 All right, can you hear me? Cool. Okay, so I'll give a second class on this fancy or funny such defecative model or models. Now, let me very briefly remind you what I said last time, and then I'll ask you if there are any questions about it. So the model, at least in its simplest incarnation, is four fermions. Again, they can be myeranas or complex fermions. That doesn't matter too much. And there is a matrix element with four indices here, and they just out of desperation assume that it's completely random. Namely, it's average is zero, and then second moment is given by the scale J. And as I tried to argue last time, the good idea is to scale it with number of fermions to the third power. So number of fermions is here. Then what we showed, that's a pretty simple algebra, that you can encode partition function or any correlation functions of this model in this type of the field theory, and this field theory will depend on two matrix fields. One plays a role of a green function and the second plays a role of a self-energy. And then the capital N, number of fermions, appears solely in this place. Namely, it's place the role of inverse plant constant, which means that N to infinity limit is relatively simple. It is a classical limit of this field theory, which is given by extrema equations for our action in terms of two fields. And these equations immediately can be read out to look like this, which is a Dyson equation with a particular form of the self-energy. So what this system of equation does, it sums up all diagrams like this, and the claim is that in N to infinity limit, that's an exact procedure. Now these equations are still not completely simple, but of course much simpler than initial theory. So something can be done. And the first thing to pay attention is that if they bravely put time derivative to zero, then equations become solvable, or almost exactly solvable. And what you find is that the green function behaves like one over time to the power one half, and corresponding self-energy, it has three green functions. So it behaves like one over time to the power three half, which being after Fourier transform, lead to self-energy as function of epsilon, being square root of epsilon, which is for sufficiently small epsilon is larger than epsilon itself, which justify these assumptions, which we did to begin with out of the blue. And with these assumptions, what you find is that your green function is function of energy, behave like one over square root of epsilon, and that's the sort of first basic fact about the model. That at certain energy scales, your green function behaves not as a thermiliquid one, but in a completely different fashion, right? So now I will start sort of from here and go on, but first let me see if there are any questions, suggestions. No, right, so first of all, already on this level you can start playing games and you can invent slightly more complicated models and using these very facts obtain some not completely trivial results. So one example which I like sort of most is given here. So this is a paper from a year ago. I don't know, I took it from an archive, but it's probably published already by Leon Balans and his collaborators. So what they did, they took this S-Y-K model, but now they put it in a d-dimensional array of dots. So you imagine that you have a array of dots and each dot is represented by this S-Y-K model. Okay, so on each dot you have precisely what we have been discussing here. They use complex fermions because they want particle conservation to discuss electron transport. And then what they add is a hopping from one dot x and x prime labels, label dots. So they add a weak hopping, so T supposed to be also random and in some sense small. And then what they discuss, let's say electrical conductivity of such array. And what they find is that in a very wide range of temperatures, electrical resistivity behave like temperature. Which sort of you can start speculating that it looks very suspiciously similar to high T-C superconductivity and other strongly correlated materials. Now all what goes into this result is essentially this green function, nothing else. So essentially all what they do, they calculate conductivity, which is given by cubo formula. So there is this diagram. In each vertex of this diagram you have your hopping, T. If you wish, this is a velocity of your electrons. And then you substitute two green functions from here. From here, rather. And you have to be careful about which one. The target advanced Keldish, so standard technology. Basically everything is completely standard. And you got that conductivity behave like one over temperature. And frequency independent and behave like one over temperature. From here you immediately conclude that resistivity behave like temperature. And since then many other people played with games and slightly different versions of the model, but it's usually everything rotates around this observation. Nori, so now I want to go slightly farther than that. And basically I tell you that this is a good answer, but it's only good in a certain range of energies. And you have to be careful about end to infinity limit versus temperature versus energy. And so there are many more subtleties. Now, if you just calculate this green function numerically, yes, so that's a numerical calculation. It's quite easy. So once you have your Hamiltonian written as a big matrix two to the power n over two times two to the power n over two, you can numerically diagonalize it. I think this is done for n equal 34. And then you can calculate your green function of time. That will be sum of all eigenstates, many body eigenstates of your Hamiltonian, ground state chi i n square e to the minus e n tau. So that's a lemon representation for your green function in terms of exact eigenstates and exact energies of your model. So that's not a very difficult calculation. So what you see is green function is function of time. And what you see is that if time is not very long, it approaches this black asymptotic line. It's a logarithmic plot. And the slope of this black line is precisely minus one half. So what we find is this result. But now if time is a little bit longer and you see that the characteristic time is of order 10, maybe 20. And I will argue that time is a little bit longer. And I will argue that it's a fodder of n actually. Then the slope changes and it approaches a quite different slope, which I will argue is actually three half, not one half. So the low energy physics, long time physics of the model is also very interesting, but it's quite different from the naive mean field classical prediction. That's the first thing which I want to tell you. And then if time permit, I will try to tell you a few things about out of time order correlation functions, which people like to discuss in the context of this model, which are signatures of quantum chaos. And maybe I'll tell you a few words about black holes, but I don't know too much about it. Any questions? Well, it will have this T characteristic value of this T, of course, square divided by temperature. So pretty much that's it. Doesn't know about interactions. It only knows about hopping and temperature. Good question. Anything else? No, right. So to understand this low energy behavior, what I already started to explain you last time is that we need to appreciate that this solution is not unique. And there is a whole family of low energy of classical solutions, which have almost the same action. This idea goes by the name of reaparization invariance that stems from an observation that if instead of this green function, you will write it in a way one quarter. So basically, if we go from tau to some other function, f of tau, tau goes from zero to beta and f of tau, at least we can think about something like this. So if instead of thinking that time runs uniformly ticking like a clock, we change the pace of time and we call it f of tau and we try our green function in the form of this and corresponding endsets for self energy, we find that it still is an exact solution of this. Two equations, again assuming that this guy is zero. And that tells you that there is out of all these degrees of freedom which are encoded in two matrices and there are very many of them. There is a particular set of the degrees of freedom which is parameterized by these functions f of tau which forms a sort of low energy manifold of relevant degrees of freedom. You may of course worry that that's not the only one and you can look for others and there is no theorem that there are no others, so maybe you'll invent something but so far nobody come with anything but this. But this story, by the way, you can also worry about replica structure of this theory for those of you who like replicas, so there are actually AB and so far we are working on in replica diagonal sector. So you can ask question what if I have a replica of diagonal or maybe replica symmetry breaking and some exotic things like that. And indeed some people discuss it. My personal view is that in this very model there is no any replica symmetry breaking and I even have some numerical evidence for this. But if you make this, you change this model slightly, you deform it, then most probably you will start having some issues with replica symmetry breaking. So life may be much more interesting than what I'm telling you now but I can only tell you that much in two classes. So let's proceed with the free replica diagonal and the set of the degrees of freedom. So as we argued last time, the actual manifold of these degrees of freedom is a defiomorphism of unit circle on itself slash SL2R, which actually states that a group of Möbius transformation leaves these green functions completely invariant. So the next step is to say, okay, we want to calculate our green function let's say I want to calculate the green function G of tau. So I will perform the savagering but instead of integrating over all possible matrices G and sigma, I will only integrate over matrices which are parameterized by the set of functions F. Okay, so then I need to substitute this guys into my action and calculate what's the action in terms of reparameterization function F. Okay, now what we already said that the moment you put the tau to be zero, the action is not. It's a solution of the settle point equations. So I change F and action doesn't change. It's a completely zero mode of my action, provided I forget about this detail, right? So therefore I have to keep this detail so I have to recall what's my action. So S, if you remember, was trace logarithm of D tau minus sigma and then I had something like sigma G and then I have something like G square four. So I need to recall about this but I will think that this is small because I want to be at small energies. So I will expand in powers of D tau so you will get something like trace of D tau G D tau G plus one half in the second order perturbation theory and then something beyond it. So this is the most interesting term you can easily calculate it. So it's in principle it's non-local. It has two green functions and it's parameterized by this function F. So I just use this green function into here. So it will give me some non-local functional which I don't want to bother you with. If you somehow, yeah, what should I say? So if you think what this bubble is it will have a form of omega square log omega and you see that it's non-analytic function so in time space it's going to be non-local. But if you're brave and you say that logarithm is probably not a function I will somehow introduce some regularization of my logarithm then omega square is already an analytic function so the thing become local in time and then what you find is that your action looks like under base assumption. You find that your action looks like integral from zero to beta D tau and then there is a funny construction I want to tell you a little bit more about it. So you have a second derivative of your reparameterization function divided by the first derivative and everything is square. So luckily this business was picked up by people who do black holes and for people who do black holes apparently this is a bread and butter thing and they know it as a Schwarzen derivative. So there is a construction in complex analysis which is called Schwarzen derivative or Schwarzen notation is S of F of complex number Z and this thing is F double prime over F prime of Z prime minus one half square. Another notation for this thing is F Z. Can you see it? Okay, so this Schwartz is actually Hermann Schwartz it's not Isaac Schwartz which you may read books on string theories. So this Schwartz was living in the end of 19th century in Germany. So what is interesting about this object for the complex analysis, it's double prime is that if F is a Möbius transformation if F belong to this subgroup then this thing is exactly zero. So it basically measures how much your function F deviates from Möbius transformation. So what you observe is basically up to the full derivative which in our context is of very little interest the action which measures the cost of the given reparameterization is exactly given by this Schwartz and derivative. So it will be proportional to integral from zero to beta d tau F Z. So the way how string people arrive to this we basically say that let's take the simplest possible object which is invariant under this corset space subgroup of transformations. It's not a group. Now as I am trying to tell you here that you can indeed derive it but in the process of derivation you commit a little bit of the crime because you substitute non-local kernel by a local kernel. Yes, sure, yes. So that's the simplest one, right? That's a good question. I think no and you will sort of see why in a sec. But it's a usual logic that you take the most relevant from the renormalization group point of view invariant object and higher power of it are probably irrelevant and they are irrelevant. But that's of course a valid concern. All right, okay. So let me go on, yes. Well, the honest answer is that I don't know. I have some thoughts about it but we can talk but it will take me too far away. Yes, but by the way that's precisely the moment when black hole people start to be interested because black holes in two dimensions are known to be characterized by two-dimensional gravity is known to be reducible to one scalar function which you can call f and the action for this function is known to be precisely this. So in some sense this is already for people who are in the business and I'm not but for people who are in the business that's immediately ring a bell that should be related to two-dimensional gravity. Again, I may tell you a little bit more but not sure. Okay, so now this thing look mysterious but in fact it's actually very simple. So what you notice that your function f better be monotonic because otherwise our time will go back and forth that's probably not good. So if this is a monotonic function then what we understand is that f prime is larger than zero and since it's larger than zero let me just say that f prime of tau is e to the sum of our function, I can always do it. Okay, now in terms of this phi the action appears to be completely non-mysterious. It's just phi dot of tau square. So the simplest possible, the simplest possible I think you can imagine. So that's observation number one. Observation number two is that to perform our calculation I need an effective measure in terms of my reparameterization functions phi and this measure is nothing else but hard invariant measure on this manifold. And that's a small exercise in mathematical physics to derive invariant measure. I will not do it if somebody is interested I can explain privately. But the bottom line is that in terms of this function phi the measure appears to be just flat. It looks again mysterious and complicated in terms of f but in terms of phi it's just a flat measure. So therefore all what I end up with in calculation of my green function I will continue to here. It is an integral over d phi with an action which is phi dot square d tau. There is a certain coefficient let me call it m over two. And this m is proportional to capital N because everything is proportional to capital N. And then there are some factors which I don't want to bother myself. And finally there is this green function. Now green function I know what it is. It's written here. Now f prime is a good thing. It's e to the phi. So what I'm now calculating is e to the one quarter of phi of tau times e one quarter. I'm sorry, one quarter phi of tau prime. And the last unpleasant thing is this denominator. Now denominator, look it's f of tau minus f of tau prime. So you can notice immediately that f of tau minus f of tau prime is just an integral from tau up to tau prime of e to the phi of tau. Since e to the phi is f prime, I integrate f prime and I get this. So what is sitting in my denominator is this integral over tau to tau prime e to the phi of tau e tau to the power one half. So it's almost a manageable exercise in functional integration, but these denominators still looks a bit ugly. Fortunately, there is a trick due to Feynman which allow me to take denominator and put it in the exponent. And the trick, do I have it here? Yes, so if I have one over x to the power p, then what you know is that this is one of our gamma function integral from zero to infinity d alpha e to the minus alpha x alpha to the power p minus one. Almost trivial thing. So with help, what I'm finding is that I have this auxiliary integral over d alpha, one over gamma function, one half whatever it is, square root of alpha. And then there is my functional integral e to the, m over two phi dot square from zero to beta d tau. Then minus alpha integral over t to t prime e to the phi d tau. And finally, phi over four tau and phi over four tau prime. That's already completely familiar. Think now, look what I get. I get a Feynman path integral for a quantum mechanics of where the coordinate is called phi. And this is a quantum, this is a kinetic energy like x dot square. And this is a quantum mechanics in a potential which is exponential in my coordinate. Now, there is a slight caveat here is that this potential is switched on at time tau and switched off at time tau prime. So it exists on the finite amount of time and then it's not there before and after. And what I need to calculate is a correlation function of this exponent with co-efficient. One quarter of my coordinate at two moments of time. And these two moments are exactly the moments where my potential is switched on and off. So what we got now is that I have this coordinate phi and I'm thinking about quantum mechanics in a potential which is given by alpha e to the phi, right? So we actually know pretty much everything about this quantum mechanics. It even has a special name. It's called quantum, Liouvillean quantum mechanics. I'm not sure about my spelling, but sorry for that. So after French mathematician, Liouville, and again this quantum mechanics, this field theory with this potential has a lot to do with two-dimensional gravity. So what people realized back in the 80s is that two-dimensional gravity is very much given by field theory of that type. Again, that's sort of not completely surprising since I told you already that they realized that it's given by Schwartzen. But for condensed matter people, we don't have to know about it. We just can solve it brute force exactly. So you understand that if you have quantum mechanics with this potential, it has a continuous spectrum which is labeled by momentum at infinity. At infinity you just have a scattering wave. It's a free thing e to the i k phi and e to the minus i k phi lane waves going back and forth. So energy is labeled by this k and it's simply k square over two m. So that's just kinetic energy. And all that we need to do is to find wave functions. So let me see. Yes. The wave function psi which is labeled by quantum number k, it depends on coordinate phi and it's given by some modified basal function k of square root of m phi over two. Nevermind. So basically, corresponding Schrodinger equation can be transformed to a basal equation and you know how to solve it. That's already technicalities. So finally, my green function which I will write down, here is be given by the integral over all quantum numbers, so sum over all quantum numbers. Then I will need zero e to the phi over four a matrix element and e to the minus e k tau. Okay, so that all you can do because you know pretty much everything. So let me just write one next step because it will allow me to discuss one more thing. So after you calculate this matrix elements, you find that this is d k a. So you know all this. So nevermind, what is important is that you find that for relatively short times, time less than capital N and capital N appears in this effective mass, you have one over square root of time and that's already result which we discussed long ago but at longer times which is larger than this integral tau. Demonstrate completely different behavior and it behave like one over t to the three half. On time scales, which is longer than number of particles, the physics is actually quite different and instead of going like one over square root of tau, it go like tau to the three half. By the same token, what you actually observe here is exact result for many body density of states. So if instead of k you use energy, there is my energy here, you can write it as d energy times density of states, many body density of states and this many body density of states is simply given by, I will explain it in a sec. So what this, again, nevermind technicalities but what this simple integration of the reparameterization invariance tell you is that your green functions are different and your density of states, many body density of states has a particular shape. So let me first mention what the green function is about. So if you do go to energy representation, so this one is already familiar one over square root of epsilon. So this one you do Fourier transform and it goes like square root of epsilon. So our non-theorem liquid actually become quite different instead of one over square root of epsilon which was used by this paper which I just showed you, you find that at the scale j over capital N behavior changes and go like square root of epsilon. So that one remarkable fact, another one is here. So let me maybe show you the graphs which I already showed you last time. Yes, so this is a many body density of states. So what we discover here is that behavior at small energy, at very small energy it goes like square root. So it's a Wigner-Dyson kind of singularity but at larger energy it grows exponentially and what is sitting in the exponent is square root of energy. So if we would have bigger, probably it's better to look here for larger capital N. So it grows exponentially here with exponent of square root of energy and that's known as a beta formula basically in any fermionic system. You expect that many body density of states increase exponentially with square root of energy. So that's good news, at least it's consistent with the general principles. Now how much time do I have? Norit, so yeah I probably need to stop here for questions. Yes, well I would call them inspirational as opposed to analytical. I don't know any hard core procedure which allow you to go from one to another but you're completely right. They are jumping on you. So my initial motivation when I started to learn about it was to see if this gravity methods can teach us of condensed matter problems and in some sense I personally failed in this pursuit but apparently the flow of information may be going cover way around. So now people in gravity think that this is maybe a good model for a black hole. So they have this hocking paradox of information and what happens with information falling into the black hole and what microscopic degrees of freedom. So the thinking is that this very simple model completely random fermions may be a model of a black hole. Now what exactly goes into it, I can tell you a little bit more but not much more. No, no, look, be careful. So this thing precisely coming from the time derivative. If you completely ignore time derivative here then the action is zero, it's completely flat. Well, okay, good. So let me try to say it in a slightly different way. So there is this symmetry which I already erased. Now our mean field solution, classical solution breaks this symmetry out of all possible expression it chooses one which is this, right? So this is a, if you wish a spontaneous symmetry breaking. Now since you have spontaneous symmetry breaking you have Goldstone manifold and this defiomorphism of S1 divided by slash SL2R is a Goldstone manifold. So now you have to be careful and you have to think whether Goldstone fluctuations restore the symmetry or does not, okay? Now time derivative d tau plays a role of sort of magnetic field, okay? So if d tau is big, if energy is big then the symmetry is not restored and you have this solution. Okay, but if you go to smaller and smaller energy that's like going to smaller and smaller magnetic field Goldstone fluctuations become more and more important and eventually they restore the symmetry. So this green function is going to zero at very small energy if you wish is a restoration of symmetry due to Goldstone nodes. So that's the physics. All right, so let me tell you a little bit more about another interesting object which people like to discuss in the context of SYK and it goes by the name of out of time order correlation function. Yeah, question? So the idea is that you want to calculate the correlation functions c of t which is many body trace or expectation value of, think like this, x of zero, y of t, x of zero, y of t where x and y are some operators. So that's not what we usually do. Usually we have some operator at time t and some other operator maybe of one or two at time t, t two, they do like this and t may be larger, t one may be larger than t two or t two may be larger than t one but that's what we calculate. Here we want to specifically calculate it one time, another time, back to the first time and then back to the second time, okay? So you can invent more complicated arrangements but that's the simplest one. Now I have no idea how to measure it and you probably need a time machine to actually measure it but you can compute it numerically. Now, why worry about it? So the motivation came at least what is quoted from Larkin and Avchinnikov in 67, back in long ago, they thought about quantum chaos in the context of superconductivity but never mind. So what they thought about is how classical chaos manifests itself in quantum mechanics and then it led to lots of studies. So as x and y, they basically thought about momentum and coordinate. So you may think about, let's say x, so let me call it p momentum. So then what up to dumb constant, what you study is x of zero p of t commutator and quare and expectation value. Now, why is it, or let me actually change notation, I will think about x of t and p of zero. Doesn't matter. Now, why this may be of interest? Look, in a classical limit, commutator is probably going to pass on bracket, right? So in a classical context, that that subject is Poisson bracket of x and p, quare. So this one will be, it's easy to calculate Poisson bracket according to textbook rule. So this will be dx of t over dx of zero quare and in some sense average. Now, if you have a quantum chaos, then you have a sort of idea that trajectories which start, so x of zero, so this is x, this is time. So if x of zero was very small, then with the time these trajectories are going to deviate from each other, probably exponential. So this thing will go like plus lambda t where lambda isn't Laponov exponent. So if the system is chaotic, you probably expect, yeah, and I forgot in this transition, I'll have Planck constant squares. So you will have h bar square e to the t lambda t, again, provided the corresponding classical system is chaotic. So if you find that this expectation value of this commutator behave exponential with time, then you can, in some sense, regard it as a signature of a classical chaos going on in your quantum system. So the expectation values like this sort of indicative of, maybe indicative of the chaotic behavior. That's statement number one. Statement number two is a theorem they presented as a theorem proven by Moldesena Stanford in 2015, I believe. So what they say is that if you have this out-of-time order correlation function, it may behave as e to the lambda t or two lambda t. And what they prove is that in any, supposedly prove that in any quantum system, the lambda should be less or equal than pi times temperature. I'll come to this in a sec question. Yes. Well, look, so far I'm not telling you anything about SWK model or what I will in a sec. I'm trying to talk generality. So hold your question for a few minutes. Yes, Zhenya. Well, that's a very good question. I mean, yes, so what Zhenya asking is why temperature? I mean, what is temperature? Because the way I presented is here as a Lyapunov exponent. That's a certain characteristic of a classical motion of classical dynamics that's not related to temperature, seemingly, right? Now, and that's a very good question to which I don't have actually a good answer. Now, formally what we have proven, that proving that this trace is a thermal trace. Now, that's also not so trivial because for normal people, like doing condensed matter, thermal trace will be like this. By some reason, high energy people don't like this and they prefer to say that this is divided by four and there is beta H over four here and beta H over four here and beta H over four here. It's, so strictly speaking, this theorem is proven for this crazy object where you take a density matrix and you split it into four pieces and you stick it between your observables, okay? Which if you think about, it's not an, it's not an innocent procedure because look, imagine that you have a very small temperature, let's say zero temperature. Then what it tells you that initial and final state should be ground state. But intermediate states here, if I introduce resolution of unity, maybe any excited state. Right, so as a virtual processes, I can probe any excited states. If I stick this beta H in the middle and let's say again temperature is zero, that will limit all intermediate states to be also ground state or close to the ground state. So if you have this slightly crazy definition or maybe not even slightly, then at zero temperature it basically does not evolve at all. The entire evolution of this particular object is due to the fact that you allow some temperature window. So precise relation between what we discuss and what I sort of mentioned in the beginning is not clear to me. Yes, there is two here, so. No, it's, it looks like smallest bazonic matzubar effect. Well, so the statement is that no matter what quantum system you take, lambda has to be smaller or equal. So those who saturate the bound where it is equal are supposedly very special and they supposedly have holographic dual. They can be described as a boundary of a certain D plus one dimensional gravity. So that's conjecture. That's, and the thing about SYK model is that that's probably the simplest model, if not the only model, which does saturate this bound. That you can relatively easily prove using this technology which I outlined a few minutes ago. Okay, now, so since I was already asked how the different energy scales show up, let me just tell you what it is for SYK model without any calculation. So here you have to be slightly careful. You have to distinguish whether your temperature is larger or smaller than the scale J over capital L. So the interesting part from a black hole point of view is high temperature, so they want temperature to be larger than this energy scale. So then this correlation function has function of time. It starts from some constant if you properly normalize it is one and then it's exponentially deviating from one. So this is this exponential behavior which goes like one over capital N E one minus one over capital N E to the power two pi temperature tau. So that proceeds up to time which you can call Erenfest time which is one over temperature times logarithm of capital N and then it changes to exponential decay E to the minus pi and then at times which is larger than this time proportional to capital N which I told you about half an hour ago, it changes to power law and then it go like one over I think tau to the power six, nevermind. So signature of quantum chaos is here in a short time limit. This quantum mechanics which I mentioned is in an opposite limit of long times. So that's this Liouville story. And there is an intermediate time between logarithm of N and N, their behavior is yet different, right? But again supposedly the fact that this exponent saturate the bound is another indication that there is a gravity dual model for SYK. Well it's what is growing is deviation from, well look it's a matter of ever you take a commutator here and then you have this one at short time, right? So if time is zero then a commutator is one. Yeah, so this one is a boring thing but deviation from one is supposedly what matters. Good question, possible answer that it doesn't because it's unmeasurable thing. There are statements in the literature that it measures entanglement propagation, whatever it means. Which can't be measured either, right? But some people like to discuss entanglement and propagation of entanglement. So another popular word which come in this respect is a butterfly. So supposedly this thing measures a butterfly effect which is, you know, but some time ago, butterfly things of a butterfly did some perturbation and with the time going with perturbation multiplied exponentially and our universe is now is completely different. So you can discuss this thing, so I'm so far in zero dimensional system but you can discuss this object in a finite dimensional system. And an interesting thing about it is that even if you take completely, you know, disordered diffusive system where heat, charge and everything else propagate diffusively with some diffusive coefficients, then this guy will propagate still ballistically with some non-trivial velocity called butterfly velocity, whatever it means. And supposedly it tells you how quantum entanglement is destroyed or spread. Somehow it measure outliers, the fastest, the particles which managed to propagate without any scattering. So that's sort of condensed matter perspective but string people like it because they have this theorem of if it saturates, then apparently should be some gravitational. All right, any more questions? Yes, well again, it's maybe nothing, right? So maybe it has no release but the hope is that since they anyway don't know how what are degrees of freedom inside of the black holes, maybe modeling them as a fermions with random interaction does a great job. And some people really hope that it does and it can tell us something about physics of black holes. But you better talk to somebody else about it. All right, I'll probably finish here. Not go any further. Yeah, sure.