 beyond, I guess. And our first speaker. Recording in progress. Adal Thierry from the University of Paris-Saclay. Cité. Cité, sorry. We're going to tell us about dynamical mean field. Adal, whenever you want. Thank you for the introduction. And thank you for the invitation to this amazing conference. So today I would like to discuss dynamical mean field theory formally starting with glassy system and discussing, in the end of my talk, also application to ecology and inference. So the study of mean field models has been instrumentally revealing the appearance of many metastable states. Slow relaxation and aging has hallmarks of glassiness. The first analysis of the dynamics was pioneered by St. Polinsky and Zipelius in the 80s, mostly focused, however, on equilibrium properties. And then later on, thanks to Culliandolo and Courchan France, Mésard, Dien, Les Dussals, Parisie, the analysis of aging was also pointed out in specific models, in specific subset of mean field spin glasses and mean field models. So these such dynamical treatments together with the Thaulis Anderson and Palmer approach that is very well known also in machine learning and computer science and the computation of critical points of the given index that can be done using the cat's rice approximation that was presented yesterday, both in its annealed and quenchant version, has allowed us to distinguish two main classes of disorder landscapes. So on one end, spin glass landscapes, which are characterized by a sub-exponential number of free energy minima, which are separated by substancy barriers, convergence of one time observable to the equilibrium thermodynamic limit. That means that if you look, for instance, at the magnetization or the energy at asymptotically, this quantity will converge to the equilibrium value. And on top of that, the dynamical transition temperature at which there is a dynamical arrest and the static or thermodynamic transition temperature at which the configurational entropy exactly abruptly goes to zero, coincide in this kind of models. Then there exists another class of landscape which belong to, which characterize simple structural glass landscapes, which are characterized by completely different features. First of all, an exponential number of free energy minima, which are now separated by extensive barriers, no convergence of one time observable to the equilibrium value. On the other hand, what happens is that the long-time dynamics tend to explore, to wander across the most numerous and marginally stable states that are called the Dingergon threshold states and that the variance with the first class of models, the two temperatures, so the dynamical transition temperature and the static transition temperature, typically in this model, do not coincide, but tend to occur over well-separated temperature ranges. So what I would like to present in the following is a dynamical mean filter approach that could allow us to complement this static version of the, and it could allow us to complement the analysis of the landscape. So first of all, I should say that dynamical mean filter is rooted in condensed matter and systems of strongly correlated electrons with potential and many application also to high-temperature superconductors. But more recently, thanks to recent developments in many different fields, so this dynamical mean filter approach has been applied to many other and various contexts that involve, for instance, the analysis of amorphous systems and the analysis of their rheological properties, ecology and evolution, and inference. So what are the basics of dynamical mean filter theory? So first of all, you need to identify the correct, the relevant degree of freedom and to treat all the rest as a thermal bath in such a way to write down the Hamiltonian of your system as two different contributions. So the first one is the Hamiltonian specifically of your system or your relevant degree of freedom plus the Hamiltonian of the environment that will play the role of a thermal bath that can be mathematically formally modulated by an ensemble of harmonic oscillators or phonons. And then you have the Hamiltonian that takes the interaction between your system and the environment into account. And then as a second step, you can consider the bath to be statistical equivalent, statistical identical to single out the degree of freedom. So given these mathematical details, given these premises, you can then study and can then tackle the equilibrium dynamics, which is quite simple, in the sense that here, response and correlation are related one to the other by fluctuation-dissipation theorem. And you can also study aging dynamics, so you can try to study aging dynamics, which is a big deal in the sense that now, the response and the correlation are not related by fluctuation-dissipation theorem, but they are extremely slow function. And on top of that, the thermal bath is aging with the rest of the system. So what you can do in this case. So the dynamical mean filter theory question simplify considerably, dramatically, and can be written in terms of a closest set of the integral differential equation just for a very narrow class of models that include, notably, the spherical P-spin model, which would be greater than 2. The truncated Schering-Tonky-Patrick model, or soft-spin version of the Schering-Tonky-Patrick model that was first studied by St. Pauliński and Zipelius and then, later on, by Bushock, Ulland, and also the problem of a particle embedded in a high-dimensional random manifold. So here, I'm referring always to the classical version of this model, but clearly, there is also the quantum analog of this model, so they have been studied a lot also in their quantum version. So what is the main purpose of today's talk? So what I would like to do is to present a new approach for retaining dynamical mean filter equation and to study the asymptotic dynamic of this model and also to go beyond these three simple listed case to basically obtain a very general formalism, a very general framework that could apply to whatever models where dynamical mean filter holds and without explicitly solving the equation on the correlation and the response functions at two times. So the starting point is the dynamics for soft degree of freedom, this SI. So the evolution of this soft degree of freedom is described by a Langevin equation, where B is a soft potential. And then the interacting part of your Hamiltonian is described by this term, where J and J are quenched random variables that are extracted from a Gaussian distribution, this one. And you also consider a wide noise, which is a Gaussian random variable with zero mean and covariance delta correlated in time. So using a filter approach, so a Martin-Sieger-Rose path integral formulation, you can basically integrate out these quenched random variables, this J and J, and you can end up with a mean field treatment, so a self-consistent stochastic process for now for the single degree of freedom. So you move from a many-body problem to a problem in terms of the single degree of freedom, S of t, that depends now on this friction term and a noise, which is no longer delta correlated in time. So the price to pay for integrating out these quenched random variables is that you end up with a memory kernel plus a noise that contains two different contributions. So if you look at the covariance of this noise, now it contains a first contribution, which is for the TTI regime, for the time translational invariant part of your dynamics, and another part that can be written in terms of the correlation to the power p minus one that accounts for the interaction with the rest of the system, and basically described, as you can see here, it has a dependence on two times k and t prime, so it described escaping from the TTI regime. So what we did is basically to use a sharp timescale separation between the TTI regime and the auto-equilibrium part of the dynamics. So you can basically split your correlation function as a function of two different times, t and t prime. In the first part, qd minus q1 times the scaling function where the quantity to be over the one is the difference t minus t prime, and so basically this first part describes the decay from the starting point to the plateau, whose height is q1, and now we have a second function, f2, which satisfy time re-parameterization properties where the quantity to be over the one is now the ratio t over t prime, and so this second scaling function basically describes the escaping, the way out from the plateau value and the auto-equilibrium dynamics. So by making use of the same timescale separation, we could rewrite this friction term which takes two different contributions. So the first one for the TTI regime where I remind you fluctuation dissipation theorem holds, and another contribution for the aging part of the dynamics. And so since fluctuation dissipation theorem holds here and we have a clear relationship between response and correlation, we can basically integrate by parts this term in yellow, and we get these two different contributions. So this one is just a boundary term, and this other one takes basically the contribution of the correlation function to the power p minus one times the derivative of this degree of freedom, S of t. And so we can exactly repeat the same protocol on the noise-noise covariance, and we end up with the self-consistent stochastic process for the single degree of freedom that depends now on this friction term, a color and noise that again can be split in two different contribution, and the slowing-evolving effective field, H of t, that embeds all contribution for the slowing-evolving part of the dynamics. So this is the first result, but the other remarkable result is that this new stochastic process can be mapped, can be associated to a quasi-stationary probability distribution, which is nothing but that the Boltzmann Gibbs distribution at a given fixed temperature t. So the main difference with an equilibrium approach is that now we have this calligraphic with this effective potential that takes different contribution into account, so the original potential, but also the contribution for the slowing-evolving part of the dynamics, H of t. So this result is perfectly in line with a similar approach, a similar strategy that was performed by Cugliandolo and Kirchhoff for analyzing the motion of a particle embedded in a random potential and coupled with two different bars at two different temperatures. And so since we also know P of H, so the distribution of the slowing-evolving field, we basically have all the ingredients, all the tools to obtain P of S, the full probability distribution, integrate our degree of freedom over this probability distribution and obtain the parameter Qd and Q1. So as I said, Qd and Q1 correspond to the starting point of the correlation function, so C at the same time, t and t, and Q1 is the decay of the correlation function, so the high of the plethora in this long-time-limit scenario. But in a replica approach, so in a static formalities, this Qd and Q1 can be rephrased in terms of the size of the deepest and the largest basin of attraction. So if you want to rephrase this purely dynamical picture in terms of the landscape structure, they basically give information on the overlap parameters in your Parisi matrix. And so now that we have a complete characterization of the dynamical process, we can think about what happens in the, we can wonder what happens in the aging regime. And so in this first part, I would like to characterize aging dynamics for models that are characterized by just once-low timescales or for people who are familiar with this other system, this corresponds to models that are framed within a one-step replica symmetry breaking. So as I said, we split our dynamics into different contributions. So the first part of the dynamics were the degree of freedom, fastly equilibrate, and the slow part of the dynamics in which there is a violation of fluctuation dissipation theorem and the emergence of auto-equilibrium feature. So for describing the aging regime in models that are characterized by just once-low timescale, the physical requirement is that the dynamics in the TTI regime has to be marginal. So this means that the relaxation of the response or the correlation function cannot occur in an exponential way, but according to a power low decay. So we use a perturbative diagrammatic approach to rewrite the stochastic process for the single degree of freedom, where these are zero as much simpler expression in Fourier domain. So we express this response function in terms of omega in Fourier domain. And so this response function in the TTI regime admits a Schwinger-Dyson equation. So it can be written as the inverse of this Bayer propagator that is graphically represented by a straight line, plus another contribution which is called self-energy and basically accounts for all one particle irreducible diagram. So it basically accounts for the diagrammatic perturbation in the couplings that are due to this term. And so now the condition embedding, so accounting for the appearance of a marginally stable dynamics, is that the inverse of, so the derivative with respect to omega of this inverse response function evaluated in the small omega limit has to be divergent, has to be infinite. And so if we look into the tails at the expression of this inverse, at this derivative of the inverse response function, we get an exact expression where the numerator does not matter because it's not singular at all levels in perturbation theorem, at all orders in perturbation theory, but the leading contribution is given by this denominator. So when this denominator touches zero, we have a condition for the appearance of a marginal dynamics. And so again, what you can do in the statics, you can write the action matrix of your free energy, you can compute the second derivative of your free energy that correspond to your stability matrix, you can diagonalize your action matrix, and you could look at the smallest gain value of your matrix that gives information on the criticality of the emerging phases. And so if you do exactly the same by using for instance a replica approach, you will see that this denominator in a purely dynamical computation precisely coincide, precisely correspond to a vanishing gain value in the stability matrix. So the appearance of a zero model in your stability matrix. But in this case, we have a more general approach in the sense that we can perform this dynamical computation for whatever models where dynamical means theory apply, and we don't need to define a free energy or an Hamiltonian. So this is specifically for models that are characterized by a one step replica symmetry breaking, so just one slow timescales. Now we can wonder what happens when the model is characterized by an infinite sequence and infinite hierarchy of slower and slower timescales which in jargon correspond to a full RSV picture. And so the different universality class to which this model belongs. So basically this described the case for instance of the Heising-Pispine with P equal to two, also called the Sharon-Tonkir-Patrick model. So the different universality class to which this model belong lead us to introduce a new rule, a new criterion for studying the dynamics, and also lead us lead to a completely different phenomenology in terms of aging and in terms of resulting features. So we need now a new rule, a new criterion that could encode this infinite hierarchy of timescale and therefore this infinite sequence of effective temperature. So as you can see here, we don't have any more a finite number of Q value of overlap parameters, but we have a continuous function Q of X where X is the violation parameter that described the ratio between T and T effective and the effective temperature. And so we have to describe, we have to find a good criterion for describing this full RSV scenario, this full RSV picture. So what we did is to study in detail H of T, so this low-ing evolving effective field that contains a first noise term for the aging part of the dynamics at the integrating response times M that can play so for, can play basically the role of a magnetization so an average spin. And so we look in this model to the variation of this low-ing evolving field in two subsequent, in two close time sectors. So we compute this variation, delta H of T, specifically a time T plus delta P, T and a time T. And so this variation between two subsequent sector takes contribution from three different terms. So we have a first term, which is the evolution which describes the evolution of the stochastic slow noise. Another term that multiply, that basically multiply your response function evaluated at the same time, T and T, times the magnetization which provides an instantaneous snapshot of your system plus this other term which is the derivative of the slow-ing evolving part of your response integrated over all previous time. And so this term you can see that immediately drops out because it provides us a bleeding contribution compared with the first two terms. And so we end up with this delta H of T which has the form of a stochastic process with a drift term and the diffusion term that have to be determined. And so to determine in detail this drift and this diffusion term, we basically use a generalization of fluctuation dissipation theorem. So we could rewrite this drift contribution in terms of this violation parameter X that I remind you since we are describing the full RSV picture, these X is a continuous variable ranging from zero to one. So we rewrote basically this drift contribution in terms of a generalized fluctuation dissipation theorem. And we notice is that the noise covariance is nothing but the different correlation, the different contribution of the correlation function in different time sectors. And now the turning point to close the equation is to notice that at a long time limit there is this one-to-one mapping, a one-to-one relationship between each time scale T and this parameter X. And so this property translates into the fact that the correlation in different time sector is can be related, can be mapped to a continuous function Q of X that is exactly the function that I showed before in the full RSV scenario. So this Q of X describes the intrastate overlaps reached at the largest time. And so by gathering all the contribution together we end up with the distribution of this low-ing evolving field H, whose evolution is now measured in terms of an effective temperature change. So we are no longer expressing this evolution in terms of the time scale, but in terms of this X, this violation parameter or effective temperature change. And so for people who are familiar with the replica computation, this is exactly the same stochastic Langevin equation that was found by using a thermodynamic approach, according to Parisi's solution and according to the emergence of a hierarchical cluster, hierarchical organization of the different equilibria in the landscape structure. So with the purely dynamical formalism, we managed to get exactly the same stochastic equation that was found in the 80s by Parisi and coworkers. And so now that we have a completely general scenario, I would like to describe, given the topic of this conference, I would like to describe also interdisciplinary application. So how this formalism can be applied, for instance, to inference, problem and ecology. So in inference problem, you might be interested in, for instance, in identifying the condition for which your gradient flow has a positive correlation with the signal. You can, for instance, define a loss function like this, which is a trivialization or a generalization of the perceptron model, as you prefer to think about, where x star is the signal and xi is the pattern, so the weight. And so what happens in terms of the landscape? So there are regions of the landscape that are quite easy in the sense that the landscape is just characterized by one single minimum or there are minima that cannot trap the dynamics and there are other regions of your landscape that are characterized by many, many spurious minima that can basically trap the dynamics and making it slower and slower. And so in the previous part of my talk, I was interested in finding the condition for aging. So what is the condition for finding the effective temperature and the out of equilibrium dynamics? Here, a natural question that one can ask is, how to derive the algorithmic threshold for the gradient flow? And another interdisciplinary application I'm working on right now is the critical ecosystem. So the description, the correct description of critical ecosystem, not only using static formalism, like the cavity method or the replica computation, but also dynamical mean filter approach. And so one of the model, one of the reference model that has been mostly used in this field is the loft-cavolterra model that was introduced at one century ago, but now it's... Go for Libyan or Lizer or Liby. Now it's... It's okay for them. Yes. Is it okay? Okay. So one of the mostly used models in theoretical ecologies, the loft-cavolterra models, so this equation I've been introduced in one century ago, but more recently I've been studied also in their random version. So the idea is that you start from dynamical equation for the relative species Abundances Ni, where I now is an index that run from one to S, and this is the total number of species in your species pool. So this evolution equation is basically described in terms of a self-regulation term that can be written in terms of a gradient of a single species potential, plus a cross-regulation term that can be written in terms of this coefficient, alpha ij. And then to complexify a little bit the scenario, you can also introduce a demographic noise that scale as the square root of Ni, plus an immigration parameter that basically couples the A-land to the main land of your model. And so the main assumption to get an exactly solvable model is, for instance, to study from a well-mixed community where you don't have space dependence to treat this demographic noise in terms of a Gaussian-wide noise with zero mean and covariance, again, delta correlated in time. But another important assumption, so last but not least, is that to tackle into account the complexity of your ecosystem, you can basically assume that this interaction, alpha ij, are random and, for instance, are extracted from a given distribution with a given mean and a given covariance. And so by performing exactly the same protocol that I mentioned before based on a time-scale separation, you can end up with a dynamical mean-field theory formalized for the single species and dot, where now you have mu and sigma, which are the mean and the variance of this, the mean and the start and deviation of this random matrix. And you get also these friction terms that as you can see, it takes into account also gamma that describe the degree of symmetry of a symmetry between your coupling. So it's a very, very general approach, according to which you can perform, you can obtain the different phases of your phase diagram and you can also go beyond the replica computation because you don't need, according to this dynamical mean-field theory formalism, you don't need to define a free energy or an Hamiltonian operator. So to conclude, I presented here a new approach that we did in collaboration with Giulio Virali and Chiara Camarota to deal with cases where you cannot close the equation on the correlation and the response functions at two time. And so basically you don't have the tool to obtain a closed set of integral differential equation. So I define in this way a self-consistence stochastic process for the single degree of freedom, which is coupled to a terminal path and a determined two different criteria for obtaining the condition for marginal stability, models that are described first by just one's low time scales so that are framed within a one RSB picture and models that are characterized by an infinite hierarchy and infinite sequences of time scales, lower and slower time scales. And so as you could see, this new effective stochastic process can also be mapped in quasi-stationary probability distribution and according to this mapping you can obtain all parameters and the whole main quantity of interest for your models. Then there are other open question that one can try to answer. And so time is running out. So I would like to thank you for your kind attention. Any question? Okay, so I'll start with mine. I would like to know, you mentioned at the beginning the limitations of this kind of formalism. What are the key components of a model for this type of formalism to apply or not to apply? Okay, so an important limitation to apply so on this model is that you have to perform an analysis in the long time limit in such a way to basically use a sharp time scale separation between a faster regime in which a fast degree of freedom, fastly equilibrate and the slower relaxation dynamics. So first of all, long time limit. So long time limit is one of the condition. And another important property that we use is that this noise is Gaussian in such a way to use Novikov theorem and particular theorems from mathematics. You can try to generalize also to other kind of noise. Clearly it's a bit more involved and a bit more complicated but you can do it in in perturbation theory. And another thing, another important assumption is that we assume that there exists weak ergodicity breaking. So I'm not describing complex scenario like mix at the spin or stronger ergodicity breaking scenario. So in the weaker ergodicity breaking scenario what you are assuming is that at a finite, at a long but finite waiting time, your correlation function eventually decays to zero. So this means that basically you are losing memory of your configuration that are reached at a finite time. So every configuration and in configuration that's reached in a finite time will be completely forgotten because of this weaker ergodicity breaking. So for stronger ergodicity breaking is much more difficult in the sense that you should keep memory of your previous configuration and so it's difficult to generalize in this case. There is a question by Jean-Christophe which is... No, they are... Okay, so... Can you read the question? Is there... Okay, aren't these equations derived for the spherical models? Is there no spherical symmetry plus minus one? You can still do all of these. Okay, thanks for the question. So here I'm starting from the simplest possible scenario where these SI are soft spin, are just soft spin which are subject to a soft potential that can be whatever. So it's not necessarily forced to be on the sphere. So what we did is basically to study also the... As I said, so we also studied the Ising model would be equal to two that correspond to the Sharon-Toky-Patrick model. To do this you can try to use a sort of truncated SK model version that means that you assume to have a double well potential with a given roughness parameter and in the end you will send this roughness parameter to infinity in such a way to recover the discrete spin version. So this can be generalized. So you start from, let's say, a double well potential and then you can recover the equation for the discrete case. And so this is basically what I discussed in the second part of my talk when I mentioned a full RSV picture. Any other question? So your solution is still on the... needs a thermodynamic limit, so can you... Yes, yes, it's in the thermo... So as I mentioned, so we are assuming a long time dynamics and also an infinite degree of freedom, yes. Yeah, so my question is about the use of that in the Lotka-Volterra model. Can you think of a system that actually has... What we are trying... Not infinite, but a lot of degrees of freedom. Okay, what we are trying to do now on the Lotka-Volterra model is to reproduce meta-community scenario in the sense that we are introducing space and so we are assuming different communities that interact according to diffusion and migration parameter between different highland. And so in this way, one can model also activity front as a function of a finite number of species. Does it work if you have this spatial structure? Sorry? If there's no random interaction between everything, there's some spatial geometrical stuff. So you have random interaction between your species, but your model system, instead of having a well-mixed community, you are modeling meta-community scenario. So you have different highland that interact. But it's still interaction with no lens scale? Sorry? You can also assume that the interaction of your variables just decay on a given range. So you can use a sort of cat's limit. That means that your interaction variables, your alpha ij, just decay on a given finite range. So they are not long range. They are not cushioned by decay on a finite range. Then we hold for short range interactions because here it's looking more like an SK. We are trying to work on this. So I don't have results yet in this direction, but we are working on this. Thanks. All right. If there are no further urgent questions, we'll think again about that. OK, there is an urgent question there. Yeah, sorry. So I think it's going to be interesting to talk. And I'm far into these topics. So what generates the long time scales in these kinds of models? Just the interactions are so complex. Is that the intuition to have? Sorry, can you repeat? What generates the long time scale? What is the? What generates the long time scales here? So it's just an assumption. So we assume that we are looking at the dynamics at very long time scale. And so it's an assumption that was firstly used by Cuyandola and Kurchan in this PISPIN model in such a way, basically, to treat, to perform a mean field approach. OK, so there are two different strategies. You can assume that you have an infinite degree of freedom, or you can either assume that you have an infinite degree of freedom, and so you can write your mean field equation, or you can assume to be at infinite times. So here we are assuming, so we are assuming both hypotheses. We are dealing with both assumptions. So long times that allow us to simplify this dynamical equation, because otherwise I cannot use this sharp timescale separation. I will get just the first part of the dynamics at short timescale. And so I would get correction that are not simple to tackle with, to tackle. OK, OK, thank you. All right, thank you very much, Ada, again.