 Now a square root of n, no, for the one, of the radius square root of n, for the one, around this real axis where you have square root of n eigenvalues, you will see some depletion of complex eigenvalues. This follows basically from this modulus of y factor which shows that when the density of complex eigenvalues goes to zero when you approach this axis, real axis. But otherwise, if we just look at typical scales comparable with the typical size of the circle, you see just its uniform. Okay, so this is more or less the picture that we have for the densities. So coming back to the question I started with, so we see that indeed square root of n is correct scaling to expect the largest real part of the eigenvalue, but now we'd like to understand how much the eigenvalue with the largest real part may fluctuate, may deviate from square root of n. And for this, this knowledge was not available in fact to me in his time, it's available now to us due to basically large deviation theory and recent developments, although it was interestingly not that much discussed in the literature as I found preparing these lectures. So part of it, of course, is well known, but the other part was less known. So let me discuss. Oops. Okay, so I'm happy that Sylvia had her lectures before mine, although originally it was, I know not planned, it just happened accidentally, but frequently accidental changes are beneficial. So I'm grateful to her since she really invested a lot of time in explaining meaning of large deviation ideas for Coulomb guesses. So after that, it will be easier for me, I can just quote a few things. So from now on, I'd like to consider most in most of, sorry, okay. From now on, I'm going to consider, it's natural to rescale variables by square root of n in order to have all eigenvalues concentrated, typically concentrated inside circle of radius one. So we consider, instead of original matrices G, we consider matrices Z, which are G divided by square root of n, and then for such a matrix, mean eigenvalue density are one of Z, which I also probably frequently denote as a row equilibrium density. This terminology should be by now known to you. It's then one over pi if modulus of Z is smaller than one and zero otherwise. So just inside unit circle. So we have circle of unit radius, and so let me just draw it again now of radius one, and we have more or less of order of n complex eigenvalues there, apart from some subdominant proportion of real eigenvalues. So as was several times discussed or mentioned many times in the syllabus lectures, typical separation between complex eigenvalues will be, okay, will be n to minus one half. It was one over d in here lectures, and here we are in two dimensions, so n to minus one half. Obviously, if you just take n points and distribute them inside unit circle. And this means that this is also the scale at which we can expect some non-trivial correlation between eigenvalues, but otherwise if I, for example, fix two values Z one and Z two with final separation and let n tend to infinity, then I should expect that basically properties of eigenvalues or eigenvalue densities, they are uncorrelated on that distance. And in particular, so if I fix, so the fact that I will use later on is that if I consider this n point correlation function, Z n for fixed Z one not equal to Z two not equal, and so on to Z n, Z n fixed afford of unity, then it factorizes in the limit, I would write it like that. When n tends to infinity, we should expect that they just factorize to the product from one to n of one point densities of r i of Z. Densities of r i of Z one. Sorry, r one of Z i. So we'll just factorize. Since arguments then will be already far apart in this scale. I will use it later on. But what we else know about the density we know really the large deviation type result which controls how the empirical density, namely by empirical density is just, I mean the density of the counting measure are n of Z just sum of delta functions, two dimensional delta functions at every eigenvalue of the matrix, one to n divided by n, one divided by n. So this is the empirical density. We know that it converges to equilibrium density, but we also know how deviations from this, what is the probability for this density or for corresponding measure whose density is this empirical density, to deviate from the equilibrium measure. And this is given by large deviation principle, which I now informally state, more or less in the form close to how it was stated in syllabus lectures. So basically, if we consider, okay, call it emu, space of probability measures, symmetric with respect to complex conjugation because we know that it's enough to consider only this because we know that all eigenvalues of real Geneva ensemble, they come in complex conjugate pairs. So it should be symmetric with respect to complex conjugation. So this and consider any measurable subset b in emu, then we know that then large deviation principle states that p, okay, I will call it pn of b, which is probability, probability that corresponding, counting measure, emu whose density is raw, belongs to the subset of eb, is approximately equal to exponential minus n squared, this famous n squared, then infimum over this subset b of some functional g of emu, this large deviation functional, and it's given explicitly, it's given explicitly by the following expression, j of emu equal to one half of integral modulus z squared d emu minus one half integral of log z minus w d emu z d emu of w minus three a, this is the result of Ben-Arus and Z-Tunin and we know that this is the result of z squared d emu, minus one half integral of log z minus w d emu z d emu of z d emu of w minus three a, this is the result of Ben-Arus and Z-Tunin 98, that for real Geneva ensemble, this is a correct form of the large deviation functional, this is nice convex, has unique minimizer, which is given by that equilibrium density, okay, I think after Silvia's lecture, there is no much point to discuss the origin of this, but basically just few words, if you look at the joint probability density expression that I explicitly given to you in the last lecture, you will see, okay, you will see the origin of that, this of course comes just from the modulus of a product of differences z i minus z j, modulus of this Vandermonde factor, you may be surprised Satya Majumdar, I'm not sure that he's here today, but I can answer his question of the last lecture, namely he noticed that exponential factor in the joint probability density contained not, which I showed to you, contained not modulus of that square as he naively expected and which is the case for Geneva with complex entries, with all complex entries, but just z itself, it was fine because we know that to every z there is corresponding complex conjugate eigenvalues, so z squared plus z bar squared is still real, so it can be a correct factor, but here we see that really in large deviation exactly modulus of z squared enters and one, it's simple exercise to show how to massage that corresponding density to get this term, but otherwise this is typical these log gases or Coulomb gases, large deviation function. Okay, so we know this, so now how it, I remind you, our interest is in understanding what is the probability for the eigenvalue with the largest real part to deviate in this scale from one to the left and to the right. Now it's clear, completely clear, that if I'd like to consider such measures for which my eigenvalue with the largest real part is to the left of one, I need to show the whole bunch, I mean the whole rest of eigenvalues to the left and basically it will be penalized precisely by this type of penalty, exponential of minus n squared, since it's basically microscopic, if I'd like to move this eigenvalue to microscopic distance to the left, I show all eigenvalues to the left and then it's penalized by really that expression where corresponding rate will be given by minimum with imposed condition that all eigenvalues are to the left. And clearly equilibrium measure is not in that set so it will be penalized with rate n squared and therefore I can write that I conclude the following simple fact which will be also useful for us later on when we study nonlinear system, surprisingly it will be very helpful or this result or rather it's analog will be very helpful. So I will write that, sorry I have problems with this mic, it goes off all the time. So if I denote now Xm, Xm is a maximum of real part of all the i's from 1 to n, so just maximal real part of all eigenvalues, then this consideration shows that probability that Xm is smaller than X is given by exponential minus n squared, some constant c of x as long as x smaller than 1 and c of x is positive, c of x smaller than 1 is positive and given by corresponding minimization problem. I do not need explicit expression, I just need that it's very severely penalized. What about, okay, this is probability that my eigenvalue, the largest eigenvalue is to the left. What about probability to the right? Is it controlled by this type of large deviation? The answer is no and it's clear morally quite clear why because in order to have such an event that just largest eigenvalue is to the right, it's enough to take a single atom of this measure, just a single eigenvalue and place it some way here. This practically does not change on scale comparable with n, the rest, it still will be almost equilibrium density minus 1 atom, so one cannot see any change in functional here. So it's not really the way to consider it shows that this is controlled by something else and now I'm going to argue that it's given by, in fact, by different large deviation type with different speed, not n squared but n, so my goal is to show that probability goal to explain that probability of xm being larger than x, larger than 1, behaves as exponential minus nL of x and I even will quote explicitly this L of x. So it's less penalized with n rather than when n squared. So how to understand this? So I just will derive basically this probability or leading term of this probability using a nice line of argument that was given for different, basically for the limiting case of this ensemble, for GOE, for limiting case of, not for real Geneva and not for family really which I'm going to discuss in my next lectures tomorrow, ensemble which interpolates between real Geneva ensemble with purely real eigenvalues, GOE, but for GOE it was known and it was given in, it was first calculated by Ben-Arus and Guillaume by using quite different technique but I think there is a very nice and simple argument which gives, which recovers their result and also gives this result which I was not able to find because it seems to be no one was interested in this result and I will give you a brief idea of how one can get it. So basically let me introduce such function, basically it's indicator function, function chi of u which is 1 if u is positive and 0 when u is negative. I know mathematicians prefer to write indicator function but I prefer to write it. This is step function. This is like to call it heavy side step function. Its derivative will be delta function which I also use and using it it's easy to write down the distribution of the largest real part just formally. So what is the distribution of, the density of the largest real part pn of x is derivative of the distribution by definition dEn where En again by definition is just expected value which I agreed to denote with angular brackets, products of just this chi of x minus xi where xi is real part of the i, product of all i to n. So what is written here is just an event that x is larger than every real part and then taking its derivative we just have the density of the largest real part. So why this is, this all is of course evident but slightly massaging it one gets interesting information as was noticed by by Peter Forster. So one writes chi of u is 1 minus chi of minus u replaces every chi with 1 minus chi of its opposite and then John expands the product. So then we get that En obviously is just product of 1 minus chi of xi minus x now. I just revert in the order product of all i and then I just expand this term by term. So the first term is product of 1. Next term minus sum 1 to n chi of xi minus x then next term plus double sum i not equal to j to n chi of xi minus x chi of xj minus x and so on, triple and so on obvious expansion. Now I will use the definition of course it's already gone. The definition of these correlation functions which are obtained marginal densities which are obtained by integrating out all variables but 1 all variables but 2 all variable by 3 it's clear that every term here can be written down using these correlation functions that's in particular why they are useful. So it will be 1 minus integral of r1 of z times chi of real part of z minus x integrated over complex plane this is this term then next term plus 1 divided by 2 factorial double integral r2 r2 of z1 z2 and then product of 2 chi chi of real part of z1 minus x chi of real part of z2 minus x integrated over z1 and z2 and so on the next term will be third correlation function times product of 3 chi and so on and so forth with corresponding k-efficient and now I will use the following the following fact in fact we'll use this factorization but first we are now discussing an event when eigenvalue really when x is larger than 1 so the density here is clearly very small it's 0 when n tends to infinity but when n is big but but finite it's clearly small with n we in fact can explicitly find how small is that if this is the case and also one can show that really these integrals are dominated by in the Lachen limit by fixed values of z1 and z2 I mean not by distances which are small so basically I can use factorization and since we know that r1 is already small clearly all other terms will be higher order of even smaller they will be like square like cube of this small so it means that the dominant term will be just this and using this fact we see that basically the probability of interest to us is simply given in terms of of the object that we spent much time discussing this mean density that one should investigate it with higher precision which does not just give zero here but just give the order of magnitude and this will give us precisely the required result so this E n is approximately one minus one minus integral of r1 of delta of re part of z minus x z plus small terms and then taking derivative ah no no it's chi here sorry it's chi here but now I take derivative P n is obtained by taking derivative over x so I get delta function rather than differentiating chi over x I get delta function so P n of x is approximately equal to integral r1 of z delta function of re z minus x so it means that I can immediately integrate over x is just what remains integral over y right so this is just integral from minus infinity to infinity r1 of x y d y where r1 I remember r1 of x y has two parts has r1 complex of x y plus delta function of y r1 real so then exercise amounts to studying this r1 at values of x outside the circle these amounts to okay we remember that the main ingredient in both r1 real and r1 complex is this incomplete gamma function so one should find asymptotic of this gamma function but this is completely straightforward using integral representation and then one recovers one recovers precisely what I promised did I promise one recovers exactly this with l of x so and one recovers that P n of x as a result of this exercise P n of x larger than 1 is equal to exponential minus n and then explicitly explicit function very simple in fact for this particular case minus log x okay and it's easy to see that this is positive for x larger than 1 and equal to 0 at x equal to 1 which is of course very natural so this is the right large deviation expression for the probability of real part to be to the right okay how much time do I have? minus 5 okay then I have to apologize I didn't notice you probably flagged me but I didn't pay attention so we'll continue tomorrow thank you