 I'm very glad to say we're starting the year with a hybrid form of conference, so we're here in the footage lecture hall with many of you, audience, a great pleasure for me to introduce... Sorry, oops, thank you. I hope you heard what I said without the microphone. Welcome for my CTP, hello for my CTP. I'm very, very glad to introduce our young, but definitely very illustrious speaker today, Alexander Buffetov. Sasha was a student at the Independent University in Moscow and then a PhD student with Yakov Sinai at Princeton University. He has then worked and visited many, many places. He's currently a researcher at CNRS in France and a leading researcher at the Steklov Institute of Mathematics of the Russian Academy of Sciences. He has won several prizes, amongst which I will just mention the Sofia Kovalev-Sky Prize of the Russian Academy of Sciences, and he was the Gabriel Lame Chair of the French Embassy in the Russian Federation in St. Petersburg a few years ago. He has written more than a hundred papers in very illustrious... Less, less, less than a hundred papers. Probably getting there, he will soon become a hundred, and in a wide variety of topics ranging from harmonic analysis to probability theory, dynamical systems, operator theory, representation theory, so really a renaissance man, I would say. He speaks most languages, certainly all the languages I speak and many more, and he will tell us he's been at ICTP for the first time in 1998. I think that was his first international conference, so he has a very strong ties and has visited and organized events at ICTP many times since then. So Sasha, it's really a great pleasure to have you, and we're looking forward to your talk on... I can't remember the title. But I will say myself. So thank you very much, Stefano, for this generous introduction. But in fact, ICTP is a very special place for me, and it is with very special emotion that I speak here today. So Stefano has asked me to give a very introductory talk. So I apologize if some members of the audience are bored. Conversely, please do not hesitate to ask me questions at any point. So I can also, obviously, the listeners online. Only I will ask, maybe Stefano, if there is a question to tell me, because yes. OK, so let me start with the example, with a specific example. And we consider a random series. Random series. So the coefficients are random. Here is the randomness. And each coefficient is gaussian, and different coefficients are independent. Each coefficient is a complex gaussian, as distribution with gaussian density, with density, with density. And coefficients are independent. So this object, for the purposes of this talk, we will call the gaussian analytic function. The radius of convergence of this series is equal to one. One can prove this without difficulty. It is possible by including some convergence factor into the definition to obtain also modifications and series which give an entire function. And this is a completely different problem. This is a completely different problem, completely different situation, which I will absolutely not discuss today. So there are also some other modifications of this example. But where we go beyond the setup of random series, on the other hand, if we want to consider random series, then random series, let's say with independent coefficients, then I cannot consider more general example. We will see why the gaussian property is very important for what I am going to say. So as I said, this series has radius of convergence one, so it defines a holomorphic function in the unit disk. And in fact, it will be useful for me to consider the unit disk as the Pankare model of the Lobachevsky plane for a reason which will appear right now. So I am interested in the zero set of this holomorphic function. So and in fact, as we shall see, the zero set is in there, it's a probability measure on the family of subsets of the disk. The zeros only accumulate at the boundary of the disk, so at the absolute of the Lobachevsky plane. And in fact, the distribution on the set of zeros we shall see will be invariant under Lobachevsky isometries, which is not obvious a priori because this distribution is not invariant. So this invariance under Lobachevsky isometries will be important for the progress of the talk. And in fact, and this is the reason why I will be very specific. So I am considering a very specific example. In fact, this zero set admits a completely explicit description, which is a theorem of Peres and Virag. And it is in fact a determinant point process with Bergman kernel. I will remind the definition a little bit later. Yes, Stefano? Sorry, Sasha, just a question because I'm not familiar with what exactly you mean by this random series and therefore the zero set. So is the zero set a zero set for each particular implementation? That's exactly right. For each particular implementation, you have a zero set. That's exactly right. And so you talk about the probability distribution of the zero sets. That's exactly right. That's quite precise. So I'm talking, that's quite precise. I'm talking on probability distribution on the space of subsets of the disk without accumulation points. So we will see how to give such distributions explicitly in a minute. I will be very specific, very precise about that. So, but okay, I will also formulate the Peres-Virag theorem, which gives very explicit description for this zero set. And in fact, the main result of the talk depends very much on this explicit description, as we shall see. But let me first precisely formulate the main result. So the main result concerns the conditional measure of this distribution. So conditional measure where I fix the configuration in the exterior of a bounded set. So I consider the composition D equals B, this joint union C. So B is compactly contained in the disk. So it's an open set with compact closure. C is the complement. So I consider conditional distribution, conditional distribution, subject to the condition of fixing under the condition of fixing the configuration in C. So infinitely many particles are fixed in C. Fixing the configuration in C. So infinitely many particles are fixed. So all the exterior of the bounded set is fixed. And I'm interested in the conditional distribution in the bounded domain, in the spirit of the Bruschen, just Lanford-Ruel, if you wish. So just this is now distribution on finite subsets of this smaller B. One can think of B as a disk. It doesn't matter if it's a disk or not, but one can think of it, for example, as a disk. So of these finite subsets of this B. So, and yes, Antonio? So if you look just inside B, say it has reduced R, is the expectation of number of zeros in R bounded? Yes. OK. Yes. Yes, the expectation is bounded. In fact, all the moments and even exponential moments of this random variable are bounded. And I will give a very detailed answer to this. I will give very explicit formulas for these expectations and even higher moments. Even, and in fact, as I say, the fact that this result is formulated in only in this specific example is due to the fact that such explicit formulas only exist for this specific example. These formulas due to Peres and Virag, which I will very shortly remind. So for instance, if you take for the AN, if you rescale the variances and take, say, one over N factorial, look at this, it's not going to work. Then we would, if I heard you correctly, then we would obtain an entire function. Yes, yes. And it's completely different. That's exactly my point. And it is completely different. It is completely different model. OK, thanks. It's precisely, so I will say a few words about this a little bit later. What did you say? What do we know about the overall set? You said something about the fact that is the set of subsets without accumulation points or what is it that you know a theory of the set? Please allow me to get back to this question in three minutes. OK. OK, yes, I will answer this. We know a lot. We know a lot. I will answer this in detail. But please let me first formulate the result about this conditional distribution. And then I will, in fact, discuss the context. So the following proposition holds. So this conditional distribution is a conditional distribution on finite configurations in B, but number of particles is not bounded. So I can write, so the conditional distribution has the form. The conditional distribution has the form. So maybe allow me to erase the formula for the Gaussian distribution. We remember it, has Gaussian, complex Gaussian distribution, OK? So has the form, OK? So I have probability of absence of particles. So I have to fix the configuration outside. Why is the configuration outside? So why, why fixed? This is the restriction of the configuration on C, OK? So there is obviously the term corresponding to absence of particles. So I denote it like this. So the probability that there is a positive probability that there are no particles. And then I have the sum of the terms. So there is a density which has determinant form. So l will depend on y. So it is convenient for a reason which will become very clear very soon to write the density with respect to the Lobachevsky area form. So just the determinant is also ij from 1 to n. So there is this determinant form. And the main point is what is this ly? So ly has the following form. And then there is just dependence on y. It is convenient for me to write it in a little bit strange way. There is I take square root of something which is itself a square. But why I do it will become clear as the talk progresses. So now the main object is this psi y. Psi y is, so psi y is the term that expresses the interaction of the particle with its exterior. I could also put psi y here, of course. And this is just the following expression. So I need to take Bleszke product. So over y and y, except this Bleszke product diverges. So I need to normalize this Bleszke product. So I have a product of terms, but this product diverges. So I, how do I say? Such products diverge because the distance of y to the boundary to the absolute decays too slowly. So one gets a divergent product. So I need to take exponential of the sum of 1 minus this Bleszke product. So and now I have divergent product and divergent sum. But this, so the terms, each individual term does not make any sense, but the product does. So this is not anything that should surprise us. In fact, one can define it, for example, just by limit transition. Just by limit transition. Just by taking this product over y's within a smaller disk of radius close to 1, then both product and sum are finite, and then taking the limit transition. It requires proof that this expression converges. I will maybe briefly explain if I have time how one proves this. There is also a constant, a normalization factor in order that this have expectation 1. And this factor can also be found quite explicitly. It is the exponential of gamma minus 1, where gamma is the Euler-Mascheroni constant. So this is the main result. These expressions of this type are called L processes by Borodin, when the letter L in the notation. And of course, by the way, from already what has been said, it is clear that eta is precisely the determinant, the threshold determinant of 1 plus L. Because in fact, if you think of L as an infinite matrix, then these are precisely its minors. And so the normalization factor is just determinant of 1 plus L. What are the Q i's? Q i's are the variables. It is a measure. Again, I am describing conditional measure subject to fixing the configuration outside. So this conditional measure is a measure on finite subsets, but which may contain arbitrarily many particles. Q i, Q n are the particles inside B. Does it make sense? So this is the main result. So now, let me get back to the question of Stefanie. Are there any questions about formulation of the main result? The notation makes sense. So now let me, if not, let me continue. So let me go back to the question of Stefanie. What do we know about the zero set of the Gaussian analytic function? And in fact, I am getting to the theorem of Peres and Virag from 2003. And before I formulate it, let me start with a somewhat philosophical question. What does it mean to define a measure on the set of these infinite subsets? I will call them configurations. What does it mean to define a measure on the set of configurations? They are infinite subsets. So it is not possible to define such measure in terms of densities because there is no natural underlying measure as opposed to here. Here we have a natural Lebesgue measure, or if you wish, Poisson process. One can say that this measure, I have computed the density of my conditional measure with respect to Poisson process. But in general, there is, of course, Poisson process also, but these measures will be singular with respect to Poisson process. So how do I define this measure? So there is a formalism, which is quite old, which is the formalism of correlation functions. It's a completely general formalism. So let me recall correlation functions of a point process. So a point process is precisely, in fact, strictly speaking, I want to say correlation measures. Correlation measures of a point process. A point process is precisely a measure on the family of subsets without accumulation points. So as in this example, let me really stick to this example without aiming for too great generality. Imagine that the space itself, in our case, the disk has an underlying measure. So Lebesgue measure, let's say. It doesn't matter if it is Lebesgue with respect to Euclidean or Lubachevskan distance, it doesn't matter for this discussion. So only the measure class is important. And then we take several small neighborhoods and we consider the infinitesimal probability that a particle be found in each of these neighborhoods. So I have this B1, B2, BL. And I have infinitesimal probability P so that there is a particle in each BI. So you might say there might be more than one particle, but in fact, probability that there are two particles in a small neighborhood decays very, very fast. So if neighborhoods are small enough, the probability that there is a particle or that there is only one particle are comparable. The probability of more particles is negligible. So precisely this expression, this probability, captures also interactions between particles. So in Poisson process, for example, it will be a product. But in general, no. In fact, the zeros of the Gaussian analytic function, the zeros repel each other. They don't like to be close together. So it is very improbable. For example, if you take eigenvalues of a matrix, it is very improbable that eigenvalues be very close. This, by the way, one sees in the paper of Peres and Virag, which is a very beautiful paper, there are also simulations. It's immediately visible in simulations. If you take simulation of Poisson process and simulation of this, it is immediately visible. The difference is immediately visible in Poisson process. There are some clusters, some parts of the phase space where there are very, very many particles. Nothing like this occurs here. Particles don't like to be close together. They don't like that. So, okay. But still there is this probability and this is precisely called the correlation function. So this probability is some expression. Well, depending on the process, obviously, dq1, so it's an infinitesimal probability, dq1, dqm. So, and this precisely is the mth correlation function. mth correlation function. Existence of correlation functions is related to the question that Antonio asked about finite, so-called intensity of the process. Is the expectation of a number of particles in a given bound, it said, is it finite? The fact that it is finite is roughly equivalent to existence of first correlation measure. So there is a little, so I write correlation function, correlation measure. This is precisely, this is just a point of terminology. This is precisely this assumption of underlying measure, the Lebesgue measure, with respect to which all these measures admit densities which are then correlation functions. So this is just a terminological point. This is the situation we always consider. But the point is that the existence of correlation measures. The existence of correlation measures is precisely the statement about existence of moments of numbers of particles in several domains. If the random variable number of particles is given, so point process, again point process, measure and a set of subsets, point process, the main random variables in this case are precisely these random variables, number of particles in B, number of points which we call particles, number of particles in B. So these are my main random variables. Once I assume that they all have all moments, I have correlation functions, correlation measures. Conversely, as the question arises, do the correlation measures determine the point process? Well, again, if the moment problem is well posed, then yes. So the correlation measures determine the distributions of these ones, which by an analog of the Kolmogorov theorem is equivalent, so prescribing finite dimensional distributions of these random variables is the same as defining the point process. Here, so in Kolmogorov's theorem, there are many delicate measure theoretic details about the sigma algebra on which the process is defined, is it sigma algebra or Borel sigma algebra in the space of continuous functions, or is it just Kolmogorov sigma algebra and there are many subtle points. None of this is relevant here because the space of configurations is in fact a Polish space, but the Borel sigma algebra is precisely the sigma algebra generated by these random variables. So there are no such measure theoretic niceties here. Just defining these distributions of these random variables is precisely the same as defining measure on Borel sigma algebra. It's the same thing. Okay, so the correlation functions in reasonable situations, particularly in all the situations of this talk, when the moment problem is well posed, determine the point process uniquely, and it will be very convenient to us to determine the point process in terms of its correlation functions. And now I'm ready to formulate the beautiful Peres-Virage theorem, which says that the correlation functions of the Gaussian analytic function, so let me introduce the notation which will look a little bit strange, but it will not look strange in a minute. So this will be law of X. What is K? It will become clear in just a moment. So correlation functions of BK of, let's say, Q1, QM is a determinant of K where K is the Bergman kernel. 1 minus PQ. This theorem is truly very remarkable. And one of the reasons why it is remarkable is that despite the fact that almost 20 years have elapsed, at least I'm not aware of proof. This seems like a very, how shall I say, very immediate result. But all the proofs that I am aware of involve quite extensive computations. So it seems very strange that there does not exist somehow a conceptual proof of this result, but to the best of my knowledge there doesn't. And in particular, so in fact, in their book, in their book the authors say, in the book on Gaussian analytic functions, they also say that when we obtain the theorem we thought that we will very quickly obtain the formulas also for the plane. So now I'm answering your first question, Antonio. Just, but for the plane there are no determinant formulas. They prove that it's not determinant. So these determinant formulas are specific to this model and to the best of my knowledge are specific in very strong sense, in the sense that I'm not aware of any other, for example, random series with independent entries that also has determinant structure. So there exists, so let me say, for this side of the formula I have a lot of flexibility. So kernel, kernel that gives a determinant point process, there are very many kernels. So in fact any kernel of a projection gives a determinant process. So the right-hand side of the identity is very flexible. You might say that the left-hand side is also flexible instead of Gaussian analytic function I can take analytic function with some other lower coefficients. It's the matching that doesn't seem to work. Change the kernel and ask yourself for the law of the coefficients. And just there, I don't know, of any result in this connection. So there are several proofs of the Perseverac theorem. My favorite is the proof of Krishnapur, which argues, let me just, this is a little digression, I will be very brief, which argues roughly as follows. So determinant formulas are very well known to exist for, for example, unitary matrices. If I consider unitary matrix distributed with respect to char measure, then the density of eigenvalues and the correlation functions have determinant form. This is because of the vial character formula, because of the determinant of the Van der Mond determinant in the vial character formula, which expresses itself in such determinants for correlation functions. But so here, obviously, we have points inside the unitary disk. So in fact, Krishnapur says, let us take a unitary matrix and let us cut a corner out of it. So if you cut a corner out of a unitary matrix, then clearly it becomes a matrix with eigenvalues inside the unitary disk. And so then, and precisely, well, at least the proof as is written in the book of Krishnapur has a quite non-trivial amount of computations, then one checks that this is what one gets, that everything matches. So, but the verification involves computations. So I think it would be very interesting to obtain some like half-page proof of Peres-Virac theorem, but at least I'm not aware of such proof. Okay, so the kernel K is a very remarkable kernel. It's a kernel of projection on the Bergman space. So the Gaussian analytical function is related to the Bergman space. So it's a kernel of projection of L2 on the disk with respect to the usual Lebesgue measure to the space of holomorphic functions on the disk. So holomorphic functions on the disk, holomorphic functions on the disk, functions on the disk. This is the kernel K. You can check this quite explicitly. So in fact, any such projection kernel, any kernel of a projection induces a terminal point process. And in fact, the main point in the proof of this result is the equivalence of so-called palm measures of this process and the initial process. I will explain this, this is what I will explain as the main idea of the proof of this result. This is obtained in joint work with Jan Cesiu. So for the time being, let me just say that here let me pursue one little digression more. So this function is quite reasonably clearly not a Bergman function. If one computes the variance, one gets the harmonic series. Because the variance of this one is 1 over n, so the variance is 1, so one gets the harmonic series, it diverges, and by the Kolmogorov two-series theorem, this function is almost truly not square integrable. On the other hand, it is just not square integrable. So the harmonic series, of course, diverges, but it, so to speak, almost converges. It diverges very slowly. Zero set does not determine the function uniquely. The fact that this function is not square integrable does not exclude a priori as the possibility of an existence of a square integrable function which would be zero at all these points. So, and in fact, it's a theorem in joint work with Jan Cesiu and Alexander Shamov, so that the set X is almost surely a uniqueness set. So Pk almost every X is a uniqueness set, uniqueness set for the Bergman space. What does it mean, uniqueness set for the Bergman space? What does it mean, uniqueness set? It means that a Bergman function is uniquely determined by its values on this set. So if a Bergman function is zero at all these points, then it is itself zero. So this is our result with Czern Shamov. And also with Czern, we have some results which are not even only if, so some results about explicit reconstruction of Bergman functions from their restriction. So in fact, explicit reconstruction is possible by Patterson-Salevan construction. So explicit reconstruction, as we prove with Jan Cesiu, is possible by Patterson-Salevan construction by writing the Poincare series. So this is possible for almost every Bergman function. So by writing the Poincare series, we recover the Bergman function. The sum must be understood here over annual I. So there are important cancellations in the sum which is why it converges in the first place. And then there are, how do I say, there are delicate problems in making this formula uniform, in making this formula uniform in the function f. So it is true that for every f, this formula holds almost truly, but it is not true. It is in fact demonstrably false that this formula holds almost truly for every f. It's not the same thing. The implied set of measures zero that we throw away depends on f. And in fact it is sort of clear why because Bergman function does not have boundary values. So this is essentially a version of the Poisson boundary value theorem. Except Bergman function does not have boundary values. Bergman function is somewhere very wild in the boundary. So, but different Bergman functions are wild in different parts of the boundary. And so just it is not possible to satisfy all of them. So such formula cannot possibly hold, we prove this, such formula cannot possibly hold uniformly for all Bergman function. It does hold uniformly for some subclasses of Bergman space. So let me now go back to the main result. And let me formulate, so let me explain, let me explain why this function, why this function appears in the first place. And here I will need to, I need to erase, wait, but I did and I lost my eraser. Oh no, here it is, yes. Just I will need to recall the definition of palm measure of a point process. In fact, I have already spoken about palm measures at the basic notions seminar here in ACT a few years ago. So there is a detailed recording. And so please allow me to be a little bit brief right now. So palm measure, palm measure of a point process. So palm measure is a conditional measure subject to the condition that there is a particle at a given spot. Such definition is, formally speaking, meaningless. So because such event they don't form, there is no sigma algebra with respect to which it is conditional measure. In fact, palm, by the way, palm, as I explained in greater detail in that elementary introductory talk, palm was an engineer, palm was not a mathematician, palm was an engineer. He worked for Ericsson and he was dealing with the problem of waiting times for phone lines. So it was serious problem because phone communications exploded and there were very serious lines for speaking on the phone, for putting a call through one had sometimes to wait and extend a period of time and precisely palm was interested in the expected waiting time for the next coming call provided that a call has just arrived. A call has just arrived, what is the time I have until the next call? This is precisely the concept of a palm measure, conditional measure subject to the condition of a particle at a given spot. So mathematical theory of palm measures was introduced by Hinchin, by Hinchin. So, and there is also a more modern and more convenient formalism introduced by Alan Berg using Campbell measure. Using Campbell measures. So, let me say that just, so how does Hinchin, yes, Antonio? Is the probability zero event? Not only, just it is possible to define conditional probability with respect to measure zero sets, but the problem is what is the, so if you have a sigma algebra with which if you have some decomposition of the space into some disjoint events. So, it can be continuous. So for example, think about decomposition of, say, the square into vertical intervals. Each vertical integral has probability zero, but it is perfectly possible to define conditional measure. The problem is when I say there is a particle at a given spot, these events, there is no decomposition of the space because particles are not numbered. It is perfectly possible if I have a measure, so if I have a measure on Rn, it's perfectly possible to take conditional measure with respect to the first coordinate. But except here, there is no first coordinate. Particle is a set, there is no first particle. So, there is a particle here, there is a particle there. But you can ask conditioning on the fact that one particle lies in there. Well, precisely, no, so, yes and no. So, yes, this is precisely what the Kalenberg formalism will do, but by passing to a different space. Let me ask a question, just maybe to understand. Instead of an analytic series, I take a polynomial of degree n. Yes. I can condition on the fact that one zero is at a specific location. Strictly speaking, not quite, because what you need to do, you need to number the zeros. Please allow me to speak for three more minutes and then let's go back to your question, because this is exactly what I plan to address and maybe then we can go back to the question, because this is exactly what I'm going to talk about right now. So, how does Hinchin define palm measures? How does Hinchin define palm measures? Hinchin, he takes a small neighborhood around the, well, the spot where there is a particle, takes the event that there is a particle in this small neighborhood, and then, and then shrinks the neighborhood. This is what Hinchin does. It's in his very beautiful book, published in the 50s by the Stack Off Institute, the first mathematical treatment of palm measures, palm Hinchin measures. This approach has advantages, but since it involves essentially taking a derivative, it makes derivations, it makes proofs a little bit long. One can see in Hinchin's book, it's very rigorous, but it requires a certain amount of effort. So, precisely, I am answering Antonio's question. So, the formalism of Kallenberg, through the formalism of Campbell measures, precisely seeks to address this difficulty by passing to a bigger space. So, if I have an infinite configuration, particles are not numbered in it. So, it's not possible to say first particle is in this space. At the same time, I don't need to number all the particles. I only need one particle. So, I can consider the space of configurations with one marked particle. With one marked particle. And this is exactly... And I can lift my measure to this, and this is exactly what Campbell measure is. Campbell measure. So, it's a measure, let us call the space, let us call E. It can be arbitrary space. Campbell measure is the measure on E times the space of configurations on P. It's an infinite measure. I denote it like this. C, P. So, I define it. So, I have E, let me draw a picture. I have E and I have space of configurations. And informally, over each configuration sits itself. So, there is a configuration. And this configuration is what... So, the fiber measure over this configuration is the configuration itself. So, this is informal definition. Formal definition is just that I take subset B, I take cylindrical subset. So, B is a subset of E, and Z is a subset of the space of configurations. And the Campbell measure of B times Z is the expectation, the integral over Z of the number of particles in B, D, P. So, precisely for each configuration, I count how many particles of it are in B. And, well, this is the definition of the measure. And the conditional measures of the Campbell measure are precisely the Palm measures. Campbell measures are infinite, but locally they are finite. Locally in the sense that if I take bounded B, I obtain finite measure. So, conditional measure with respect to fixing some Q is precisely the Palm measure. So, by definition, I can write down this definition. Definition. Conditional measure. Conditional measures with respect to fixing the first coordinate are by definition Palm measures. There is just one little point. So, I do not have an answer to your question, Antonio. No, no, let me answer it. Let me answer it. Yes. Take a Gaussian polynomial. Now I want to condition on the event that it has a zero as a fixed location. Yes. So, polynomials that vanish at that fixed location, they are an hyperplane inside the Gaussian space. So, for... Excuse me. You are saying that in the space of random series I can consider the subspace of functions which vanish at a given point. Yes. But I am considering... I am considering just the space of zeros. At the same time your question is... In fact, this question I have not answered, but I will answer it now. Your question makes very absolutely perfect sense because what you ask is how does Palm taking Palm measure interact with determinant property? And the answer is that it doesn't interact very much and it's a theorem of Shirayin Takahashi Shirayin Takahashi which says that the Palm measure of determinant point process Palm measure of determinant point process is again determinant point process whose kernel is exactly what you said. So whose kernel projects on subspace of functions projects on the subspace of those functions which are zero at Q. And once this theorem has been formulated the proof occupies really one line because you can see it on the level of correlation functions. What is correlation function for palm measure? What does it mean correlation function for palm measure? It means correlation function for the initial process but with one more particle Q. So fifth correlation function of palm process is sixth correlation function of initial process with Q substituted. Well then we all know that area of a parallelogram is side times height. Well but this is precisely the height. So you have the side and then you need to take the perpendicular but perpendicular is precisely the projection on the orthogonal complement of the value at Q. So this is just the Shirei Takahashi theorem comes down just to saying that area of parallelogram is side times height. So let me just make one little remark and then formulate one more theorem and then close just that here obviously by such definition the particle Q it always is in the support of the palm measure. So palm measure by definition is supported by configurations which contain particle in Q. So it is convenient to throw away this particle Q. We remember, so we just throw it away. So it's otherwise all results would just have a little bit measure formulation. So for example Shirei Takahashi theorem would be not what I said but would say palm measure is this measure plus particle in Q. So we just throw away this particle Q. This is called reduced palm measure. So the particle in Q so the palm measure what's its difference with the original measure that we just threw away. This particle in Q repels the particles. They don't like the point Q. They don't like it. They don't like to be there because even though we threw it away it is there it repels the other particles. So the question arises whether the palm measure is or is not equivalent to the original measure. And so this is the result in joint work with Shirei and with which I close that the answer is yes. So and in fact so this is joint work with Shirei and in fact the palm measure the derivative of the palm measure with respect to the original measure is just this Psi. Psi so of Y my notation in this sense is a little bit strange because here Y was a parameter and here Y is the variable so but yes but this is the formula in our joint work with Yajitsu. So and once we have equivalence between palm measure and original measure we also can compute the conditional measure and obtain the main result of the talk. Thank you very much. Thank you very much Sasha. Are there any other questions? If our guests online want to ask a question you can either intervene or you can type it in the question box. Any other questions? When processing variant under the isometrics of the Lobachevsky plane that's quite right. Does it follow immediately from that or not? Yes from Peris Paxner of course, yes. In fact, yes, yes, excuse me thank you very much I should have said this more clearly. Yes this determinant is invariant under the isometrics of the Lobachevsky plane. Which is not obvious. Yes, yes, precisely, thank you very much for I should have said this more clearly. Yes, it is immediate from Peris Paxner that the that the process is invariant under isometrics of the Lobachevsky plane. So in fact all these formulas are also invariant under such Lobachevsky. But the Gaussian function is not right. The Gaussian function is not but there is no contradiction because the zero set is and by the way this is the key difficulty improving this result because usually improving uniqueness one uses the fact that our set is separated in Lobachevsky's sense when one proves uniqueness for the disk it is very convenient to be able to assume that the set is separated except this one is not this set is not separated in Lobachevsky's sense. In fact so even though the particles repel but there the event that there some particles are very close has positive probability and by by invariance and in fact ergodicity of the action of Lobachevsky isometries this event will reproduce itself infinitely many times. So this set each individual realization is not separated. So a natural question arises so if you have some results that you proved almost surely there is always a natural question that can you say that if realization X is such, such, such, such, such, such then the result holds and then also say well these properties are satisfied almost surely. But in fact in this case I cannot, I do not have we do not have such explicit characterization it would be very nice to say so in some sense this set is very close to being separated in the sense that these clusters are extremely improbable they arise very rarely but is it possible to say that if a set is such, such, such then it is uniqueness so how to say without expo so to describe the full measure set in a certain explicit manner and we are not able we do not have such description it is for us it is really essential it is essential it is in fact the the the determinant property so the probabilistic how to say we, so our description is something like that the Fubini theorem that some conditional measure theorem holds for this so I mean it is not an explicit description so it is a full measure set in the sense that it is a full measure set so it is not thanks thank you any other questions maybe if there are no other questions just one last very naive question from someone who does not it is not familiar with this field at all so besides the fact that of course in mathematics every question is legitimate but this particular question about the conditional distribution given some configuration does it have a particular motivation because one wants that it is unique maybe with these that there is uniqueness of the Gibbs state but this I don't have proof you mean that some configuration determines the other that's exactly right that do this so this is question which also maybe I should have formulated more explicitly do these conditional measures determine the process imagine I have a process with such conditional measures does is the process determined by them? such results by work of Kaoula and Minadius exist for sign process but in this case no thank you very much Sasha anything else there is a thank you from Ahmed in the chat and so let's thank the speaker again thank you for coming