 Okay, let us continue so We discussed this representation for the density of Complex eigenvalues and I mentioned that there is similar formula for the density of real eigenvalues Of these elliptic ensemble and now why it's called elliptic If one considers it's Slightly more Involved but not really much more involved using this integral representation of Hermit polynomials substitute here we can find asymptotic expression for For the kernel and eventually then using a large argument behavior of herific function to find the density of Complex and real eigenvalues and Basically it turns out that The density in the large and limit Basically eigenvalues fill in the interior of an ellipse And this is the name this gives the name to the ensemble of the semicircle or sorry of the semi-axis of the Longer semi-axis along x With the right end position one plus tau square root of n and This point the Semi-axis along y direction is at one minus tau square root of n Of course When tau equals to zero we are back to the circle of radius square root of n as we know for Geneva, so this is generalization also in the limit tau approaching one this vertical direction shrinks and All eigenvalues eventually become real When tau is equal to one otherwise There is a gradual process of Transforming this so one can guess that When one minus tau becomes of the order of one over square root of n So there is an untrivial crossover regime, but I'm not going To discuss it now And one also can find that the density of real eigenvalues. It's uniform And depends on tau, but uniform Here and the now for any fixed tau not equal to one The number of real eigenvalues Is of order of square root of n whereas the number of complex eigenvalues is of order n Okay So what about other properties that we? We learned in the previous lectures about Geneva ensemble eigenvalues all of them have their straightforward analogues and for For example The large deviation result for So if we see if one scales Which is natural to scale these matrices x by square root of n by x divided by square root of n then the density Converges to equilibrium measure when n tends to infinity One should just suppress then the square root of n there will be an ellipse with some excess of auto one We always from now on will Consider this rescaling done in order to have well-defined limits and For example large deviation result is practically of the same form so probability of that The measure counting measure whose density is Empirical density to belong to some subset Set B is Exactly of the same form with rate and square and then We should minimize some functional J of Mu Which can be found in the lecture notes. I won't write it. It's Seem quite similar With some modifications including this new parameter tau, but it's quite similar in its properties to the corresponding functional for Geneva ensemble and Basically the same Properties that it's it has unique minimizer and this minimizer is Or the equilibrium density this elliptical equilibrium density. What about probability so of Probability that The eigenvalue with the largest real part exceeds Value x which is now larger than one plus tau So what is this right end probability to be? found somewhere here to the right of this equilibrium density edge The formula or Have the same The same Of the same type as of course for for Geneva, but just I'd like to present explicitly this right This large deviation functional it generalizes known result for For Gaussian Artigonal ensemble and for Geneva so gives true inter interpolation between them. So generalizes Known result in the literature does not seem to be available in the literature. So I In fact just worked it out Preparing this lectures But the method is just generalization and just basically the same method that works for Geneva just slightly more Complicate so This is explicit form of the functional Satya showed us. I think this expression for tau equal one He mentioned g u e but I mean it's not that much important It's practically the same a function. So for tau equal one it goes to known large division function first found by Benarus Demba and G&N then rediscovered by by Satya This is right deviation function Okay Properties are natural at the age So for x larger than one plus tau this is positive function for at the age it's zero as Expected And now Let us rewrite The In terms so now we understand that we should express everything in terms of this Metrices x or rather in terms of this reskilled Ensemble so when I rescale I'll call it new matrix that And what are the formulas that we'd like to? To evaluate so this is mean value of this counting Of counting Over the number of Equilibria I put here index Sigma just which will take two values just to write it Just one formula one value is I will call Sigma takes two values one is EQ which means equilibria it means total number of equilibria and I will use that another index ST for stable equilibria, so what is these in terms expected value of this in terms of Parameters of our problem and properties of this ensemble, so it's one divided by M to N This M to N If you remember in the formula that I showed to you was factor one over mu to N But now I use the may parameter may parameter They are not any longer here, so I will Probably Reintroduce it so may parameter is mu divided by its characteristic value given in terms of variances of gradient and non gradient part of the field So one M to power n and then integral over the real axis then exponential minus and I've X minus M squared Divided by 2 tau Then Dn sigma of X Dx over square root of 2 pi divided by N and Where Dn's are these expected values of Of the Modulus of characteristic polynomial of these Jacobian matrix Namely Dn equilibrium is just Expected value of determinant of X Okay, X delta ij. I will write Minus z ij where z is this rescaled elliptic ensemble rescale to have equilibrium density in interval of order of unity Okay over that and similarly not surprisingly Dn stable is The same basically the same determinant The same determinant, but just Condition chi k of that Where chi is in the indicator function, which is one if If all a real part of Eigen values smaller than X So that's why it bears Index X and zero otherwise so Okay, just a few words how this we remember that it was just expectation of Ensemble of Jacobian matrices and there was no integral. So how this integral appeared this integral is just Basically rescaled integral just averaging over this normal variable psi Since Z is basically Rescaled X So we still need to take into account that our Jacobian matrix Ensemble of Jacobian matrices contains this extra single variable and this is precisely What these gauss and integral it's just That variable in in after shift and disguise Okay so how to evaluate this Expectations of determinant the first thing which I just like to mention that In fact, the first of these determinant can be evaluated exactly Exactly for any n exactly any n any Size of the matrix This is just a fortunate observation. I won't go this route Although I maybe mentioned some results which up to one obtains in this way later on if I have time But what why it's possible to evaluate if you remember? In my lecture yesterday, I discussed the relation between the density of complex eigenvalues and Basically expected value of squared Determinant or basically of Geneva ensemble. This was obtained by employing Incomplete surely composition assuming that the matrix has complex eigenvalues now Complex eigenvalue and then you calculate density of this eigenvalue now Absolutely the same trick can be done assuming that the matrix has one real eigenvalue and then finding density of this real eigenvalue and It turns out that basically Formula for the density of real eigenvalues It's simply proportional to Exactly to this object. It's just okay for the Ensemble with reduced size by by one this time by one in complex. It's by two here It's by one. It's clear from this block structure so since There are explicit formulas due to Forrester and Nagao for the whole I just forgot to mention that First these formulas worked out By in the very nice work by Forrester and Nagao just extending their work earlier work on the Geneva ensemble So these formulas are explicitly given It amount therefore one can calculate just this basically by analyzing Asymptotics of this kernel and in all regimes that we need and this can be no Symptotics can be done, but also I mean for finite and you just explicit result So mean number of equilibria can be found in fact in this model for any N This is we will see when this is helpful, but I'd like today but this method obviously won't or at least I Won't say obviously this method is not obvious how to apply to Evaluate the number of stable equilibria similar problem for purely gradient For purely gradient flow for tau equal one Number of stable equilibria in that situation which are basically Minima counting minimum of this lapen of Lapen of functional which Drives gradient dynamics this can be done as also exactly because there is a mapping of this condition Determinant in that case only in that case One can find a mapping to the problem of the largest eigenvalue exactly to the largest eigenvalue of GOE I'm not aware and I tried The fact that I failed does not mean that you more clever people won't find such a relation But so far I was not able to find any trick which Would help me to evaluate this condition determined and and in our first paper with Boris Korojenko We just left this as an interesting problem, but later on joining forces with Gerard Benaruz Basically, we realized that large deviation theory gives Gives a nice way and simple way in fact of extracting asymptotic Lugging and most interesting behavior of this but still not for finite N. So let me proceed describing this this route wire large deviations and In order to do this let me There was a lot of discussion of large deviations in Silvia surfati Serious of lectures, but I think maybe I'm wrong But I think that she did not mention one fundamental fact about large deviations which is Important here and this is really what is known as Laplace Varadant theorem I think she did not make any use of this or maybe implicitly and therefore did not mention it So what is this statement about so let me remind you that say a random variable X Has large deviation Satisfies large deviation principle if probability, okay, it's indexed by say some parameter n which is eventually tends to infinity and We are interested in this probability of X belonging to some set and informally if this Behavior is exponential with the rate Which I call alpha n for example for in the matrices that we discussed it was for the left alpha n was n squared for the right It was n. So it just speed I Don't know what what is The function is called rate function, but this is called speed. Maybe someone tell me what is correct name for this parameter which governs Really the order of magnitude of this probability and then We we have some rate function And now Laplace Varadant theorem Basically discusses the following consider a function f of variable x which is we will consider to be bounded Bounded function and then consider The object which is natural to call it's a functional of f Which is natural to call Partition function and the frequently it has the meaning of partition function, but not all this But it's basically Just this type of integral so it's Basically, we are averaging we are averaging exponential of this function f of x with some Different speed or in general different speed it may coincide with alpha, but maybe also smaller Or larger than alpha so different speed and we are interested in basically behavior of this as n tends to infinity and The statement is that if alpha so these behavior symptomatically as Exponential minus alpha n then One should minimize Of f plus i Where I Where I is this rate functional for for for the probability of large deviations This is the case when alpha n and alpha are of the same order asymptotically. They are the same And it's especially easy to understand when this pn has a density Which behaves then necessarily exponentially then this is just Laplace evaluation of this integral That's why it's called Laplace, but one are done extended it to much broader class For example, it's x can be not just simple relevant variable But random measure and even more complicated object measures on measures. So it's really extension of These two physicists would call to functional integrals But but basically this is what it is and also Suppose now a second case is when alpha n tilde is smaller than I mean it's sub leading grow into infinity Slower than alpha n then in this case Obviously what happens that the integral will be dominated then the result is then just Exponential of minus alpha and tilled f at I will write it x star where x star is just Minimizer of of this I it's obvious also because then Alpha n tilde if it grows not as fast as alpha n then the integral will be dominated basically by a mini by minimum of the rate function and You just substitute it there and this is the leading behavior. So I will need this Okay, Laplace-Varadan theorem is more general statement, but this is what I need from it and why This is really the helpful bit Which helps to evaluate these dn's? Okay It's very simple now. I need to Get rid of this What is written here? So basically the idea is to represent this determinant in the form when we can apply Basically this Laplace-Varadan Evaluation so how it's done And simply so if we introduce again this empirical measure which is sum of delta functions of senders the eigenvalues of our our metrics and Consider the corresponding measure for which this is density as mu mu n Then one can just literally rewrite dn equilibrium. We start with dn equilibrium This is just then exponential so what I do I simply and right instead of Mod that I just Exponentiated I write it exponential of log of mod that and then Basically, it's immediately clear that I can rewrite it in the following way Phi x of mu Where Phi x of mu is? Just what sometimes is called logarithmic potential for the measure logarithm of x minus z D mu z D mu nz Right, this is just rewriting Exponentiating taking log and then rewriting it but now let us start with With equilibrium and then we will see in the end how to deal with with a number of stable points so We see that this integral basically is is a variant of of this type of expression where Expectation is with With With the law of Of the measure of the counting measure and for for this measure we know that this corresponding speed Is n squared so we are here in in second situation. We are here So we immediately can conclude that asymptotically This will be nothing else So asymptotically applying Laplace-Varadan, okay, there is immediately you can You should tell me you are cheating and I need I'm cheating because I mentioned that Laplace-Varadan is At least as I stated it is valid for bounded function here. We deal with log Which is potentially unbounded? But fortunately the people who are in this business of large deviations and special internet methods they know how to tame This behavior of log at large and small arguments where it potentially dangerous So virtually this is a bounded function or at least one can show that This unboundedness does not produce any problem So I will pretend that it's you can apply straightforwardly this Laplace Varadan and then You just should conclude that leading term is just n phi of x where phi of x is Phi x of mu equilibrium Phi x of mu equilibrium and Then it can be calculated if you see if you take this equilibrium density inside Ellipse you can integrate it with log useful exercise and one gets that phi of x Phi of x is equal to okay. I will write it explicitly Expression maybe not that that Nice but still you can work with it in some sense. It's not short, but it's nice You Can work with it quite efficiently, okay Log divided by two For x smaller than one plus towel So this happens when a parameter x is Under x is real is inside this support of the equilibrium measure Just no wrong when it's outside when it's outside the support of equilibrium measure and inside it's very simple in fact Silver mentioned that it's always inside equilibrium measure. It's basically quadratic. So it's quadratic Okay, it's just it's just quadratic polynomial in x with Trivial coefficients. I won't write it explicitly. Maybe I'll write it when I use it Okay, so If this is correct, we can proceed and So remaining job is to perform this now this integral, but this is relatively straightforward Let us do it so we need to into let us So I will write this An equilibrium Equal to one over M to n integral over all our Exponential minus n. I just deliberately write it l equilibrium of x We were L is obtained by adding up five of x and and this bit here Dex and L is just x minus m squared divided by two tau minus phi of x Okay, so how to Evaluate a symptom of this integral. Obviously one should look for I mean we play in Laplace method to it, but one should do it accurately Really there are because of Different behavior at x larger than one plus tau and a smaller one plus tau one needs to subdivide real axis into three Regimes or three intervals one plus tau minus one plus tau so This for internal part To use one expression and here outside to use different expression for phi and then Proceed evaluating it. So it turns out that really Okay, let me write down consider first but I will call it a a part a so this this integral There L L is given by Okay Now you will see what is a phi inside the support because I can write it here My so it's this quadratic piece, which I mentioned two over one plus tau Plus one half so nice quadratic expression one finds that differentiating and setting to zero one finds that there is Stationary point a stationary point x which is Stationary point of this L where derivative vanishes equal to m times one plus tau So in order To dominate integral inside a one should ensure that m is smaller than smaller than one So if m is larger than one, this is outside the domain of integration So integral will be dominated by the right end one control by the right end Otherwise for m smaller than one it will be dominated by this settle point not settle point to minimum and Calculating everything one gets expression for NA. Okay. I will call it average and a Contribution a contribution from part a Which is different for smaller or larger than m smaller larger than one and This is okay and Expression is exponential minus n Here is M squared minus one over two plus log m a log log m