 Okay, so hello everybody, and thank you for the invitation. So I'm Nicolas Leverny, I'm currently in Marseille, and I will talk to you today about the impact of initial perturbations on the first upper passage time of random walks. So basically, I will consider a random walk, pretty general, looking for a target here in red. And I asked a question similar to what Satya talked before. How long will the worker take to find the target? You can rephrase the question, how long will the worker find the target for the first time? So that's why it's a question of first passage time that I will abbreviate FPT in the following slides. So this question of first passage actually appears in many domains, so you can ask the question what's the time for a predator to find a prey, or also what's the time for a transcription factor inside the nucleus to find a specific part of a strain of DNA. But also in a totally different domain, what's the time for a given asset price to reach a threshold. So this question are first passage time problems. A few definitions first. This first passage time is naturally random variable because the walk is itself random, so it has a given distribution that I will call f and a given mean value that is called the mean first passage time. And it is known that in case the walk is non-Markovian, the FPT statistics is ideal and trivial to determine it. And that's the purpose of my talk actually. So generally speaking, you can separate this FPT problem in two big classes. In the first class, the walker is not confined, so you can go everywhere before finding the target. And in this case, typically the density of FPT is algebraic with a decay exponent that is called the persistence exponent in case of Brownian motion, this theta is equal to one-half. And due to this heavy tail, the mean first passage time is infinite. And in this case, in this point, typically one is interested in the value of this exponent, this persistence exponent. On the other side, if your walk is confined, then your walker cannot go very far before touching the target and the FPT density is typically exponential with a finite mean first passage time and one is interested in the computation of this mean first passage time. And actually this second part will mainly be the purpose of the talk of Thomas Guerin tomorrow. And in this talk, I will focus on this first part, unconfined random walk. And in particular, I will focus on this exponent, theta. And if you want a nice review about this exponent and the different way you can use to calculate it, you can read this good review about it, written by Satya in particular. Okay. So the motivation for the walk I will present to you today actually is this paper, published at the end of the 90s, about fluctuating interfaces. So we consider an interface that is fluctuating with classical dynamics with the deterministic part here. So h is the height of a given interface. You have this fractional operator here. For example, if you take h equal 1 divided by 4, you have an Edward Wilkinson membrane or in the context of polymer, it's just a rauss polymer. It's a simple spring polymer. And driven by a Gaussian white nose and we take the temperature as t equal 1 as a reference. Then you can pick one point of one special abscissa and you ask the question, what's the time for this point to cross the level y equal 0? And as you said, this persistence exponent. And what was known actually at the end of the 90s that in case you take initially an interface that is a thermo equilibrium, then this exponent is 1 minus h. It has been proven particularly rigorously at the end of the 90s by this mathematician, Molchan. But actually, you are not constrained to take initially an interface that is a thermo equilibrium. For example, you can take an equilibrium for a different temperature. For example, you can take the interface perfectly flat, which is an equilibrium for t equal 0. And in this case, the interface will slowly relax and this slow relaxation can interfere with the FPT statistics. And actually, what happens in this case, so in this paper, the authors have shown that if you take this flat interface, you get a persistence exponent that is different from 1 minus h. And actually, this is pretty general. We have done it for actually many initial temperatures and you see, so this is the survival priority, so just the integral of the first passage time density. And you see that the slope, so the persistence exponent, is different for any initial temperature. This is actually pretty surprising because even though the interface is slowly reaching an equilibrium, you still see an impact of the initial quench on the FPT statistics at arbitrary large times because it changes the exponent. So what we wanted to know is, can we predict the influence of the initial preparation on the FPT statistics? In particular, can we predict the value of this exponent theta? And this is actually an important question because in many experiments, for example, you have to prepare your sample to be able to affect the kinetics. This is because your interface has a zero mode. It becomes arbitrarily slow. No, it's... What do you mean? Oh, okay. I think it's pretty general. We can discuss about it. Okay, so we have addressed this question for a special class of processes, a lot of Gaussian processes. Gaussian is nice because they are entirely determined by two quantities. The first one is the correlator and the second one is the mean value. And we assume that the process is not biased so that the mean value is constant. You have no drift in the motion, so it's equal to the initial position of the random worker. And I also assume that at large time my process becomes scale invariant. So you have this scaling for here. I also assume at the beginning that the motion is not confined. It implies that this exponent should be positive because the mean square displacement should diverge with time so that my process is not smooth so h should be smaller than 1. And I have a last hypothesis. It is that at large time my process should become a process with stationary increments meaning that if I wait a large time t and I start to observe my process starting at t, then I have a process that does not depend on t anymore. And actually you see that with this hypothesis the only possible correlator for a long time process is the one of fractional Brownian motion. So for those of you who are not familiar with it, fractional Brownian motion is a generalization of Brownian motion but non-Marcovian. Actually it's exactly Brownian motion for h equal 1.5, but for smaller h we have anti-persistent motion and a really sub-diffusive motion and for h larger than 1.5 you get a persistent motion which is a strongly super-diffusive. These are examples of process in this class so you have this poem of quenched interfaces that I talked to you in the slide before and also the fractional Brownian motion that is constrained on its path for example you can assume that you have a fractional Brownian motion that is constrained to be constant for t negative and then you let it evolve but as you have this constraint this will affect the posterior probabilities and actually these both examples they are Gaussian, they are scaling variant and they are reaching some stationary increment process. All right, so now I want to compute this persistence exponent and to do so it's not an easy task but we will use another a method that we have actually already used in the case of confined random walk to get the mean first passage time and the key idea was to introduce this quantity y pi, actually it's a stochastic process and it's the position of the worker the FPT. So we have a trajectory that starts at the initial position touch here the target and then in red you have the future trajectory and you can sum over all possible first passage time and you have this process in the future of the FPT and actually in the case of confined random walk we make a direct relation between this mean first passage time and the statistics of this process and we wanted to know if we can do the same for unconfined process to get persistence exponents. So I won't show you all derivations we start with this equation that is actually just like a renewal type equation saying that if my worker is on the target at time t at least once before you touch the target and you make a partition over this first passage time tau so it simply buys formula then I introduce my process in the future of the FPT that is defined in this way so in the position of the worker after the FPT and you sum over all possible FPT I make the difference between the two which I integrate over time and this is actually what you get so this is still not very defined but what you can show is that generally this quantity and you make t large you can see that this scales as t at the power 1 minus theta minus h and this behavior actually constrains the left hand side and you can show that first both terms should scale the same way for large t and also that the difference between the two is to scale as 1 t at the power h plus theta and this is the key observation because it shows that actually theta is related to the speed at which the probability density in the future of the FPT converges toward the initial probability density so actually with this this equation you have a strategy because now to get theta we want to know actually this p pi so how to do so actually we will use an approximation which is that the process in the future of the FPT so this y pi is a Gaussian stochastic process with an unknown covariance function sigma pi so you can see on the right that this approximation is pretty precise and this sigma pi actually you can get a closed equation for it so still it's not easy to solve but it's a closed equation and actually what do matters actually it's only this equation for large time because we want to get this deviation between sigma pi and sigma naught and with this you can do by studying this equation so first we use the scaling variance of the covariance by introducing this function z and we expand everything for large time because it should become negligible compared to the initial coefficient sigma and at the end of the day what you get is an integral equation for this auxiliary function z theta and you get an integral for actually any choice of theta and you can solve this equation numerically and what you get now is this kind of thing so this is a resolution for different theta and you see that this theta, this function z is always diverging for small arguments and we argue that the correct criterion to select the solution was that z should be regular for small arguments and you can show that this is actually necessary in particular for theta minus 2h negative because if you don't have this condition rho which is the difference between sigma pi and sigma will diverge for small arguments ok so this is a good criterion and you can see here that theta should be between these two curves so in between 0.27 and 0.3 and you can also check directly the numeric simulations where we have observed this process in the future of the SPT so these are the simulation you measure the covariance and this is the dose that you get which is that in between these two curves it's consistent alright so at the end what you get for the persistent system are these curves so for point interfaces the first part is the same one as in Kruger and collaborators paper it's an interface that is flat initially with different earth exponent h and you see that our theory the black line is in good argument with the simulation in blue and the second part is a fixed h and it should be divided by 8 for different initial temperature and you see that once again you have a good argument and so these are non perturbative results but you can also have perturbative results around burning motion and you get this formula so it looks a bit complicated but the first part actually 1-h is what you get if you have a stationary process so if you have no aging in the in the process this is much a result if the second part here actually you see that it is consistent but Kruger and collaborators get for t equals 0 which is absolutely not obvious because the methodology is totally different but you can recover this formula and you get also the second order term so these are for the other kind of process so it's exactly the same it works very well so it's a small difference between the two is that now the difference between c time 1-h only appears at second order in h minus 1 half so these we have tried for all possible parameters and you see with the theory and the stimulation and you see that the argument is really good except in this small box here for which actually I've told you before that maybe the criterion to set the execution is not a good one so to conclude first you have seen that transit aging naturally arise when you prepare a non Markovian process and this can affect deeply the first passage term statistics link to the way the process in the future of the FPT converges to the initial process and we have developed a non participative scheme to compute this theta so I want to thank my collaborator so Thomas Guérin with you and his PhD student Tony Mendez and also my previous advisor in Paris Olivier Benichu and Raphael Vatery and I thank you all for your attention thank you very much for this nice talk you most likely already said it but as far as I understand you take an infinite system yeah so what does T equal 1 mean I mean why does this relax in a finite time the big T was the temperature if you look at the relaxation time of the system you say it doesn't matter which initial condition you take it doesn't matter actually but doesn't the relaxation time of this membrane diverge with the length the membrane is infinite so it has an infinite let's assume now I prepare an initial condition where I put my membrane with two gaussians separated by some delta with this initial condition there is no gaussian distribution at any time and you cannot describe a system with a single variance so how does how do this apply as far as I understand you said you can treat general initial conditions so I need to have a Gaussian process at the end so we need to start with a gaussian Gaussian and scaling variance okay so the initial condition must be gaussian okay then and the other thing because this is so understand that if there is no relaxation time then you never forget the initial condition yeah so this is the message yeah okay thanks oh it's fine yeah that's true are there any other questions or online okay a brief question so I see you use this process Y which has no drift right so it's the expected value in the future is the same as the initial value so this is like a martingale process do you are you familiar with this because the fact that Y is equal to consult a Y by the expected value of Y without constraints equals to its initial value this appears a lot in martingale theory I don't know if you know you're aware of this I assumed that the meaning is constant but there is no drift after T so if I understand correctly the martingale would say that if you condition the evolution on T at time T plus tau the value would be T so not at zero because he says it's at zero maybe this is a corollary of the general no I was wondering if you made any connection with martingales in terms of this stopping time theorem no I don't maybe it's not true are there any more questions then let's thank you again