 Yes, thank you very much wait. No, it's it's it's working. Can you hear me? Yes, okay So let me start with a very simple example today just With example of discrete sign process So this is just a measure on On Zero one also, this is Stefan or now I can take this opportunity to thank the organizers and Just it's a very great pleasure to be here and in fact so much. It is true that My very first international conference in my life was in this very hole in 98 So the time is going fast. Okay. Yes, so it's a great pleasure to be here again okay, so now This is just a measure on the space of binary sequences and I will use I Will use slightly unusual notation. So let me first write it in usual notation, but then in a little bit unusual notation so the probability of a Cylinder given by So many Ones at such at given positions. I am equals one Is given by determinant and then determinant s Alpha I K I L K L from one to N and S alpha of X Y is by definition equal to Sine Alpha X minus Y over pi X minus Y So this is understood as By so to speak lopital rule even if X and Y are integers alpha over pi if X equals So clearly the kernel S alpha only depends on the difference of the Values and so this measure is by definition stationary. It's not obvious that such measure actually exists So that this formula actually gives a measure. This result is called my key Soshnikov's theorem for French Physicist Odil my key and Russian mathematician Alexander Borisovich Soshnikov so, okay, so but such measure exists and And this is the first example that we shall study today Let me say how this measure Appears it appears in a study of young diagrams. So if one considers large young diagram Large young diagram and for example one one draws it in the so-called Russian way so like this emphasizing the symmetry between columns and rows and if one considers the Configuration so what a young diagram one can assign binary sequence. Is it possible to see when I right there? Is it okay? Not down. Okay. Let me be let me draw it here. So if to young diagram One assigns configuration So by putting a particle or by putting the symbol one in Position where the graph of the diagram goes down Where the graph of the particle goes down Then one obtains distribution on space of binary sequences and in the limit So if now one considers the limit as n goes to infinity Of so-called plancherelle measure in the limit here In the bulk of the diagram one gets this discrete sign process Which is how it was introduced by Borodino-Kunikov and Dalshansky. So discrete sign process is of course very close relative of the usual sign process of Dyson and In fact the sine kernel is very similar to the the discrete sign kernel is very similar to the usual side kernel So one takes function f one takes the Fourier transform One restricts to the interval from minus alpha to alpha and takes the inverse Fourier transform. So this is the Form of this kernel Okay, so a lot is known about the discrete sign process. This is a very interesting dynamical system it is Chaotic, so it is it's Kolmogorov and in fact even Bernoulli so Kolmogorov by work of Kolmogorov system by work of Lions Russell lions and in fact even Bernoulli by work of lions and stifle Just excuse me one second. Yeah, sorry. Excuse me. What? That's excellent question. So alpha that's excellent question. In fact, I did not say this But in fact alpha is the position of the observer. So the law the Limit depends on where you place yourself where you look at at the limit and in fact if you place yourself at position alpha square root n Let's say a square root n Then this alpha is there So what is it the cosine of alpha is to a Okay, so one has to be careful that this a should be strictly between minus two and two So if it's equal to two or minus two then it's different situation, but here it's in the bulk of the spectrum So alpha is the position of the observer Okay, so In continuous case, of course all there is no alpha because there is homotity So all these kernels are equivalent up to the homotity, but in discrete case, they're all different Okay, so so it is called more often Bernoulli at the same time its spectral density has spectral density of Just the characteristic function of one has a Zero at zero so this is the form of the spectral density in the neighborhood of zero so in particular the variance the variance of the number of particles number of particles in Minus n to n Grows as logarithm of n Grows as logarithm of n so grows very slowly Precisely because there is zero of the spectral density Okay, on the other hand the number of particles number of particles Minus expectation Over square root of the variance of course satisfies the central limit theorem Which is a result of Kostin-Lebovitz and also Soshnikov so In greater generality so in fact the Fact that this density has a zero that spectral density has a zero It is relatively easy To show and this was in fact done by Kolmogorov in the 1920s so that the position It's I think 1927 That if I consider the stationary process in broad sense corresponding to this system So I just consider the sequence omega n just considered as elements of L2 Then omega zero belongs to span of omega to span of omega n and nautical n non zero So this is quite Unusual situation So let us look at this property more attentively if I know The configuration in the complement of zero so it is convenient for me to Represent configuration as a collection of particles so binary sequence So in position or if omega n equals to one then this means that I put a particle here So if now at zero So if I know the configuration in the complement of zero so beyond zero Then this configuration Determines in fact what is here at zero almost surely whether it's particle or whole So and this follows again it This follows from the form of the spectral density, but in fact this was this is a result of gosh and Paris from 2000 so in fact in this case it's result of gosh and in a different case It's result of gosh and Paris from 2012 So he gosh essentially reproofed use the argument reproofed Komogorov's lemma in this situation More generally one can Take a bigger window so window from minus n to n Bigger window So and then The config it follows from the work of gosh and gosh and Paris so that the Configure the configuration outside of the window Determines so this is what they call rigidity rigidity So configuration outside the window and Again configuration I just mean binary sequence where I put particles in positions where the binary sequence has one configuration outside the bounded window The bounded window Determines determines the number of particles The number of particles Number of particles inside of particles inside, so this is the result of gosh So in fact, this is a little bit paradoxical property of this or How to say property of this model that goes in the opposite direction of chaos, right? So it is not possible to put one more particle. It is not possible. So the configuration outside of the Outside of finite window determines the number of particles inside So this raises a natural question. Is there anything else are there? Is there anything else that the particle outside knows about what is going on inside? and so the main result of this talk is that no there is not and So now I am ready to formulate the main result in this particular case main result. So this is on the archive So and it is that the measure PS alpha so this measure that I Defined here. Let me write it like this PS alpha is quasi invariant is quasi invariant Under the action of the infinite symmetric group the action of s infinity So the infinite symmetric group the group of finite permutations the group of finite permutations of So the infinite symmetric group it's a countable group it naturally permutes the entries of the Sequence and So it naturally acts on the space of sequences and The statement is that the measure is quasi invariant under this action It's quasi invariant under this action. So Radonnik-Adim derivative will be found explicitly I will find it explicitly and to motivate The analogy just I will give explicit formula Okay, let me let me first write the formula and then I will explain the analogy with Gibbs measures So the Radonnik-Adim derivative is the following. So if I act By permutation, so let's Sigma PQ be just permutation of two C of two positions Sigma PQ permutes P and Q So and let's assume that So Sigma PQ permutes P and Q. So then the Radonnik-Adim derivative So and it is here that I want to start changing notation a little bit. So instead of Denoting binary sequence by the symbol omega I will denote So again, I will use a slightly different notation because I will also use it in the in more general situation Of which this is particular case. So I will denote My binary sequence or my configuration by X. So X so this space of binary sequences it is space of configurations and so Configurates on that and I configuration will be denoted by X by X is a collection of particles So particles are positions where the ones are and so instead of this I will write like this I 1 I am belong to X So in other words the measure is just measure on subsets of that Configurations are simply subsets of them. So the formula doesn't change but it will be more convenient for me to write it in this notation So at configuration X is equal to some constant Times product X minus P over X minus Q square where product is over X in X so product is over all particles and So which one goes on top which one is at the bottom. So if Q is in X and P is not in X and obviously the product is Taken by all particles except Q Okay, so this is the formula for the Radon Nicodim derivative So there is Question here in what sense does this product? Converge so it converges in principle value So what do I mean by this clearly this product? So it's product over all particles X in my configuration, but the configuration is realization of stationary process So particles have some frequency Clearly this product diverges absolutely because harmonic series diverges But this product does converge conditionally in the same way as for example, one can write some 1 over n over n in that equals 0 So in this sense this some Diverges absolutely, but it but it converges in principle value So precisely in this sense of principle value is understood as this product Okay, so let me Formulate let me explain the analogy with Gibbs measures So analogy with Gibbs measures The constant does not depend on X the constant depends on P and Q of course The constant is just constant of normalization The constant is just constant of normalization constant is just constant Okay, so clearly if P and Q both are inside the configuration or both do not belong to the configuration then this quantity is just one But when when one so when the permutation actually does something as in this example then this is so to speak the price to pay So analogy with Gibbs measures so if I have Gibbs measure corresponding to Hamiltonian H Then so if I have P Gibbs measure corresponding to Hamiltonian H Let's say of pairwise interaction with Hamiltonian H Then the corresponding Radon Nicodem derivative will be Just exponential Well up to a constant it will be exponential of the sum H precisely Px Minus H Qx Overall X X obviously not equal to Q So precisely for Gibbs measures how do I say? the permutation will if I if the particle goes from Q to P as here then of course interaction with all other particles How do you say the potential of the interaction precisely will enter the Radon Nicodem derivative? so then this result can be understood in the sense that it is some analog of Gibbs property with potential to log X minus Y with Hamiltonian to log X minus Y so this is this is The meaning of this result So I formulated this result in This specific case, but in fact the result is much more general so there is there is first let me say very briefly about Previous work and about why this How to say why it is it could be natural to expect such formula for Radon Nicodem derivative So the main example that motivates study of determinable point processes let me erase here is the case is the One of the main objects of let's say theory of random matrices the Orthogonal polynomial ensemble Orthogonal polynomial ensemble which it is possible to consider in both in discrete and in continuous case DXI in continuous case, but it's also possible to consider in discrete case So these are orthogonal polynomial ensembles In the limit they give they give Determinant open processes so in particular sign process arises as the limit of such orthogonal polynomial ensembles with hermit polynomials Other processes arise in different situations, but it is clear that if one considers Such orthogonal polynomial assemble even maybe in discrete case Let's consider some orthogonal polynomial assembly in discrete case and one considers the action of the permutation sigma pq Then it is clear just from the formula that the Form of the Radon Nicodem derivative will be like will be this so it is just clear From definition, so I substitute x1 equals p. I substitute x1 equals q and I divide Just this is just the form of the Radon Nicodem derivative So and in fact as I said this result has predecessor In the case of so-called gamma kernel gamma kernel Determinant process with gamma kernel which is not stationary which is just a different example of the terminal point process Gregorio Alshanski proved This was invariance in 2011 so in 2011 proved was invariance for gamma kernel and Roughly the idea of Alshanski's argument and he obtained similar expression for Radon Nicodem derivative It's similar, but there is also one important difference. I will explain it a little bit later The expression is similar, but different in important way. I will explain this so but It is so to speak ideologically very similar. So Alshanski's argument Alshanski approximated gamma kernel by finite dimensional approximations of this kind and Obtained the formula for Radon Nicodem derivative by passing to the limit of finite dimensional approximations So the difficulty with this and in fact one can see The paper of Alshanski there are it's long and there are many computations The difficulty with this is that it is very non-trivial to pass to the limit in a statement that two measures are equivalent To measures may be equivalent, but after passage to the limit they can stop being equivalent So it's not observed that the Radon Nicodem derivative is not bounded It's not bounded. It can be arbitrarily large. So the fact that this Statement survives limit transition requires a lot of effort So and Alshanski did it in the case of gamma kernel So the argument in my paper is completely direct. It avoids limit transition and it works in Substantially greater generality. I formulated the result for sine process Just for concreteness, but in fact the result works for a very wide class of determinant point processes and Namely determinant point processes with integrable kernels so again We keep our eye on the orthogonal polynomial ensemble here. So It is well known that such Measure is determinant. I can write this like this KN XI XJ where XI where KN is Christophel Darbukernel Is Christophel Darbukernel? But Christophel Darbukernel, it has its kernel of the projection on the first n orthogonal polynomials, but Christophel Darbukernel Has integrable form It is after some constant Pn of X Pn minus 1 of Y minus Pn of Y Pn minus 1 of X over X minus Y. So now the point is that In many examples while this is not There is no reason for this to hold a priori, but in many examples This integrable form is preserved under limit transition So in many problems It is very typical that the kernel of our operator has integrable form in the following sense. So and again, there are many examples Of course the sine kernel which we just saw for example the airy kernel Where it's The functions a and b are the airy function at its derivative the best cell kernel which arises In the study of singular values of random matrices so where the best cell functions arise as the in the kernel In this integral for so the gamma kernel of Olszanski where a and b for so for gamma kernel a and b are Expressions involving gamma functions a is a And b involve gamma functions. Let me not write the exact formula It's just some ratios of gamma functions with some shift involved Okay, so these kernels arise in many examples. So And the result again is completely general. So it exists in Discrete and continuous case So let me start with formulation in discreet case and then I will give formulation in continuous case as well So in discreet case In fact, there is very little to change. So Let pi the integrable kernel integrable kernel of projection operator And then Then the result stays the same But there is one important detail So the important so the result and the result stays completely the same The result stays completely the same, but there is one important detail namely this detail with principle value so the definition of this multiplicative functional is the following so the Meaning of multiplicative functional is the following is that one takes So what does what is this by definition? by definition This is the limit as m goes to infinity Of this product X minus p over X minus q square Over X less than n Divided by the expectation of the same Divided by the expectation of the same. So this is just normalized product and then this limit This is exactly what we consider I should say expectation with respect to what there is little technical detail expectation with respect to what not with respect to the measure With respect to the conditional measure Corresponding to the condition that there is a particle in q. Let me write it like this so this constant still remains but it's Immediately computable expectation with respect to this so in this sense the result stays completely the same I would like to stress however that the meaning of the limit transition Can be very different depending on the specific model and in fact, I will illustrate this precisely by the example of the gamma kernel So gamma kernel arises again in the study of young diagrams But in fact, I just redraw this picture in fact by very definition of young diagram So it is clear that configuration which codes young diagram Has very special structure. So clearly there are since We put Particle when the configuration goes down So and we put whole when configuration goes up like this. So clearly by definition Beyond the young diagrams, there will be only particles will be only particles To the right beyond the young diagrams, there will be only holes so in on the right on the right I have not very many particles in a sea of holes and To the left I have the opposite not very many holes in a sea of particles Okay so When one takes the limit the limit Which leads to gamma kernel This situation Remains to some extent so gamma kernel is not stationary and it is true that the sum of Inverses of particles converges on the right and some of inverses on holes converges on the left and therefore the meaning of this let me just write g of x The meaning of this normalized functional is Completely different from what it was for sine process for sine process normalized functional It was just the usual function. So This quantity is just a constant and in fact the limit if in the case of the same process this limit It was just the limit But in the situation with gamma kernel, it will be completely different. It will be the following it will be product of g of x over x greater than zero particle times So this normalized product times. So in fact, it will not be x and x anymore because it will be Product of g inverse of y Times y less than zero and y is a hole Pardon me g is any function For example g in this example g is x minus P over x minus q, but it doesn't matter g is any function of the form one plus something small So but the point is that How to say in gamma so in the example of the gamma kernel This in the example of the gamma kernel This procedure gives a complete this procedure of regularization gives completely different result So it gives product of particles over positive particles product of over holes on the negative side Because of course so to speak product over product over holes How do I say product over particles in finite in finite situation product over particles times product over holes Right, it's is equal to a constant Is equal to product of g over everybody, right? So constant So clearly product over particles is up to a constant the same thing as product of the inverse function over holes But this quantity survives limit transition while this one doesn't Okay, so up to this important Remark that the meaning of principle value depends on the model the result is completely the same and in particular I stress that the form of the radonikadim derivative does not depend on specific process So it's the same for all process with integrable kernels Okay, so now let me formulate the result in continuous case. So the result in continuous case It's actually very similar There is in continuous case main result So again So now it will be a measure on configurations on our So Continuous case of our or maybe our plus. So again pie is projection operator by locally trace class projection Trace class projection on our Projection Hey Whose kernel has integrable form Kernel has integrable form and again The examples apart from the sign process Examples are for example every process or vessel process and I let me note that for these results for these process rigidity Also holds and this is a little note from 2015 so Rigidity it also holds for all these processes So, okay has integrable form Then the measure This by P is quasi invariant under the group the group of Defiomorphins of our with compact support With compact support under the group of different more things of our with compact support and The formula for the radon nicotine derivative is has similar Gibbs form. So I take some diffiomorphins with compact support I Take some configuration F F has compact support So let's say F is equal to identity to the complement of some open set V Okay, so then The formula is the same Again product is regularized And I put X minus F P X minus P Square so P is in V P in X Intersection with V and X not in X But not in V. So and the formula is completely the same One can ask It's possible to choose the set V in different ways It is possible to choose the set V in different ways and so in particular so to speak to develop some If I may say so parasitical points, which are actually fixed points. So when f of P is equal to P, but this will not Imply any change in the form of radon nicotine derivative. So this much so this similar result holds again with this clarification on principle value holds For the case For the case of continuous continuous determinant process. Yes P is in V and X is in complement So excuse me. Oh, maybe it's not. Okay. Let me rewrite this P is in V and X small is in complement. So it's just the same as for Gibbs measures interaction of Those who move with those who do not move Okay. Okay. Okay. There is of course also. There is of course also ratio of There is of course also ratio How to say interaction of those who moves of ratio of interaction of those who move between themselves and also Of course the product of derivatives F prime of PI Okay, there is of course interaction of those who those who move between themselves. Okay, there we go P f of PI Okay, so let me very quickly say Just two words about the proof the proof uses The main point is the equivalence of so-called palm measures for Determinant all processes and the main point is that It is in general difficult To check that it is not easy to check the two measure I'm sorry to say such a banality, but it's not easy to check that two measures are equivalent So that just if you have two measures given a different way how to check that they're equivalent It's not immediate But It is possible to check in simple way that two determinant all measures are Taken one into another by multiplicative functional and This is again a little observation from russian's medical service 2012 so that just if I take multiplicative functional So multiplicative functional being just product of that of values of a function over particles and configuration so it It's possible to think of it as Laplace transform of additive functional usually one can say that it functional But in fact determinant all measures interact very well with multiplicative functions Okay, so then the statement is that if I consider determinant all measure and I multiply by multiplicative functional And I normalize so I normalize so that this probability measure then in fact It is a gain determinant all measure again determinant all measure where kernel Transforms in very simple way Namely if P is projection onto l onto some subspace l then PG is projection onto square root of GL and Again, it's very clear if you one thinks about Orthogonal polynomial ensemble if I multiply orthogonal polynomial ensemble by some multiplicative functional Then this is just Product I put G in the weight So precisely this will be the say so what is what is Christofield double projection? It is projection onto polynomials times the square root of the weight So precisely if I multiply measure by multiplicative functional it comes down to Multiplying the subspace base square root of G which is exactly what is written here But in fact, it's true not just for some polynomial ensembles But for all this is true in complete generality for all determinant open process. This is just completely general fact With very short. It's an observation with very short and simple proof so I Would like to stress that it is very important that here it is convenient to work with range of Operator it is not convenient to work with kernel I would like to stress that even if I consider some Superclassical orthogonal polynomial ensembles the one you like most of all Jacobi are meet legend or whatever you like And I multiply the weight by some function G It is not possible to say how orthogonal polynomials get deformed They get deformed in some very complicated way So only if G is some very simple special function for example fraction linear something like this It's possible to say how they deform but in general this transformation completely harsh They destroy the kernel if I have some nice kernel then For if P some nice kernel with some super nice formula when I modify it by G there is nothing There is no nice formula anymore On the other hand it is possible to say this so this statement It is possible to verify directly and this verification is what lies at the bottom of these Points all of these Proofs so one uses the result of Shirai Takahashi from 2003 That palm measures of the terminal point processes are the same are themselves determinant corresponding to subspace of functions which are annulled at a given point and Then so if I for if I set LP is subspace of functions in L Which are take value zero at a given point You might think I'm writing some kind of nonsense What does it mean L2 function which takes value zero at a given point But in fact since my function is range of operator admitting a kernel It's fine. It's this is well-defined and so this point is reduced to Verification of this kind L of p is equal to x minus p over x minus q L of q something that is completely immediate for Polinomials in fact, we all know that polynomials that take value zero at given point are divisible by x minus q But in fact it's also possible to prove in much greater generality In fact in the generality of operators with integrable kernels and just to finish I want to formulate one more case of this result Just so one other situation where Such result clearly holds is the situation of Hilbert spaces of holomorphic functions so and In the station of Hilbert spaces of holomorphic functions, this is joint work with Yanxi shiu from Senres so in 2015 so The result is the following one can consider Hilbert space of holomorphic functions in two situations So to speak on Euclidean plane and in the Lobachev ski plane so on the sea and on the unit disc so in the first case this is Geneva ensemble and this is just the space L is just Fox space and the Colonel is this and In this case, this is Bergman space and the Colonel is the Bergman colonel So and this determinant of point process by the way was studied by Paris and Virac This is process of zeros of Gaussian analytic function So this process of zeros of Gaussian function are detrimental point process with this current and so in this case with Yanxi shiu We obtain similar results. I should say that in this case Using approach of Olshanski essentially using approach of limit transition in this case There is work simultaneous work a bit earlier of Posada and Shirai But in this case and I want to finish with this in this case Everything is the same Except the formula for Radonnikadjim derivative. So formula for Radonnikadjim derivative It becomes different So instead of so in fact such formula does not make any sense does not make any sense in case of Disc just this product is just completely meaningless. So in fact in this case one has to consider ratio of Blyashki products In the same in the same way, let me not write So the chair is looking at me with reproach. So let me not write it completely But let me just say that instead of usual product. There is Blyashki product. Thank you so very much