 Доброе утро, друзья и джентльмены. Спасибо за инвитацию, что пришли в это очень красивое место. Все было прекрасно организовано. И даже еще есть еще российские вкусы, когда вы пришли к кофе. Во-первых, вы пришли к кофе с российским чайом, а потом кофе с какой-то стороны. Так что сейчас... Окей, я думаю, что это очень временная митинка, которая создала новую тренду, которая появилась в нынешнем году, и уже есть вот этот очень полезный workshop, который позволяет увидеть различные коммунити. Так что я покажу, что я от лабораторфии, у нас есть новая физика Теорика Тулус, у нас есть квантовая группа с Клаусом Фрамховым, с частью группы, и он есть здесь. Мы работали в классическом каусе, в квантом каусе, а потом мы пошли в мезоскопические системы, где Клаус участвовал в таких экспертах. Карла Бинакер и Жан-Луи Пишар. И потом мы сделали много тела, интеракционные эффекты в этой системе, с андресом и локализацией. И это всегда работало с большими матрицами. И в том числе, у нас был квантовый компьютер, в котором матрица в размере увеличилась в количестве кубиц. Иногда мы спрятали в эту сторону, а потом, может быть, еще одна апплика была в гугл-матрицах, которая тоже очень большая, с марком, шеейом и сцеплением. И в какой-то смысле, мы всегда были держать глаза на интеракциях, и дизод, и это был вот этот пример из СВК-модели, появился, может быть, из-за 1-2 года, может быть. И я покажу последние результаты, которые мы сделали с Андреем Каловским, мой коллаборатор, который работал в Российской Академии в Краснеярске, и тоже из филта Кантум Кеоса. Итак, там будет какая-то интерсекция с 2 линии, с полной линией, когда вы говорите о Кантум Доте, и я скажу несколько слов о истории, как вопросы появились там, и там будут идеи динамичных термолизаций, которые идут обратно к вопросам дисплитов Больсмана и Локшмит, как вы можете иметь груз в энтропию, если вопросы динамичные, и ответ был, конечно, появлением динамичных термолизаций. И с этим сомнительностью и практикальным способом вверх. Даже если какие-то эксперименты для атома-велосипеда были реализованы экспериментально, мы называем Атомы несколько лет назад по Хуберленде и Нью-Зеландии. Здесь также идея термолизации, главная линия термолизации, не должно появиться, потому что вы интервете скалпленово к басу, и потом у вас есть Квантум гипс, дистрибюшен, дистрибюшен к квантуму уровнях субсистем, что может быть квантум дот, или бэкхол, но идея это, что система должна стать термолизацией, потому что интервью, нолиняритез, которые происходят там, и, может быть, главная линия здесь, чтобы сказать, и, может быть, для квантум дот, ок, это, может быть, нужны экспериментальные эффекты, чтобы сделать это хорошо изолировано, но это, в какой-то extent, было сделано в экспериментах. Но, квантум дот, бэкхол, должно быть, также, в первой опроксимации, выглядело как изолированный объект, который, если у него термолизация, там, должно быть динамичная термолизация, из-за интервью. Там. И здесь, я думаю, филт квантум-каус будет очень полезен, потому что, ок, в этой extended version, потому что оригинальная идея квантум-каус, ну, ну, бэкхол, в frequencies of paper of Einstein in 1970, when he asked question, how to nucle Police systems which are not integrable resistance on recent results of Poincare. And there was long progress starting from 70s, 80s of development of quantum chaos, where we understood from works of Gohzvillir Chirrikov Mohvious, обозревает, если у вас классическая динамичная хаотика, и как вы должны квантовать, и как вы должны быть хаотиками, и, конечно же, потому что в ближайшем системе, как СИН-АИ-БИЛЬ-ЛИ-АРТОЛ-ЛЕВАЛСА-ДИСКРИД, так что вы не будете иметь кондиниум-спектрум от движения, и это будет quasi-дискрыт. Но мы понимаем, как эта корреспондентная принципа для борьбы будет оплачена. У нас есть первое время, и мы понимаем, как транзиция идет от одного к другу. Но это была часть один-партикальный квантовый хаотик, который сейчас, я бы сказал, в глобальности есть, конечно, математические вопросы и какие-то рефайманские вопросы, которые еще, может быть, сделаны. В различные направлениях, но в глобальности это мы понимаем. Но есть еще один момент, когда мы говорим о многих телах. Каос. И здесь я думаю, что development of the concept will go to this dynamical thermalization, which I hope will be useful for different fields. Quantum computers, quantum dots, solid state, mesoscopic objects, and probably for quantum black holes. So, this is one of the examples I will discuss later in detail. Some probability on states k of a given many-body quantum system. This is energy. And the image colors show coarse-grained probability distribution on levels of white. Some eigenstates you get for this many-body system with k indices, k one-partical orbitals, which are presented here. These are many-body energies. You coarse-grained some intervals for each k and then you have probability distribution. And if you suppose that on this one-partical k orbitals you have Fermi Dirac distribution, then left is, if I remember, the result of numerical diagonalization of this system. And right is what you should get from Fermi Dirac distribution. And they are very close to each other. There is no thermal bus. This is a close quantum system. But due to this dynamical thermalization we reproduce Fermi Dirac distribution. So, this will be maybe the main line. So, a few words again about SYK model, which was very nicely represented by Timur on the first day, showing various links with different fields of physics. And this, I think, is some important point. Because in this way when we speak about concrete model it's easier to understand different members of different communities. You have some the same single model, so you try to analyze by your methods each community may have differences, but then we can speak about the same model. So, we try to apply method and discuss the results. That's common understanding we'll emerge from that. Hopefully. So, we will use the form of proposed by the Sparrox of Subir-Sajidiv. So, interacting fermions not myerano fermions as discussed by Ketayev, it's a bit closer to our understanding of mesoscopic systems and it's linked to the history of the development of this what we analyzed. So, there are a lot of publications listed here with people coming from on the top mainly from quantum gravity community and maybe I will stress not so many numerical simulations and they came from the field of people who worked on random matrix theory quantum chaos partially so this at Stony Brook, with his collaborator Garcia Garcia from Cambridge. They did two recent papers the one is here and they did quite good numerical studies I would say we have something different they looked on myerano vision of SWK model we look on fermionic but we also have different approaches even the part of results they have very nice but also there is so this is some slide from the parix of Sajdev with this links between showing duality between this quantum gravity and this SWK version of interacting fermions Hamiltonian was discussed here already few times so I will not stay much on that and unfortunately I am not expert in quantum dots so this left version is closer to my heart so I will stay more on that left side but we will discuss the applications in fact as was claimed and it is also pointed by in the works of Werberschott that this version of Sajdev there was the PRL with Ye about interacting spin system which was mapped on interacting fermions and led to the version of SWK model in fact this SWK model was invented by people from just two rivers or one river station from here Arieaux-Bahegas and Flores there was also parallel one more group Flores French and Wonka so this was 71 and at those times ideas were around random matrix series invented by Wigner so they were asking that but random matrix is full matrix so they ask the question what will happen if you will take in H we have only two body interactions so let's take Hamiltonian of fermions with orbitals and we put all two body matrix elements to be random of course in this case the matrix will be very far from random matrix because the size of the matrix grows exponentially with the number of fermions suppose you are somewhere at half feeling so the number of fermions is roughly half of the number of orbitals and mainly I will present results for this case it's not very important but it will be like that so the size of the matrix is exponential number of fermions but the number of non zero matrix elements is only goes like power 4 of number of fermions not maximum so very sparse matrix but and they did numerical simulations which computers available at that time and number of fermions available but the main statement was that you will get on the left example of random matrix or Gaussian orthogonal sample and approximately matrix of the same size for this they were calling this two body Hamiltonian with some random interactions maybe I will call this two body random interaction model Tiberim and the model was studied by two groups I showed one of Bajigas results similar results of Gaussian Planck so in some sense the model appeared in the field of nuclear physics much before such difference but there is a link and there may be interest physical understanding of importance maybe of course was not there even if people feel that it's interesting model so let's and we slightly generalized this model assuming that this is the part of SYK the usual in the usual form I have one particle orbitals they can be degenerate or quasi degenerate in the limit of SYK and I have earlier spin polarized fermions usually I will consider the case around half feeling but I add one particle orbitals so which can be not non degenerate so and this this term VK or square root of M to normalize some things like this in the case of SYK this is the usual normalization of SYK up to some maybe numerical coefficient depending on what is the feeling factor but then I have this one particle orbitals which may be not degenerate and the spacing then I can say what is the effective spacing and what are interactions between particles so I say that VK on average 0 but then they are distributed with some width V and then since I have M states I will get roughly the spacing between one particle orbitals V divided by M in power 3 over 2 and the ratio of this one particle spacing to the two body matrix element U which appears in this SYK part will be dimensionless parameter which appears in this modified model which we also call Tabarim because then the ratio delta divided by U is I will call J and it's V over J J from usual SYK and V is the amplitude of fluctuations of one particle values VK so what is the matrix this is expression for the matrix size for the whole Hamiltonian M factorial divided by L factorial M minus L factorial and then I have the number of nonzero elements in a line which goes in this way with the number of particles and let's say the number of particles approximately half of number of orbitals this will be the model and I will say that it's interesting to generalize and this model will have implications for solid state quantum dots and in the language of the quantum dot if I have some system of finite size with some number of levels in the energy B energy band total energy band of this quantum dot for one particles then usually what people say in mesoscopic physics that this dot of size L in dimension D it can be characterized by conductance of this dot G which is tau less energy divided by level spacing in this dot so for me delta is one particle energy spacing in this dot and E C is tau less energy it goes from tau less paper PRL 1977 where he discussed conductance such mesoscopic dots and in fact the conductance he showed that there is number of conducting channels which are important to this energy divided by spacing and the meaning of this tau less energy is H bar divided by diffusion time to cross the dot and then diffusion time tau C here is this is comma size of the dot L square divided by diffusion diffusion is fermi velocity square multiplied elastic collision time so everything is well defined for the dot and then there is another important point in this metallic dot if these are non-interacting fermions like that this is conductance dimensionless you assume that it is slouchier to have good conductivity metallic region but then if there are interactions then due to screening and other effects effective interactions in this dot of two-body matrix elements behaves like this delta divided by J this is effective interactions in the dot and then you come to the question what will happen if I look on excitations so J slouch interactions are relatively weak and that's maybe the interest of this parameter which we introduced here so we may have transition from SYK K7J goes to zero very strong interactions between fermions or J is slouchier then we have weakly interacting fermions in the case of practically non-interacting fermions in mesoscopic dot this is characterized by one parameter J now what is the question which immediately appears and it goes to many most many-body quantum systems and it appeared because I was working at Novosibirsk Institute now Institute of Nuclear Physics Nuclear Physics community and there were also hot debates you say that Fermi system number of level, number of states with excitation energy delta above the Fermi level growth exponentially then the spacing between levels drops exponentially so then the question when this will we will mix all these levels by interaction the first answer which continued up to very long time even there is a review of guru Mühler Greilin Physics reports 97 and this question was even at that time not clarified even the first answer appeared so in fact you ask the question when you will mix this many-body quantum levels by interactions and in your hands in nature you have only two body interactions so in the first glance people will say of course we immediately mix these levels because spacing is exponentially small but in fact the answer is different the answer is that we will mix these levels only when our excitation energy above Fermi level exceeds some chaos border and this excitation energy is proportional to j in power two-thirds multiplied by delta and since our j is larger this will be excitation energy and we may say that this is the statement that above for energies excitation is lower than this and j is our parameter and in fact the system was in many respects the view two lines maybe one going from nuclear physics and another from mesoscopic experiments the experiments of Urusivon flew from Technion when he did me spectroscopy of such a quantum dot with conductance about hundred and what he saw in his spectroscopy when he connected the leads he made some kind of spectroscopy moving on the Fermi level in the leads he saw that there are some peaks of current which appears at some energy discrete levels in this quantum dot for which he is doing spectroscopy but above for some excitation energies he have seen most like continuous spectrum so this discrete resonance is disappeared and this is the manifestation of this idea of dynamical thermalization that above for some energy the quantum states become a sense of quantum mechanics so there will be Wigner-Tysons or RMT study, random matrix statistics for this level dependent on symmetry and then all thermodynamic properties will appear so in fact there are now many discussions about eigenstate thermalization hypothesis many body localization and all this goes to this question when you start to mix many body quantum levels and I should say that some arguments and first numerical simulations were done by Sven Berger Swedish professor in Lund and he has this publication of Perelman 1990 where he effectively proposed this criterion why you are at that excitation level your density of states are roughly like exponent of J so you have huge number of states the spacing is enormously small but you don't mix them because the explanation is that you have like selection rules due to this 2-body nature of interactions so you should take into account not the spacing between nearby levels but the spacing between levels which are coupled by 2-body matrix elements and if you do this I will not enter in detail we developed this it works for 3 particles first you can make analytically and for many particles we did this with my collaborator Philip Jocon in 1997 and confirming by numerics and these arguments that you will get this body but still now there are debates and discussions about this body there is group Mirlin-Polikov in Kalsruye which derives this now in more analytical ways but essentially what is the element why I will not enter in the detailed explanation why there should be J in power two thirds but effectively you say that you excite your levels and then the number of effective fermions which are interacting above the Fermi level only those which are excited are interacting and their number is energy temperature divided by delta so you determine effective number of electrons above Fermi level which are able to interact those which are below the Fermi level they are blocked by Paul principle and then for them you write this what we call now aber criterion comparing spacing between interaction of two-body matrix elements this level spacing given by this two-body transitions this leads to this criterion which at least we confirmed in various numerical simulations with this Tabarim model but also for model of quantum computer and okay this of course this is maybe more rich model because we have one particle orbitals which are not quasi degenerate but S y k will be reproduced when J goes to zero yeah Do you get that criterion by applying estimating the size of perturbation theory in terms of okay I will tell how I get the estimate that my I should say that delta C should be order of in two-body two-body matrix element and delta C I should estimate what are the spacing between energies which are coupled by these transitions from energy band how many two-body matrix elements I have I take this energy band I divide by number of transitions I have and then I will come this is the criterion so yeah this is counting but counting in some self consistent way because if I when I am in the quantum dot in the middle of the band to count but because everybody is excited so as I wrote I know the size of the matrix I know the energy band I know the number of non-zero matrix element due to two-body interactions but it's a bit more tricky when I am close to the Fermi energy because I should say that I excite only those which which are electrons which can interact they are only in the if I say suppose they have temperature T then the number of excited fermions will be T divided by delta and then I do self-consistently and in this way I arrive to J2C G, sorry Vladimir was already doing J, yes sorry G and J is this one capital J for SYK sorry Russian pronunciation so then I have my epsilon K levels of one particle orbitals then I should apply my thermalization on that dynamical thermalization if I have these levels fermions are relatively weak then I write a Fermi Dirac distribution on these orbitals I say I have some temperature T bit is one over T then I can compute the energy I can compute okay this is a von Neumann entropy which is different from fermion entropy but okay we used somehow this quantity for comparison and then if I have temperature I have immediately the implicit curve because entropy depends on temperature and energy depends on temperature so I can draw the curve entropy as a function of energy and why it's very useful for numerics because well and had we had Yuri Rumer who was in fact friend of Landau and they both were put in jail at the same moment Landau went out one year after but Yuri Rumer spent 15 years there he returned and he was working in institute and he was giving first lectures on thermodynamics when I entered at the university so he like to say that entropy is easier to understand than temperature because it's additive and he has example like has if you have some stadium there are many La Pen running around but when you open then they go away entropy is growing so they are additive so for numerics it's very good they will be self-average so this is one of the example this was at the time when we moved to the field of quantum computer so this was one of the example of quantum computer with imperfect qubits so let's say we have gamma i with typical space in delta zero but with some small fluctuations delta i which are of the amplitude delta and then there are two qubit couplings j ij on 2d lattice on linear neighbors and these are j ij so randomly by amplitude equal to j so you can ask then what will be the border of quantum chaos and what we do here we fix the total momentum of the system let's say half of maximal momentum and then we say that delta zero is very large so in some sense you have maximal spin half and you are in one layer of these spins so if you want to switch to put this change spin from up to down there should be another spin which also change direction down to up to keep to stay in the same energetic layer which width is much smaller than delta zero so in some sense this couple of spins they behave like fermions and that's why you are getting here the statistics of Fermi Dirac if you work in one layer which is very good approximation if j and delta much smaller than delta zero and the border this chaos border is for delta divided by n where n is the number of spins you have here number of qubits which speak about quantum computer and this was verified pretty well numerically so here I show for two eigenstates couple of eigenstates with distribution of one particle orbitals of this Fermi Dirac statistics so if j is below this border this distribution is very far from Fermi Dirac which is drawn here by a smooth curve so there are two examples for two eigenstates but if for small energy excitation inside this layer but if I go above then it's one eigenstate with given energy which here excitation energy above the ground state and it's projected on one particle orbital I see that I am getting pretty close distribution which is pretty close to Fermi Dirac and then I can have some effective temperature and the first those times we estimated what is approximate dependence of temperature of energy and the curve numerical curve was pretty close to the theoretical maybe we were not so clever those times that it is better to look on entropy and energy I will arrive to that shortly this is the first example of demonstration of Erty Ash in 2001 that one eigenstate for Fermi Dirac term of distribution now we go to this Syk model with interaction and there are two values of G in left column G equals 0 so it's exactly Syk case and G equal to root of 14 on the right it's not very big conductance around 3.5 but already we start to enter this more kind of one particle orbitals and quasi particles so in both cases this is density distribution in one case and another but of course density is not important because we know from the results of Anderson localization and Anderson transition that when you have Anderson transition the density of states is practically not changed but the nature of eigenstates goes from exponential localized to diffusive quasi diffusive in the localized metallic phase density it's not important we have complex matrix element so that's why we have GUI distribution with a square repulsion and this is integrated green curve with result of Poisson and blue curve with numerics and the theoretical red curve is hidden behind there is practically a complete overlap okay we are not the best in these results in fact this repetition of Bachegas flores maybe a few months before us with much better statistics but then we go to test a few eigenstates in this regime when G is square root of 14 so we have well defined quasi particles and we see that approximately with fluctuations but we recover Fermi-Dirac distribution shown here by a blue curve and blue stars show what you would get for given energy corresponding to one particle orbital energy probability here and numerics is our result projection of one many body exact eigenfunction probabilities projected on one particle orbitals and okay with fluctuations but points approximately follow this now as soon as we said that we have about 21 particle orbitals we have the energies then we have immediately Fermi-Dirac on that give us all the dependencies it give us how bit is inverse temperature here so how chemical potential depends on this inverse temperature and how temperature depends on energy inverse temperature so this we will have cases of negative temperatures if we go above half energy excitation being above the middle of the energy band then the system will have negative temperatures and okay this was discussed in the book of Abraham for spin systems it's well known that in speck nearby in sakle it's known quite known that you may have work with negative temperatures and this is the curve as entry piece function of energy for few cases we have here A is 5 fermions on 12 orbitals the matrix size is around 800 but already we see that we start to reproduce this red curve which is the curve you take one particle which you have and you trace the fermi dirac ansatz for entropy and energy and blue numerical points about 800 points shown here because each eigenstate will have entropy and eigenenergy they trace like that and we see that we reproduce here other cases this larger matrix size number of fermions matrix 3000 but if I take j which is small j was 1 and 1 particle g was about the square root of 14 here g let's say v is like g when j equal to 1 but if I decrease j to 0.1 this is the case 2-body matrix elements are weak so I am very far from thermal distribution so you stay it's like analogous of Kalmogorov-Arnold terium for non-dynamical systems if perturbation is small then invariant curves are not destroyed here you say that if my perturbation due to 2-body interactions are small then I will be forever my eigenstates will be close to product of 1 particle eigenstates which is unpertop solution non-interacting Hamiltonian so this is the situation but if interactions are relatively strong then I thermalized and again there is this border g in power 2 over 3 multiplied by delta above which I should go if g is very large then I should look rather high energy excitations to obtain this thermalization and the entropy is in equipartition is number of electrons fermions divided by number of logarithm of the ratio l over m m is number of orbitals so this is more sophisticated case when I take this are my m orbital 1 particle orbitals I put the index of them k and then I have my many body eigenstate energy so I can make some cells of finite energy size this is density distribution for this example concrete realization of 16 orbitals with 7 fermions and then on the left is theoretical Fermi Dirac distribution which will give me the coloris again show the probability average given cell of energy and for given value of k so white is maximum so black is 0 so I have maximum at low energies some finite temperature effective temperature here and on the right I do the same but with numerical eigenstates which I obtained from diagonalization of my matrix matrix size is 11000 approximately so this g conductance is equal to 2 4 here so we see that they are pretty close to each other but here I did some coarse grain so my fluctuations are reduced while before when I show these curves here I plot the characteristics of a given eigenstate without any coarse grain of course there are fluctuations deviations of the theoretical curve given by thermal distribution but in this way I check each eigenstate without any coarse grain and I see that all of them are well thermalized practically all eigenstates now we go to the regime of yk taking g conductance g equal to 0 so they degenerate and here are the two cases which are practically the same with the number of comparable number of fermions on the right I have g like 3.5 something like that and I see that red keof is my fermi dirac ansatz black points are numerical values of entropy and energy which I obtained from diagonalization but here I have this g one particle orbitals and here is the case of syk so I see that for different number of fermions when it starts to grow we go from matrix size for fermions m equal to 10 orbitals this matrix size 230 then we increase going up to 11000 and we see that the maximum entropy is growing in syk regime so clearly this is no comparison with this keof we are getting from fermi dirac distribution when one particle orbitals we are available but here there is a question what we should compare with we don't have for syk one particle orbitals so what we did is the following we say we know the energy band and this is for black cases this is the largest matrix so we don't know what are quasi particles they are hidden but let's assume that there are some of them and that they are one particle energies somehow homogeneously randomly are distributed in a given energy band of syk model with given energy band which is available from numerics and then still we will get this parabolic keof this typical maximum which hits the entropy of syk model but which the ends of the band is of course very different and the flat behavior of entropy because the derivative of entropy or the temperature is like inverse or the energy is like inverse temperature so if it's flat that means that your temperature is infinite so the question and this comes out so approximately the maximum value we reproduce reasonably and this agreement maybe some other people like Wehrmarschott who computed the entropy but the logic we have is different all logic are going from the works of Moldochena Stanford what they were doing they were saying we have many body states of syk and then let's write quantum Gibbs on that with some temperature and we compute entropy of this distribution but this assumes that you are in a contact with some heat bus and I think it's not at all reasonable to assume that if you have isolated quantum black hole or if you have of course Hawking radiation which is since probably in the first approximation it's only small coupling which we can neglect and quantum dots also show experiment of Uri Sivan at some extent we can see this discrete quasi-particle levels before the dynamical thermalization we can have isolated quantum dots or it can be some quantum computer with isolated qubits which if pretty long there should be dynamical thermalization which appears like dynamical chaos in the classical system but for this concept of dynamical thermalization then we have a problem with this syk regime what we should write here maybe we should find some other quasi modes or other things in this sense this is in the first work of such different years when they say this is Hamiltonian which have no quasi-particles so what we should do with thermalization concept in this closed strongly interacting many body system this remains some open problem I think now let me Father now there are different questions which about remains open about syk as I understand from the pre-prince of quantum gravity community there are somehow expectations that you go to very low energy excitation above the ground state and you will assume they somehow assume that spacing will be still inverse size of the matrix so exponentially small because size of matrix grows exponentially with the number of particles in syk the energy band is fixed so low energy excitation are exponentially small from the viewpoint or inverse matrix size or exponentially small as you want but the question is when we will get appearance of statistics of Wigner-Daisen random matrix series dependent because it depends on symmetry in my run case and this so for us when we are in the regime of quantum dots so our conductance is g is few units like we saw here we have that the spacing will decrease polynomial with the number of particles while in the case of syk when g goes to zero you expect that spacing low energy excitation will drop exponentially with number of particles so there are different regimes and there should be some transition between 2 regimes at which values of g conductance g our dimensional parameter this will take place this open question and also open question not only when the behavior of spacing with number of fermions will be changed but also because here we have a gap on this manifestation it's first level spacing for Fermi system all levels filled up to Fermi level and then first excitation level spacing on this Fermi system so this drops polynomial with the number of particles due to size of our system all this normalization coefficient which we assumed but in syk it will drop exponentially then another more important question when we will have Fermi Dirac we know that usually at the end of the bands if we take Anderson model 3D Anderson model with disorder being smaller than 16.5 in the units of Hopin then in the center of the band we will have diffusive delocalized states with Wigner-Dyson statistics but at the end of this spectrum close to the first excitation energies disorder will be effectively very strong and there are even mathematical theorem which prove you that you will be localized there Eigenstates exponential localized level spacing statistics but Syk where this model in what regime it is if you go very close to the exit ground state some indications from Garcia-Garcia-Werbarshott the recent preprint they say that if they take low energy on the infrared part of the spectrum of about 1.5% of the whole states then they still have good let's say GeoE statistics for this number of my run fermions it's numerically pretty good but the question remains open what will go for asymptotically for large system size that is interesting question from both sides of community so the question is now maybe I go to the end of my discussion so we want to have a system which is chaotic at very low energies do they exist in classical dynamics yes they exist for example classical color dynamics of homogeneous Young-Mill fields this work of Martinian-Savidy 81 and Viscirikov we started this numerically you have positive Lyapunov exponent and in some sense this is rather similar because you have X to the 4 so it's like your operator A in power 4 and in this system it's clear from it's homogeneous in energies so immediately you write scale independence Lyapunov exponent H 1 quarter energy in power 1 quarter for this system and I think something like there is excitement about correlators in SYK but my feeling is that it's very similar because you have A operator in power 4 so this is your energy so some kind of this scaling is very similar to that case it's simply renormalization of time this energy these three colors dynamics almost everywhere chaotic there's some small island may be related to Kapitsa type pendulum solution but it's only three degrees of freedom now we can go to higher number of freedoms and there are some even known for children this Newton-Krad also called when you collide both in fact the interactions of both locally it's like many both and the displacement the energy on displacement difference is delta x in power n with n for Newton-Krad it is 2.5 but you can style more realistic maybe quantum case n to the 4 that's we did the work of Gerov and Pikovsky and in fact the interesting point of this system for n equal 4 also exist for arbitrary small energy going to 0 but if we quantize that what we did it is not the case chaos disappears in quantum case version you have phenomenal type of excitations life is more complicated and in fact this model of quantum Newton-Kradle was realized with with called atoms so other physical candidates to realize this YK model it's possible to have systems with J smaller than 1 in solid state they produced chains of sine billiards with small narrow connections so principle this J G smaller than 1 possible to realize there are also other examples of systems with exponentially small energy excitations like voletic called ions or Wigner crystal in a periodic potential realized by from MIT with few ions so these are interesting questions which are generated by this SYK model so thank you for your attention