 I was going to say it's working in free probability and as Percy Diakonis said last week I think everybody who works in random matrices should know free probability and the simple reason is that often if you meet one random matrix you will meet another one and once you have two you need to know some free probability to know what to do. So hopefully Dima will explain you how to do that. All right, well thank you very much again, I think in the interest of time we should just get going and then once the boards are up they could be placed. So at the end of this morning's talk you honored Amitru, she told you that if you take A, a self-adjoint Gaussian random matrix, so it was like that, with Gij normal, right? And they're as independent as they can be. Then you can actually make a computation. You can look at the expected value of the trace, the normalized trace of some moment of N and this she computed and that was the Catalan number, but also it was computed as the number of certain paths. So you started with one M and then you looked at certain walks in this graph, right? And then the idea was that this walk could only backtrack once, right? So every edge could be traversed exactly twice. So there is another way of thinking about such a thing. This is simply the number of ways to parenthesize M variables. So what do I mean by that? So I am supposed to take two M parentheses, half opening, half closing, and then I can just, I'm allowed to write any legal expression in terms of these parentheses, okay? And if you remember how these walks were described saying that this is the same thing as doing a kind of the depth first traversal of a tree, then you see that this is exactly how a computer algebra system or something like this parses an expression like that, right? The various parentheses tell you how deep inside must you go inside the recursion. And now this is just the same thing as the number of what are called planar non-crossing pairings. So what these are is that you look at M points, one, two, up to M, and you're looking at all ways to pair them up in such a way that these pairings when you draw them are non-crossing. So an example of such a pairing, let me just try to do something that corresponds to that bracket, that bracketing there is something like this. You see, all I've done is I've just recorded which bracket cancels which bracket. And it's not very hard to see that if you actually have a diagram like this, which tells you who cancels whom, and a diagram like that, which tells you which bracket closes which, then they are the same, right? And precisely, the fact that this is a legal bracketing states that this pairing is non-crossing. If you try to write down a bracketing expression corresponding to a pairing like this, you quickly see that you end up with too many closed brackets for the numbers that you've opened, right? Because you open one bracket, open another one, but then you try to close this one, right? Which is prohibited. Okay. Now, as Ali said in her introduction, where free probability starts is where you have more than one matrix. So imagine now that we have a whole bunch of them, say a1 of n, ad of m of n, so they're d matrices, and they're entries as before. So what I mean is that each individual matrix is just an iid copy of this matrix here, okay? So the entries of the first matrix are exactly like described here. The entries of the second matrix are independent of the first one and so on, okay? Well, you can do the same thing. You can try to look at the expected value of some kind of a power of a matrix, but now you have d matrices to choose from, and you can write a much more complex power than that, like that. Namely, you can look at something like you can take one of them, say the a first matrix, and then you can put it to some power, let's say m1, and then you can put, say, take some other matrix and you put it to the power m2 and so on. So you can do some computation like that, all right? And again, you would take the product of these matrices, you take their normalized trace, and you look at its expected value. And what you find is that again, this converges, I'm sorry, this should have converges as n goes to infinity. By the way, please interrupt me because I sometimes, by accident, sometimes purposefully will make mistakes. So if you're not paying attention, then, you know, anyways, what will, it will converge to is a very similar kind of thing. You will have, you'll be counting some kind of diagrams drawn on, so let's say that m is the sum of these mk's, it's the total degree, so we have m points over here. And then what you're looking at are exactly these kinds of non-crossing diagrams, so maybe this is paired with that, and maybe this is paired with that, and maybe this is paired with this, and maybe something like that. But now, all the points receive colors. So point number one receives color i1, point number two receives color i2. This is color i3 in all the way to, oh, you know what, let me erase all the m's because it's equivalent, I can just repeat the i's, let me do that. So m is just k in this example, i4 and so on. And then the rule is that the colors must be matched, so you must match colors, okay? So now you're counting not all diagrams that you can draw in a non-crossing way, but only ones which are compatible with a specific coloring of the partition. So for example, if all the colors here are different, what is the number of partitions like that? Zero, right? And indeed, if all the numbers are different, that means that if you look at any entry of this k-fold product, it's going to be a product of k independent Gaussians, and of course that's going to have zero expectations, so for sure the trades will have zero expectations. Anyways, if you think a little bit about what happens in this case, is that we have defined for each p, what's called a non-commutative polynomial, we have defined some expression which will write tau of p, and this will be by definition the limit as n goes to infinity, the expected value of the normalized trace of this p evaluated in my matrices a1 through ad, okay? Right? So here the specific polynomial I took was p, let's say the variables are t1 through td is ti1 through tid, okay? So the point is that random matrix models, assuming convergence, give you these kinds of linear functionals on spaces of non-commutative polynomials. So we have a linear functional tau from all non-commutative polynomials in the variables to c. So formally this is very reminiscent of probability theory because you can in a very algebraic way interpret a usual expectation functional as a way of assigning a number to a polynomial or to a function, a more general test function. So this leads to the definition of what's called a non-commutative probability space. So we are dealing with some kind of a unit algebra, one example of course is this one, the algebra of non-commutative polynomials that I've advertised like over there. But what is very often done is to make the assumption that this algebra is actually represented on a Hilbert space. So why is that a good thing to do? Well, if your algebra is an algebra of operators on a Hilbert space, H is a Hilbert space, B of H is the algebra of all bounded operators in that Hilbert space, what is nice is that this allows you to apply complicated functional calculi to your elements. So for example, you know that if you take an operator in B of H and F is analytic on the complement of the spectrum of T, then it makes sense to talk about F of T as a contour integral by a Cauchy integral formula, right, where this is the spectrum of T and you have chosen some contour gamma outside of that spectrum, OK? So let me think here on a neighborhood of the spectrum of T, right? And then if moreover if T is self-adjoint or more generally what's called a normal operator, then you have the full spectral theorem. You can write T as an integral of spectral projections where E from the spectrum of the operator, which is a subset of R, to projections in when your Hilbert space is a projection-valued measure. It's a measure that assigns to Borel set a projection and so this is the formula for your T. And then you can apply any Borel function on the spectrum by simply this formula over here, OK? So the moment that your algebra is represented in a Hilbert space, you're able to talk about complicated functions of the space going beyond polynomials and that's very useful. Just to give you one example, we can talk about resolvance. So Z minus T inverse is defined for any Z not in the spectrum of my operator. So for example, if Z is sufficiently large and my operator is bounded, the spectrum is some compact subset of complex planes. So that's immediately well-defined because the function 1 over Z minus lambda is analytic for lambda sufficiently big, OK? All right. Now, this linear functional that we have, tau, we will often assume that it's actually implemented by vector. So we will assume that this algebra A, M, which I call in the slide, is sitting in B of H and I have this tau from M to C. I will make the assumption that tau is given by evaluation on a vector. So here, C is some normal unit vector in my Hilbert space. What this ensures is that this plays very nicely with functional calculus. So for example, if I have this decomposition of T, then if I'm interested in tau of F of T, then I can just apply this expression to a vector over here and I see immediately that this is the integral of F of lambda C E D E lambda of C, which is the integral of F of lambda D mu of lambda, where mu is simply tau composed with D E, right? So if I compute, compose this projection valued measure with this vector linear functional tau that I've written down, then I will get a scalar valued measure, OK? And this is very useful because knowing that the functional is given by vector evaluation tells you that everything here actually converges. So we don't have any problem with writing this integral or anything like this because of course this is a bit of a subtle matter as to what this integral actually means, right? And then the last thing that we will often assume is that our linear functional restricted tau algebra is a trace, so tau restricted to M satisfies tau of x, y equals to tau of y, x. One motivation for it is that if you actually look at things that come from random matrix theory, of course that does work because of course anything that you write down will have the cyclic permutation property. If you take a monomial and you permute it cyclically, the answer will be the same because the trace in matrices does this, right? But there are other reasons just simply from the point of view of the general theory. In this case you can attach a very nice LP space to M, and if you assume also the tau is faithful with the definition written above, then actually you have that M sits inside the LP spaces and so on. So you can even talk about the space of measurable operators. These are called sort of measurable relative to M. These are unbounded operators for whom all the spectral projections live in M. And it turns out that in this case this is a very nice algebra which is in the Abelian case like the algebra of measurable functions. Which I should mention by the way, in the Abelian case this is just the usual probability spaces. That's the case where I take M to be essentially bounded functions on some measure space which I can view as bounded operators on L2 of that measure space by sending a function to the operator of point-wise multiplication by that function. And in that case my expectation functional is just again point evaluation. So in this kind of abstract situation we simply are looking at a generalization of the Abelian situation where we're permitting algebras like that to the situation where we simply permit non-Abelian algebras. So in my notes I've written a kind of a lecture zero which I'm not going to give which tries to give you a little bit of the flavor of what are called phenomenon algebras which are the correct framework to work in in this case. These are exactly the correct subalgebras of bounded operators in the Hilbert space. And actually what surprises me all the time is that if you want to define what a classical measure space is it's actually faster to do it using phenomenon algebras just to say that this is an Abelian subalgebra of bounded operators in a Hilbert space satisfying some properties than to go through the usual axioms of measure theory. It's quite fun to actually see it done that way. Anyway so the the carry out from this is that there are these non-commutative algebras like the algebra of polynomials. We typically want them to be represented in a Hilbert space and what we're interested in is we're interested in computing things like that. The trace of some polynomial or more general function like a resolvent of operators in that algebra. Okay so one thing I want to define is what it means to have the to what a non-commutative law means. In one way to say what is a non-commutative law is to say what does it mean to be the same in law. So what does it mean classically that two random variables x and y have the same law? Well one way to say it is that if you apply arbitrary test functions to them then you get the same answer when you take the expected value. So if you take the expected value of f of x that's the same thing as the expected value of f of y okay for all f. Now what functions can you take? Well the largest set of functions you can take is simply all functions that have expectations which is just l1 of your probability measure but in reality if you take a sufficiently big set of functions that is dense inside l1 in the suitable topology then that's enough. So often often enough to take f a holomorphic for instance f of z sorry f of t to be one over t minus z for z fixed or even take f to be polynomials. Another popular choice is to take f of t to be e to the 2 pi i t theta the Fourier transform right because if you take the expected value of this with respect to t you get the Fourier transform of my measure at theta. So there are various sets of functions that you may want to take as test functions and well the first two are enough always to characterize the law of your variable. This is enough to characterize the law of your variable if the measures say compact is supported or if you have some growth conditions. So you define two n-tuples of variables in this non-commutative space to have the same law if when you take the trace of an arbitrary function of them you get the same answer. And for all f here again can be qualified if x's and y's are bounded operators then it's enough to take polynomials this is like this compactly supported case or more generally you may have to take some kinds of resolvents or products of resolvents and so what's nice is that if you know that two such families are the same in law then actually you know that the algebras they generate are isomorphic namely if you simply take x's to y's this map extends to an isomorphism of algebras. So and we already talked about the case of a single variable if you take a single self-adjoint variable then what we mean by laws is nothing more than the usual classical probability measures right because this is just the spectral theorem it tells you how to evaluate the trace of an arbitrary function it will be integration against some probability measure okay all right well this sounded very abstract let me just give you some examples right away so the first example I want to give is that of a discrete group so let's take as an example gamma to be the integers z then I can look at l2 of gamma so this is the Hilbert space whose basis is my group gamma okay so in this case this is just the the orthonormal span of things like delta n n in z right I have a an orthonormal basis element for every element of the group now every group acts on its l2 space I have that's called the left regular representation from gamma to l2 of gamma it's defined by saying that the action on the delta function of h is just the delta function at gh so in this case if I look at lambda of 1 this is simply the shift that takes delta n to delta n plus 1 right the bilateral shift on l2 of the integers right yes yes to bounded operators and l2 of gamma thank you okay um so lambda of 3 is the shift by 3 right and so forth uh so now if you look at the algebra generated by all these things by all of these lambda of n n in z you can identify them if you if you take the Fourier transform map and identify it with l2 of the circle using Fourier transform then this algebra can be identified simply with trigonometric polynomials because this this shift right corresponds to multiplication to e to the i theta and so a linear combination of such shifts would correspond to a linear combination of e to the i n theta which is a trigonometric polynomial um now the thing that we often do here is to look at what's called the phenomenon algebra generated by all of these lambda n's n in z this is what's sometimes called the group phenomenon algebra l gamma and uh that algebra will correspond to a certain closure of these polynomials in fact it will correspond to all essentially bounded functions on the circle okay so you see why taking this bigger algebra is useful it allows you to play with more functional calculus right I mean you cannot apply an arbitrary function to compose a polynomial with an arbitrary function you won't get a polynomial but if you apply an arbitrary function to polynomials you get things like that and finally there is a a preferred linear functional there's a preferred vector in l2 which is the delta function at the neutral element of the group and so you define tau of x to be delta e x delta e and if you work out what is tau of lambda of g you see that well let's work it out it's delta e lambda of g acting on delta e so delta e gets shifted to delta g so we get delta e delta g which is one precisely when g is the neutral element so this linear functional just records whether you are a trivial element of g or a non-trivial element of g and in this isomorphism by the way the trace tau corresponds to integration against the uh harm measure on the circle right this is what it does if you take a function trigonometric polynomial such an integral gives you the trivial I mean the constant coefficient of this polynomial and all the other coefficients are killed now um this setup is is very often used actually in probability maybe not quite said this way but nonetheless let's do it suppose we take s inside g inside gamma a symmetric generating set the symmetric simply means that together with every element the inverse of that element is there so for instance we could take s to be plus minus one inside z these are two generators of z and i'm put plus one and minus one to make it symmetric and then we can look at this kind of random walk Laplacian it's called which is in this case going to be one half of lambda of one plus lambda of negative one right and so now it turns out that if i'm interested in things like tau of lambda to the m which is delta e lambda to the m delta e then this is something that is very natural has a probabilistic interpretation namely you start with uh the your z this is zero one two minus one minus two and so forth you start with some point let's say it's zero and then you toss a coin and depending on whether it comes out heads or tails you walk either forward or backward in other words you pick one of these elements of s and you either and you add that element okay so if you're here with probability of half you jump here probability of half you jump there having jumped here you do the exact same thing again you take another independent coin you flip it and depending on that you might jump forward or back so let's say you jump forward and then maybe you jump back and maybe you jump forward and maybe you jump back and jump back again so in fact this is going to be the probability of return to the neutral element after m jumps and what this general theory is telling you is that you can actually write down this probability uh the style l l m you can write it as the moment of some probability measure and one of the exercises that are in the notes is to actually compute what that measure is it's not very hard given all the identifications here but it's some exercise that you should do you know to understand how this works okay now there is a a notion of independence that you can talk about in this very general framework of non commutative probability spaces so here it is if i have two sub algebras a one a two in a probability space they're independent precisely when first of all they commute and secondly you have this funny law that the expected value of the product of two variables is zero and i wrote it in a very funny way provided that each variable has zero expectation and provided that they come from different algebras this condition here that i one is different from i two simply saying that a one comes from either a one or a two and a two comes from a different algebra than a one came from okay and you can check that this is exactly the same thing as the usual way of saying uh independence which is that the product of sorry the expect expectation of a product is the expectation of is the product of expectations the only thing that i've done is i've centered the variables but that can always be done because if you re-center the variables here in here you see uh that the general case can be reduced from from the centered case there's just a little algebraic manipulation that you can do so one example of such a situation would be to take a product of two groups let's say instead of taking a random walk in gamma z i could i look at z squared right then i will have generators corresponding to uh i will have things like lambda of one zero lambda of minus one zero lambda of zero one lambda of zero negative one and the point is that these guys are going to be independent from these as it's very easy to check okay and now what's amazing is that in this non-commutative framework there is a new parallel notion of independence which is called freely independent and the rule that this time is a little different so again i have two sub-algebras a1 a2 and i say that they are freely independent if no condition on commutation anymore so they do not commute but then i want to say that a product of a whole bunch of variables has expectation zero provided that first of all each one of them is centered so it has expectation zero and the consecutive ones come from different algebras so if the first one comes from the algebra i1 and the second one comes from the algebra i2 the third one comes with the algebra i3 then the first one comes from an algebra that's different than the next one and this one comes from the algebra that's different from the next one and so on so this looks very much like the previous one we had only there you could get by with just products of two because of commutation here you have to deal with products of just of n and one example which you will work out and written in the notes is the case of a free product of two groups so instead of z squared you could look at z free product z so that would be a free group on two generators and you can also look at generators of that you will have let's say four generators to corresponding the first copy of z to corresponding the second copy of z and these things are now going to be freely independent now this random walk for z times z will be a random walk on z squared so on the integer lattice the random walk here will be on a by on a on a tree on the calligraph of this group which will be a certain tree okay all right so there are there's a couple of main theorems that you can prove which are very simple the first one says that if i know that my two of my algebras a1 and a2 are independent so let's say a1 a2 are freely independent freely independent and i know the restriction of my i know how to take expectations in a1 and i know how to expect to take expectations on a2 then i i know how to take expectations on the algebra generated by a1 and a2 and here's the trick suppose i want to prove figure out what does tell of a1 a2 a3 and so forth an well the first thing i can do is i can assume that each one of those a's lives in different algebras why well let's say that a1 lives in a1 and let's say that a2 also lives in a1 well then their product a1 a2 also lives in a1 because it's an algebra