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The right way to set up the classical limit theorems of probability theory is to start with a triangular array of real random variables

Your first exposure to the law of large numbers and central limit theorem likely involved a different setup, namely an infinite sequence

of iid random variables, whose partial sums

are the main object of interest. There are some non-negligible advantages to the triangular array setup, for example the fact that the rows do not have to be defined on a common underlying probability space. The behavior of the row sums

is the topic of slightly more advanced and flexible versions of the central limit theorem. This result and other classical limit theorems apply to the situation when the random variables in each row of the triangular array are independent or nearly independent.

We are interested in triangular arrays of random variables as above which arise from an underlying deterministic data set

The triangular array of random variables associated to this data has rows defined by

where is a uniformly random unitary matrix of rank Thus, the random vector is a random linear transformation of the deterministic data Because the entries of are highly correlated (rows are orthonormal, columns are orthonormal) the random variables are far from independent. They are, however, exchangeable.

There are two ways to think about a Schur-Horn array. Geometrically, is a random point inside the permutahedron in generated by Algebraically, is the diagonal of the random matrix , where

We are interested in proving limit theorems for Schur-Horn arrays. Such theorems will describe behavior of the rows of the -array in terms of prescribed behavior of the rows of the -array. In particular, we are interested in the situation where the empirical distribution of the vector

converges in moments as where is a scaling exponent. The empirical distribution is the probability measure on which places mass at each coordinate of the vector If we think of these data points as beads on a wire, then the empirical measure of is the fraction of beads which lie in Convergence in moments of the empirical measure means that the the limit

exists for each fixed where is the power sum polynomial of degree in variables.

Let denote expected value, and let

be the characteristic function of Because the distribution of in is compactly supported, is an analytic function on The Maclaurin expansion of the characteristic function is

where

and the sum is over all vectors of nonnegative integers whose coordinates sum to Thus, is a homogeneous degree polynomial in variables whose coefficients are, up to a transparent factor, the degree joint moments of the random variables . Since the logarithm of of the characteristic function is defined and analytic on an open neighborhood of the origin in We write the Maclaurin expansion of as

where is a homogeneous degree polynomial in variables. The polynomials are called the cumulant polynomials of the random variables They are determined by the moment polynomials according to a certain recursion explained on pages 2 and 3 of these notes. We write the cumulant polynomials in the form

where the quantities are the degree joint cumulants of the random variables as ranges over nonnegative integer vectors of sum . Note that this is a definition, not a proposition. In statistical mechanics and quantum field theory, joint cumulants are referred to as connected correlation functions.

The main fact about Schur-Horn arrays that allows us to understand them is a second formula for the cumulant polynomials of the random variables in terms of the moments of the empirical distribution of the deterministic data

Theorem (Cumulant Formula for Schur-Horn Arrays). For each we have

where the sums are over nonnegative integer vectors and with non-increasing coordinates which sum to and and are the number of nonzero coordinates of and respectively. For each such pair the polynomials and are the corresponding products of power sum symmetric polynomials, i.e.

Finally,

is a convergent series in which the coefficients are positive integers called monotone Hurwitz numbers which admit an explicit combinatorial description.

The cumulant formula for Schur-Horn arrays looks complicated, but it is fairly easy to use in practice. For example, let us set our scaling exponent to Then, we for any fixed we have

As we saw (twice) in lecture, you can calculate the limit of this polynomial. Assuming and the only nonzero limit is

This means that

unless is a nonnegative integer vector with total sum and only one nonzero coordinate. From this we conclude that for any fixed the random variables converge in joint distribution to iid standard Gaussians.

When our scaling exponent is we do not get such a simple Gaussian limit. Rather, we find that

for each fixed where

where Thus we still have a sum of pure powers indicating asymptotic independence between the random variables but the cumulants of these random variables are asymptotically We are going to see that

is the free cumulant sequence corresponding to the moment sequence

Today we want to prove the “inverse Wigner theorem” we started discussing last time. Let be a sequence of traceless deterministic matrices, and for each let be a uniformly random Hermitian matrix isospectral with Suppose that the empirical eigenvalue distribution of converges in moments as meaning that the limit

exists for each Equivalently, we have

with any enumeration of the eigenvalues of and the power sum polynomial of degree Let denote the diagonal matrix elements of the random matrix so

with a uniformly random unitary matrix.

Theorem 15.1. For any fixed the random vector converges in moments to a vector of iid centered Gaussians of variance More precisely, we have

for any vector of nonnegative integers.

Here is a corollary which further explains why we are calling Theorem 15.1 an inverse Wigner theorem.

Corollary 15.1. For any the principal submatrix of converges to i.e. to an random Hermitian matrix following a centered Gaussian distribution of variance

Problem 15.1. Deduce Corollary 15.1 from Theorem 15.1.

The rest of the lecture consists of the proof of Theorem 15.1. First, let us explain the assumption that our deterministic data has trace zero. The random variables are identically distributed (but not independent), with first moment

So, assuming is equivalent to centering

Now we start the proof. The joint distribution of is a compactly supported probability measure on — indeed, it is supported on the compact convex set whose extreme points are the distinct permutations of the deterministic vector Therefore, the Laplace transform

is an entire function of the complex variables i.e. is an analytic function on Note that if we set and restricting the remaining variables , the Laplace transform specializes to the characteristic function of so that it completely determines the distribution of this random vector. The variable is redundant, but it is a convenient marker of homogeneity: we write the Maclaurin expansion of as

where is a homogeneous polynomial of degree . Indeed, this polynomial is

which in the standard monomial basis of homogeneous polynomials is

the sum being over all vectors of nonnegative integers whose coordinates sum to , aka weak -part compositions of We call the degree moment polynomial of because its coefficients in the monomial basis are the degree joint moments of these random variables (up to a multinomial coefficient). So our objective is to understand the behavior of this polynomial restricted to the -dimensional subspace of spanned by the first coordinate vectors, where is arbitrary but fixed. To be completely explicit, this is the polynomial

the sum being over weak -part compositions of This is an important simplification, since the number of variables of is fixed, i.e. its domain does not vary with

In fact, it is more convenient to work with the log-Laplace transform of More precisely, since there exists a simply connected open neighborhood of the origin in on which is non vanishing, and there is a unique analytic function vanishing at the origin which satisfies

Let

be the Maclaurin expansion of this analytic function, so is a homogeneous degree polynomial. This is called the degree cumulant polynomial of the random variables . According to the Exponential Formula (aka moment-cumulant formula), these cumulants are recursively determined by the moment polynomials by a triangular system of equations, the first three of which are (suppressing the variables )

We can write the polynomial as

except for the subscript on the expectation-like quantity $\langle X_{N1}^{\alpha_1} \dots X_{NN}^{\alpha_N}\rangle_c$, which is known as a degree joint cumulant of Note that this is a definition, not a proposition – the relationship between the Maclaurin expansion of the Laplace transform and the log-Laplace transform implicitly defines these statistics.

Problem 15.2. Show that is the covariance of and

Our discussion so far pertains to any finite family of random variables with compactly supported joint distribution. Now we will start invoking special features of the family we are actually dealing with. The first of these is that these random variables are exchangeable, which is equivalent to saying that the cumulant polynomials are symmetric polynomials. This means that we can also expand them as linear combinations of Newton polynomials, and as previously discussed have a powerful understanding of this expansion.

Theorem 15.2 (Leading Cumulants Formula). For any we have

where each sum is over the set of integer partitions of and

is a convergent power series in whose coefficients are positive integers.

The combinatorial coefficients are quite intricate and there is no simple closed formula for them (you can read this if you want to know more), but luckily we do not need to know very much about them in order to prove our inverse Wigner theorem. All that matters for us is that this magic formula is expressing the cumulant polynomials of in terms of the moments of the empirical eigenvalue distribution of our deterministic data matrices in a quite transparent way.

To leverage this, let us restrict to the cumulant polynomials of , which just means setting the last variables of equal to zero (this eventually makes sense because is fixed and increases without bound). We write the resulting polynomial in the form

Ignore the factor of for a moment and just think about the behavior of the double sum as . The number of terms in this sum is independent of , which means that we only need to understand the asymptotics of each individual term

Reading from right to left, we have that as By hypothesis, we have

Finally, since

we have

as Thus, the whole double sum is as so now is a good time to remember the prefactor and immediately conclude that

as , uniformly on compact subsets of for all For we have

uniformly on compact subsets of and since and this simplifies to

Finally, the polynomial is identically zero by our trace zero hypothesis. What we have shown is that all joint cumulants of converge to zero as except for pure cumulants of degree two,

each of which converges to

Problem 15.3. Compute the joint cumulants of a finite family of centered iid Gaussians and complete the proof.

In this lecture I wanted to connect our (admittedly meandering) discussion so far to the standard approach to limit theorems in random matrix theory. Let be a random Hermitian matrix with characteristic function

This means precisely that the distribution of in with Hilbert-Schmidt scalar product is the standard Gaussian measure on this Euclidean space.

Definition 14.1. The Gaussian Unitary Ensemble is the sequence

The word “unitary” here means that the distribution of is unitarily invariant. Let $B_{N1} \geq \dots \geq B_{NN}$ be the eigenvalues of

Theorem 14.1 (Wigner). For each the limit

exits and is given by

In Theorem 14.1, is the power sum symmetric polynomial of degree , and is the th Catalan number. There are two things we should discuss: what the result means, and how to prove it.

Concerning the meaning of Theorem 14.1, the random variable

is the degree moment of the empirical eigenvalue distribution of which is the random discrete probability measure on which places equal mass at each eigenvalue. Thus, Theorem 14.1 is saying that each moment of the empirical eigenvalue distribution of the scaled random matrix converges in expectation to an explicit limit. It is therefore natural to ask whether is the moment sequence of a probability measure on To answer this, one can recognize that the exponential generating function

is essentially a Bessel function. Then, a classical integral representation for Bessel functions allows one to determine that is the moment sequence of the probability measure on which is supported on with density

You can find the details of this derivation here, towards the end of Lecture One.

Now we discuss how one can prove Theorem 14.1. The proof uses the fact that power sums in the eigenvalues of a matrix are also traces of powers of that matrix. This means that we have

the point being that the right hand side is also a polynomial in the elements of the Gaussian random matrix . Indeed, by matrix algebra we have

where the sum is over all functions

Using the fact that the matrix elements of are Gaussian and independent (up to selfadjointness), the following can be established.

Theorem 14.2 (Harer-Zagier). For each we have

where is the number of ways to glue the edges of a -gon in pairs so as to produce a compact orientable surface of genus The proof is a beautiful piece of mathematics and a nice treatment can be found here. Theorem 14.2 implies Theorem 14.1 because the number of ways to glue a sphere from a polygon with sides is and only the genus zero term survives in the limit.

There is something unsatisfying about the above sequence of ideas. We know that the entire distribution of is completely determined by the joint distribution of its diagonal matrix elements

which are simply iid standard Gaussians. In fact, at this point in the course we know much more: the spectral analysis of unitarily invariant random Hermitian matrices is equivalent to the analysis of pairs

and

of triangular arrays of real random variables related by

where is a uniformly random unitary matrix. Initially, were viewed as the diagonal matrix elements of a random Hermitian matrix with unitarily invariant distribution and were the eigenvalues of this selfadjoint matrix. But, we can in fact forget about this original setup entirely, and simply be probabilists thinking about two randomly coupled triangular arrays of real random variables. The question then how to recover Theorem 14.1 simply from the given fact that our -array has rows of iid standard Gaussians.

This point of view is why the paper of Olshanski-Vershik is so relevant for our purposes, and so far ahead of its time: they are proving inverse theorems of this kind. More precisely, they are considering the case where the -array is a given deterministic data set, and proving limit theorems about the corresponding -array. A special case of their main result is the following “inverse Wigner theorem.”

Theorem 14.3 (Olshanski-Vershik). Suppose that the limit

exists for each Then, for any fixed , the random variables converge in distribution to an -tuple of iid centered Gaussians with variance

We have proved the one-dimensional version of this Gaussian limit theorem (the case ), and we will finish the proof of the multivariate version next time (the case ). In our approach to these inverse results of Olshanski-Vershik we are using a combinatorial result which is analogous to Theorem 14.2, but arguably even more miraculous: a universal genus expansion for joint cumulants of the -array in our theory of unistochastically coupled triangular arrays.

Let be a sequence of deterministic Hermitian matrices, and let be any enumeration of the eigenvalues of Consider the corresponding sequence of random Hermitian matrices defined by where is a random unitary matrix whose distribution in is Haar measure. We then have a corresponding triangular array of real random variables,

whose th row

consists of the diagonal elements of the random matrix In terms of our input data we have

This is superficially similar to the setup for the Central Limit Theorem, but it is different: are exchangeable but not independent (make sure you understand why). Thus our first objective is simply to understand the asymptotic distribution of the -matrix element of a uniformly random Hermitian matrix with spectrum

In this lecture I will use the angled brackets notation favored by physicists to denote expectation. I will also use the angled bracket with a subscript “” for cumulants (the subscript could also stand for connected). So for example the variance of is

Problem 13.1. Prove that Also show that for we have for any constant (this translation invariance is a general property of cumulants).

Our Optimized Leading Cumulants Formula says that for any we have

where

and

is times a convergent positive series.

What is nice about the Optimized Leading Cumulants Formula is that it almost immediately suggests the possibility of Gaussian limiting behavior: because of the factor in the formula, it looks like cumulants of degree should vanish in the limit, which is the Gaussian signature. To actually establish this we need to determine how the rest of the formula behaves as grows large.

This is not so difficult. We have the finite sum

and the number of terms in the sum has no dependence on . The quantity is as (make sure you understand why). This leaves the fraction

The denominator is literally just a power of . As for the numerator, we have the bound

where is the spectral radius of Thus, we have

We therefore have the cumulant bound

where depends only on

Theorem 13.1. If the input sequence of Hermitian matrices is such that for some constant and all and if exists, then the centered random variable converges to a centered Gaussian of variance

Proof: By definition the first cumulant of is zero for every and from translation invariance of cumulants and the bound established above we have that all cumulants of degree converge to zero. From the Optimized Leading Cumulants formula and the estimates above, the only term in the second cumulant is exactly Therefore, converges to a centered Gaussian random variable of variance as and the convergence holds in distribution by the Moment Method.

-QED

This week we will finish working through the Olshanski-Vershik limit theorem, and explain how the tools we have developed along the way have set us up to prove other kinds of limit theorems about random matrices.

Recall the setting of the OV paper: we have an infinite triangular array of real numbers,

Assume the entries are sorted so that each row is non-increasing from left to right and let

be the th row. Thinking of as a diagonal matrix, define a corresponding random Hermitian matrix by

We know that in the characteristic function of

we can assume without loss in generality that is diagonal, so what we are really looking at is the sequence of characteristic functions

corresponding to our data set given as

or equivalently

where the diagonal matrix elements are exchangeable (but not independent) real random variables. Part I of the Olshanski-Vershik theorem is concerned with finding necessary and sufficient conditions on the triangular array such that the characteristic function

of a single diagonal converges as Note that extracting the -entry on either side of we have

showing explicitly how is built from the uniformly random unit vector in and the data

The characteristic function of has the power series expansion

where are the moments of this real random variable. Olshanski-Vershik being by finding sufficient conditions on the data such that, for each fixed the sequence of numbers converges to a limit as In other words, they begin by finding conditions on the data such that the random variable converges in moments. We will do the same thing, but in a different way, using the following consequence of Theorem 10.1 in Lecture 10.

Theorem 12.1 (Leading Moments Formula). For each we have

where the inner sum is absolutely convergent and is the number of -step monotone walks on the symmetric group of degree which begin at a permutation of cycle type and end at a permutation of cycle type

The most efficient way to use this result is to take a logarithm, i.e. to work with the cumulants of . This is also the path followed by Olshanski-Vershik, but the method they use to do the calculation is different.

Let

be the log-characteristic function of i.e. the cumulant generating function. The cumulant sequence contains the same information as the moment sequence via the moment-cumulant relations

The cumulants encode the “connected” version of the combinatorial problem encoded by the moments

Theorem 12.2 (Leading Cumulants Formula). For each we have

where is the number of walks counted by which have the additional feature that their steps and endpoints generate a transitive subgroup of the symmetric group.

The Leading Cumulants Formula is identical to the Leading Moments formula except that the walk count has been replaced with the connected walk count . This seemingly minor change is in fact very advantageous.

Theorem 12.3 (Riemann-Hurwitz Formula). The number is nonzero if and only if there is such that

Due to the Riemann-Hurwitz formula it is more convenient to parameterize the numbers by the parameter i.e. to set

Theorem 12.4 (Optimized Leading Cumulants Formula). For each we have

Now we are in position to prove some interesting limit theorems. We begin by proving one such theorem corresponding to an explicit choice of the dataset (this corresponds to Remark 4.5 in Olshanski-Vershik).

Let be a positive constant, and suppose our dataset has th row given by

where the number of positive and negative entries is as near to equal as possible – equal if even and differing by if odd, where we assume the data favors positive numbers in the latter case. In this situation we can establish the following limit theorem for a single diagonal matrix element of

Theorem 12.5 (Gaussian Limit Theorem). Any diagonal matrix element of converges in distribution to a centered Gaussian random variable of variance

Proof: Let be a partition of Then, we have

and therefore

Thus, from the Optimized Leading Cumulants Formula we have

showing that converges to zero for all This means that converges in cumulants (equivalently, in moments) to a Gaussian random variable whose mean and variance remain to be determined.

The first cumulant of is

and this converges to as For the second cumulant,

and

Thus, we have shown that converges in moments/cumulants to In fact, this implies in distribution as thanks to the Method of Moments.

-QED

We start with a brief recap of what we are trying to achieve – since things are getting complicated it might be helpful to step back for a moment before stepping further in.

Let be the Eucliden space of Hermitian matrices, and let be the compact group of unitary matrices. We have a group action

where is the orthogonal group of the Euclidean space , defined by

This group action is called unitary conjugation. Let

denote the orbit of a given Hermitian matrix under unitary conjugation.

Theorem 10.1 (Spectral Theorem). For every , the orbit contains a diagonal matrix which is unique up to permutations of its diagonal elements.

Let

We will refer to this orthant of as the Weyl chamber. This term comes from Lie theory – we are not doing Lie theory, we just need a name for . By the Spectral Theorem, the orbits of under unitary conjugation are indexed by points of the Weyl chamber: we set

where on the right hand side we are viewing as a diagonal matrix.

A standard problem in linear algebra is the following: we are given a matrix meaning that we are explicitly given all entries of this matrix, and from this information we are asked to find such that The Spectral Theorem guarantees that this problem has a unique solution. The coordinates of the solution are the eigenvalues of the given matrix

We are interested in solving a generic version of the inverse problem: given determine the matrix elements of a uniformly random point Since is given, we are being asked about the random Hermitian matrix where is a uniformly random unitary matrix from This is something we know how to define precisely (Haar measure).

We want to solve this inverse problem by computing the characteristic function of which is

We know that it is in fact sufficient to do this for being a diagonal Hermitian matrix, and we have introduced the notation

in order to emphasize the fact that we can view this characteristic function as defining a bounded symmetric kernel

Furthermore, is actually the characteristic function of the diagonal of so a solution to our problem is the same thing as determining the distribution of the diagonal of a uniformly random Hermitian matrix with prescribed eigenvalues.

We can view as a generating function for the joint moments of the diagonal matrix elements , meaning that

where the internal sum runs over the set of weak -part compositions of the number which are simply vectors of nonnegative integers whose coordinates sum to The coefficients corresponding to a given is a very simple quantity, namely the multinomial coefficient

But, it is Haard (sorry, couldn’t resist) to see how to directly compute all mixed moments of the random variables Indeed, we actually have a formula for each of these random variables, namely

which is obtained from simply by matrix multiplication. This formula more or less shows that a direct computation of, say, the moment sequence of just involves computing a family of Haar integrals over which we don’t know how to contend with.

But is a moment generating function in another sense: we have

where the internal (double) sum is over the set of Young diagrams with first row of length at most ; such diagrams are pictorial representations of partitions of with largest part at most We know that such an expansion exists by a “soft” symmetry argument combined with a classical theorem of Newton: we use that is invariant under independent permutations of the coordinates of the vectors and The moments involved in this expansion are the moments of the (non-normalized) empirical eigenvalue distributions of these vectors, i.e. the discrete measures and on which place unit mass at each coordinate. The moments of these measures are exactly sums of powers of the coordinates of and

This leaves the problem of computing the coefficients in this nonstandard moment generating function. This can be done using representation theory – in fact one needs to know quite a bit about representations of the unitary and symmetric groups, as well as about relations between them the relationship between them. We will mostly black box this to avoid going too deeply into algebra (though an indication of what needs to be done was given in the previous lecture, where we discussed Schur polynomials). Instead, we will focus on two facts: first, the coefficients have an intrinsically interesting combinatorial interpretation, which is something that often happens with moment generating functions of interesting random variables (we discussed the example of Gaussians in lecture); second, this combinatorial interpretation will allow us to prove probabilistically meaningful limit theorems. The simplest such theorem is the result of Olshanski and Vershik on the limit joint distribution of where is arbitrary but fixed (you should be actively reading the Olshanski-Vershik paper while we are doing this).

Theorem 10.1. For every integer and every pair of Young diagrams , we have

where the series is absolutely convergent and is the number of monotone -step walks on the Cayley graph of the symmetric group which begin at a permutation of cycle type and end at a permutation of cycle type

Clearly there is much to be unpacked here, but ultimately it is not so complicated, and moreover only a small amount of the information contained in this statement is needed to prove limit theorems about random matrices. A very basic observation is that in the range the set is simply the set of all Young diagrams with exactly cells.

Now we have to explain what we mean by the Cayley graph of the symmetric group, and what we mean by a monotone walk on this graph. For this I am linking to a paper which hopefully provides a reasonably clear account of this combinatorial construction (all you need to read is Section 2.1 and 2.2, just a couple of pages). After you have read this you will know what the Cayley graph of the symmetric group is, what is meant by a strictly monotone walk on this graph. Furthermore, you will know the remarkable fact that there is a unique strictly monotone walk between any two permutations, and this walk is a geodesic.

As explained in lecture, it is often the case that the logarithm of the characteristic function is easier to work with than the characteristic function itself – it is a generating function for cumulants rather than moments. Let us consider the logarithm of the characteristic function of

This week we are looking at a landmark paper by Grigori Olshanski and Anatoly Vershik, “Ergodic Unitarily Invariant Measures on the Space of Infinite Hermitian Matrices.” This paper is among the first to consider characteristic functions in the context of random matrix theory. Namely, it considers the problem of approximating the function

defined by

where the integration is over the unitary group of rank against Haar measure. We have referred to this function as the “quantum Fourier kernel,” but as we have stressed it is also the classical characteristic function of the diagonal of of a uniformly random Hermitian matrix with deterministic eigenvalues The paper of Olshanski-Vershik is about approximating as which probabilistically means that we are trying to understand how the joint distribution of the diagonal matrix elements of a very large uniformly random Hermitian matrix with prescribed eigenvalues is determined by the vector This is an example of a high-dimensional problem: we are trying to approximate a function: our goal is to approximate a function

of real variables as the number of variables goes to infinity. Typically such problems are much harder than traditional problems of asymptotic analysis, which consider the approximation of some parameter-dependent family of functions defined on a fixed domain as the parameter approaches a given value.

One way to simplify the question is to consider the restriction of to the subspace with fixed. That is, we consider the characteristic function

of only the first diagonal matrix elements of the random Hermitian matrix Recall that we have previously shown that all diagonal matrix elements of are identically distributed, but not necessarily independent. The paper of Olshanski and Vershik shows that if the eigenvalues , scaled by a factor of converge to a point configuration on as then the random variables converge to independent random variables whose characteristic function can be calculated explicitly in terms of this point configuration. We will try to work through this result, which I believe to be the first high-dimensional limit theorem in RMT obtained via characteristic functions.

The Olshanski-Vershik approach of computing the asymptotics of the characteristic function

is based on series expansion of this matrix integral. I am a big proponent of this approach, but it does require some quite extensive background from representation theory and algebraic combinatorics which is not traditionally found in the realm of probability. We will try not to get too bogged down in the algebra, and in order to do this it will be necessary to accept some results from representation theory as black boxes.

First, let us analytically continue the characteristic function to a function

by declaring

where is a complex number and are complex vectors. Thus, by compactness of , we have that is an entire function of complex variables and the Maclaurin series of this function, which is a multivariate Taylor series centered at the origin, converges uniformly absolutely on compact subsets of We recover our original characteristic function by setting and restricting to .

The analytic function has various symmetries that allow us to present its Maclaurin series in a more advantageous form. More precisely, is invariant under independent permutations of and , and also stable under swapping these two sets of variables. This means that the Macluarin series of can be presented in the form

where is a homogeneous degree symmetric polynomial in and also a homogeneous degree symmetric polynomial in . Therefore, by Newton’s theorem on symmetric polynomials, we can write this partial derivative as

where is the set of integer partitions of the degree which we identify with the set of Young diagrams with cells, and and are the Newton power-sum symmetric polynomials corresponding to For example, if and then

Our objective then is to calculate the numerical coefficients of the polynomial which we will refer to as its Newton coefficients.

Problem 9.1. Show that and use this to show that

What is good about the Newton polynomials is that they connect us to point configurations, which is what we want. More precisely, if is a real vector, then we can view it not as a point in -dimensional space but as a configuration of particles on the real line. The empirical distribution of is the discrete measure on which places unit mass at each particle, and the degree moment of this measure is

This means that writing the Maclaurin expansion of $K_N(z,A,B)$ in terms of Newton symmetric polynomials is exactly what we want to do, because it is expressing this function in terms of the empirical distributions of its arguments , and the empirical distribution of a point configuration is a natural way to describe the “shape” of the configuration in the infinite particle limit where

Unfortunately, there is no obvious way to calculate the Newton coefficients What we can do is compute the Maclaurin series of by repeatedly differentiating under the integral sign with respect to . Equivalently, we just expand the exponential function of in the inetegrand and then integrate term-by-term, which gives

This gives us the formula

which at least reduces our initial problem of computing an exponential integral over to computing a polynomial integral over

but still there is no obvious path forward.

At this point we should ask ourselves what integrals over we actually can compute. To make the discussion as simple as possible, take so is just the unit circle in One thing we can confidently state is the orthogonality relation

Representation theory gives us analogous formulas for all namely a system of multivariate polynomials which are orthogonal polynomials for integration with respect to Haar measure on Let be the set of all Young diagrams with at most rows.

Defintion 9.1. For each the corresponding Schur symmetric polynomial is defined by

where are the row lengths of

Out of the many equivalent definitions of Schur polynomials, we are taking this one because it is a fairly explicit and elementary formula. Indeed, the determinant in the denominator is the Vandermonde determinant, which has the property that it divides any other antisymmetric polynomial in to yield a symmetric polynomial.

Problem 9.2. Let be the single-cell Young diagram. Compute the Schur polynomial directly from Definition 9.1. Do the same for a few more small Young diagrams.

The disadvantage of Definition 9.1 is that it makes the orthogonality relations for Schur polynomials, which are really consequences of the fact that these polynomials are characters of irreducible representations of into a theorem whose proof would take us to far afield. So we will have to treat this property as a black box. Given a matrix let us write for the evaluation of the Schur polynomial on the eigenvalues of

Theorem 9.1 (Schur orthogonality). For any two Young diagrams and any two matrices we have

and

where is the number of semi standard Young tableaux of shape with entries from

We also need a remarkable “multinomial formula” due to Frobenius. Let be the set of Young diagrams with exactly cells and at most rows.

Theorem 9.2 (Frobenius formula). We have

where is the number of standard Young tableaux of shape

The reason the tableaux counts and are denoted this way is that the former is the dimension of an irreducible representation of the symmetric group and the latter is the dimension of an irreducible representation of the unitary group At this point we have enough material to explicitly calculate a series expansion of . This expansion is the starting point of Olshanski and Vershik’s asymptotic analysis, and is stated as Theorem 5.1 in their paper.

Theorem 9.3 (Schur expansion of ). We have

Proof: From the Theorem 9.2, we have

and applying the first integration formula in Theorem 9.1 this becomes

– QED

Problem 9.3. Compute the Schur expansion of up to This is straightforward but will give you a feel for the structure of the formula.

Let us include the analogue of Theorem 9.3 for the quantum Bessel kernel

where and the integration is against Haar measure on the product group Here are real vectors viewed as diagonal matrices in Since is invariant under signed permutations of the coordinates of and , we can assume without loss in generality that these are non increasing lists of nonnegative numbers. The quantum Bessel kernel is the characteristic function of a uniformly random rectangular matrix with singular values Once again we can analytically continue this to

where and are viewed as complex diagonal matrices in the integral. The following expansion of is not in the Olshanski-Vershik paper, but it can be obtained in essentially the same way; a more general result is proved here.

Theorem 9.4 (Schur expansion of ). We have

We close this lecture with a pointer to interesting work in the engineering literature which considers exactly these integrals in an applied context.