 Adriano, thanks to Organizer for inviting me. So today I will talk about some recent result that we have obtained in collaboration with Diego Alberici, Pierluigi Contucci from Bologna, and Adriano Barra from Salento University. So if you want to find some details, there are two papers on the archive. Here is the index. So basically, here is the outline of the talk. I will introduce what I mean for deep enough. And I will show how to control the annealed region and then give also a replica symmetric bound. So there are related words on this kind of model, in the case of bipartite models, when we have only two layers. The first work is by Adriano Barra, Giuseppe Ginovese, and Guerra. This is the first work, I guess, on the spin glass in bipartite setting. Then there is a word from our finger and chin in the spherical case that actually was completely solved in 2017 by Beik and Di, using random matrix theory. And I want to also mention that in the Bayesian, what we call Bayesian transsecting, you can completely solve the model using the technique introduced by Sean Barbier-Macris and Milan. So we will talk about deep, sometimes called restricted Boltzmann machine, introduced by Impe and Salo Godinov in 2009. And we will focus mainly on the deep architecture of this model that we can think as a linear chain of Boltzmann machine. And we look at the model from the point of view of spin glass theory. So we fixed the coupling to be pointed and tried to understand the phase diagram, basically. So what I will show is a complete characterization of the annealic region and the replicasimetric bound. So here's the definition of the model. So you take N-capital and spins, and you arrange them in K-capital K layers. The size of the layer P is N-P. The capital N is the parameter that controls the thermodynamic limit. And we assume that the relative density here, as I will define the thermodynamic limit with lambda P. So clearly, you have this condition for the lambda P, which is the sum of lambda P equal to 1. So basically, lambda P belongs to some k-dimensional simplex. So Jij is the coupling between the spin i and j. And we assume that this is a standard Gaussian random variable. And we can also imagine to have some external field acting on the layer P. We have no special requirement on the distribution of the random field. Only we have absolute, final absolute moment. And then we can also, let's say, fix some parameter beta beta, which is a vector, that basically it's the variance. You can think of beta P as the variance of this Jij. Maybe a picture is more clear. So you have this k-layer. And each spin, sorry, the interaction is allowed only between spin from one layer to the next one. So we not allow interaction between the same layer. OK? So the Hamiltonian is defined in this way. So given the configuration of spin is the sum of all this contribution, which basically is the interaction between the layer P and the layer P plus 1. The factor 2, it's only for math convenience, as you can imagine. And you see here the role of beta P. You can think of beta P as like a local temperature for the two consecutive layer, OK? So a central object in a spin glass theory and for this model in general is the overlap. Here in this case, you have several kinds of spin because you can identify a spin with respect to the layer that it belongs. So you have for any P, which is the number of layer, you have the overlap of the system, which is the normalization 1 over NP sum over all the side in the layer, sigma i tau i, where sigma and tau are two configurations of spins, OK? So the covariance matrix of the Gaussian process with the Hamiltonian you can think as Gaussian process is basically a quadratic form. It's n times, so it's extensive in the volume, which is a good thing, times this object with order 1 and this is a quadratic form built up by defining a vector of the overlap, which basically each component is the overlap associated to the P layer. And the quadratic form is associated this matrix M1. And then 1 is defined in this way, where lambda is the vector of the relative density and the matrix M0 is this one. So behind, let's say, the main characteristics of this matrix is that it's three diagonal and this is exactly the reason why we can solve, we can say something, let's say, about the annealant and the replica symmetric regime. So if you think in the case of k equal to, which corresponds to the bipartite case, you simply have to look at this little minor here and this corresponds to the bipartite case. So OK. As usual, the statistical mechanics, the first thing to do is to look at the partition function and even more interesting, maybe, it's the coincident pressure, which is the 1 over n, the expectation over all the jj's of the logarithm of the partition function. So here, the partition function is the exponential of the Hamiltonian. We keep the action of the magnetic field in the measure, if you like, only for convenience, of course. Anyway, we are interested to this object. This is a spin-glass model and the good news and also the bad news, it's not easy to treat with classical spin-glass technique. What I mean is that if you try to use interpolation, let's say, for example, to prove the thermodynamic limit because of this structure of the matrix interaction, then you have some problem because you see that this matrix, for example, in the case k equals 2, is not positive defined. So we have some issues if you try to use the interpolation method in the usual way. Nevertheless, we can still do something. And the idea is that, OK, usually in the interpolation, if you think of the Schrodinger-Farcky model, you compare the two-body interaction, which is the characteristic of the Schrodinger-Farcky model, with some one-body interaction that you can tune a little bit and play. Here, we do something a little bit different. We compare the pressure of our model with some convex combination of SK model with this symbol, pskn. I denote the pressure pressure of a Schrodinger-Farcky model of n particle, where the temperature of this Schrodinger-Farcky model depends on a parameter. One, so there are positive numbers plus something which is a function of a. So a is my free parameter that I can tune for my convenience later. But you fix a and you define this object, which is basically this part, which is a convex combination of SK models plus something. Here, the temperature of the SK is given by this function theta p of a, which is defined below. What is the point in defining this object? Actually, you can prove that this object for any choice of your a free parameter gives a lower bound for the pressure of the deep-molson machine. OK, just an idea to how to prove this inequality. You do an interpolation between your model, the deep-molson machine, with k layers. And then you take k independent Schrodinger-Farcky models, then you do interpolation, and you can control the derivative observing that this object is always positive. OK, clearly you have to play a little bit with constant, but it's not hard to show this inequality. And this, basically, is the starting point of the work. Because the idea is that, OK, since now we have, let's say, a good control of what happened to the pressure of the SK model, so we can control the left-hand side by controlling the right-hand side. So in particular, you can choose a to infer something about the pressure of the deep-molson machine. So the first thing that you can do is to completely characterize the annealed region in this way. So first, you set the external field equal to 0. And just the definition of the annealed pressure, instead of taking the expectation of the logarithm, you take the logarithm of the expectation of the partition function. This is the definition of annealed pressure, clearly because that is the sum of Gaussian, this can be computed exactly. Actually, it's independent on n. And that's this expression. And also, by Jensen inequality, you have that this object is always greater than the pressure of the model. Notice that here, I put the limb soup instead of the limb, because actually, we are not able to prove the existence of the limb of this object. The reason is the same that I just mentioned before, because the matrix that tuned the overlap, the covariance matrix of the process is not positive defined. So you cannot apply a logarithm in the argument, something like that. So we say that the system is in the annealed regime if the pressure of the model, the quench of the pressure of the model equals the annealed one. This is just the definition. And as I mentioned, we have to exploit the control of the annealed regime of the SK model. Because if you remember here, in the right hand side of this inequality, there is a convex sum of SK models. So just to remember what happened for the SK model, there is this result, which I guess it's the first rigorous result on the SK model due to Eisenman, Lebowitz, and Ruell, which is the pressure, the quenchate pressure of the SK model equals the annealed one for beta small enough. So for right temperature regime, this is the result for the SK model. Then you can use it to control the right hand side of this inequality. Notice that the annealed pressure is always an upper bound. The annealed pressure is always an upper bound for PDBM. Here, you have the reverse inequality. So it's reasonable to opt for something. In fact, what you can obtain is that if you consider the system of inequality and you define this subset of the space parameter, which is basically all the beta lambda, remember, is the collection of local temperature. Lambda is the relative density, which belongs to some simplex. Such that there is a solution for this system. So this is my set capital A. And I did not buy the bar capital A. It's the topological closure of AK. What you can prove, if you take a point in the phase diagram beta lambda that belongs to this set, then you have a quality between quenchate and annealed pressure. Lale, sorry if I interrupt you. You have just roughly a minute. I have just roughly a minute. You are out of time. OK, so very quickly, this condition for AK it's very implicit because you see it's a mess, if you like. Anyway, there is an interesting connection between this region, AK, and some matching polynomial that comes from monomer dimer models. So basically, the condition that beta lambda belongs to AK is equivalent to some other condition that are more explicit in the parameter beta lambda. So again, exploiting the fact that now we have a good control on the SK model, we can use the previous bound to obtain a replica to obtain, sorry, a lower bound, which is given in terms of a replica symmetric functional. But the point is that this bound is not totally known for all the choice of the parameter beta lambda H only, but only in a suitable region that you can determine. This suitable region is basically the analogous of the Almeida-Tarres condition for the SK model. So very quickly, I go in the definition of the replica symmetric function. So this is the function that you have to look if you're OK. This is the replica symmetric function that you obtain simply comparing your model with some one-body interaction. And the stationary condition is this one recently has been proved by Genovese that actually this condition is obtained by a min-max procedure. And the main theorem is that under some condition on the parameter beta lambda H of the phase diagram, this replica symmetric function is a lower bound for the thermodynamic limit of the pressure. So I can stop here if I have, because actually the last two slides are only corollary of this main theorem. Because again, this is not so explicit, but if you work a little bit, you can find condition that are more explicit in the space of the parameter. So that's all. OK, thanks a lot.