And they just show that numerically. And what we think is that, so here when we are doing that our clusters are size-biased, because I conditioned them to be at a certain point. And we think that there are also a certain amount of really small clusters, kind of a dust. And that makes that the number of clusters is larger than r over log r. Is there any other questions? Well, if not, I have a question. So what made you to make this choice of model? Does it compare well with the real data? So in fact, no. The thing is that we know some biologists at INRA who were trying to work on this type of model because they wanted to do selection tests without mutation, so the thing I mentioned. So we started working with them. So they had some simulations and there were things that they could not explain, so we started working about that. And then we met that guy at UNS who had experimental data who were just fitting our model. But the thing is that in the experimental data he gave us. So the experiment, it's not at the time where it fits our model because as I said in my model, I wait until my publication reaches fixation so that each individual is exactly the same. And here you see that each individual is still different. So we still have to wait a little bit until we can compare our data to his experiments. So there are small worms that are transparent and that have only a thousand cells or something like that so they reproduce quite fast. Yeah, people use them a lot in genetics because they don't have many cells and they develop very quickly and they are transparent. So that's nice for biologists for experiments. So let's send the speaker again.