Fluctuations of particle systems determined by Schur generating functions

This article is about the random $N$–particle configurations on $\mathbb{Z}$ and their asymptotic behavior as $N\to\infty$. For each $N=1,2,\dots,$ let $\ell^{(N)}$ be a random $N$–dimensional vector

(1.1)

$$\ell^{(N)}=\bigl{(}\ell_{1}^{(N)}>\ell_{2}^{(N)}>\dots>\ell_{N}^{(N)}\bigr{)},\quad\ell_{i}^{(N)}\in\mathbb{Z}.$$

Our aim is to deal with global fluctuations of $\ell^{(N)}$. One way to make sense of those is to take an arbitrary test function $f(x)$ and consider linear statistics

(1.2)

$$\mathcal{L}_{f}^{(N)}=\sum_{i=1}^{N}f\left(\frac{\ell^{(N)}_{i}}{N}\right).$$

We mostly deal with the case when $f(x)$ is a polynomial (or, more generally, a smooth function), yet if $f(x)$ is the indicator function of an interval, then (1.2) merely counts the number of random particles inside this interval.

Since by its definition, $\mathcal{L}_{f}^{(N)}$ is a sum of $N$ terms, it is reasonable to expect that it grows linearly in $N$. And, indeed, in the class of systems that we study, $\frac{1}{N}\mathcal{L}_{f}^{(N)}$ converges as $N\to\infty$ to a deterministic limit depending on the choice of $f$. We will refer to such a phenomenon as the Law of Large Numbers, appealing to the evident analogy with a similar statement of classical probability dealing with sequences of independent random variables.

The next natural question is to study the fluctuations $\mathcal{L}_{f}^{(N)}-\mathbf{E}\mathcal{L}_{f}^{(N)}$ as $N\to\infty$. Such fluctuations would grow as $\sqrt{N}$ in the systems arising from sequences of independent random variables, but the scale is different in our context. We deal with probability distributions coming from $2d$ statistical mechanics (lozenge and domino tilings, families of non-intersecting paths), asymptotic representation theory, random matrix theory, and for them the typical situation is that $\mathcal{L}_{f}^{(N)}-\mathbf{E}\mathcal{L}_{f}^{(N)}$ does not grow as $N\to\infty$. Nevertheless, in all cases the fluctuations are asymptotically Gaussian, which justifies the name Central Limit Theorem for these kinds of results.

The main theme of the present article is to develop a new toolbox for proving the Law of Large Numbers and Central Limit Theorems, which would be robust to perturbations of $\ell^{(N)}$. It is somewhat hard to concisely describe the class of systems where the toolbox is helpful. One reason is that we believe our conditions to be in a sense equivalent to the LLN and CLT, see the end of Section 1.4 (which, of course, does not make these conditions immediate to check). Yet we list below an extensive list of available applications.

Again coming back to the classical one-dimensional probability, a universal tool is given there by the method of characteristic functions. For instance, it can be used to prove that averages of independent random variables converge to a Gaussian limit under very mild assumptions on the distributions of these variables, cf. textbooks [Ka], [Dur].

In our context the characteristic functions were not found to be useful, mostly due to the fact that the dimension (number of the particles) grows with $N$, while the individual coordinates $\ell^{(N)}_{i}$, $i=1,\dots,N$ are very far from being independent. Therefore, we suggest to replace them by a new notion of Schur generating function which we now introduce.

Recall that a Schur function is a symmetric Laurent polynomial in variables $x_{1},\dots,x_{N}$ parameterized by $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{N})$ and given by

$$s_{\lambda}(x_{1},\dots,x_{N})=\frac{\det\left[x_{i}^{\lambda_{j}+N-j}\right]_{i,j=1}^{N}}{\prod_{1\leq i<j\leq N}(x_{i}-x_{j})}.$$

Let $\delta_{N}$ denote the $N$–tuple $(N-1,N-2,\dots,0)$ and note that the map $\lambda\to\lambda+\delta_{N}$ makes the weakly decreasing coordinates of $\lambda$ strictly decreasing, as in (1.1).

For a random $N$–tuple of strictly ordered integers $\ell^{(N)}$, as in (1.1), its distribution is a function $\mathfrak{r}_{N}$ of $N$ weakly decreasing integers given by $\mathfrak{r}_{N}(\lambda):={\rm Prob}(\ell^{(N)}=\lambda+\delta_{N})$.

In [BuG] we showed how the Law of Large Numbers can be extracted from the asymptotic behavior of Schur generating functions. Interestingly, the answer, i.e. the exact formula for $\lim_{N\to\infty}\frac{1}{N}\mathcal{L}_{f}^{(N)}$ depends only on $f$ and the $N\to\infty$ asymptotics of $S_{\mathfrak{r}_{N}}(x_{1},1,1,\dots,1)$, that is, all variables except for one can be set to $1$ prior to the asymptotic analysis. A similar phenomenon was also found in [MN] by another method.

Here we make the next step and address the Central Limit Theorem for global fluctuations, the precise statement in this direction is Theorem 2.8. In fact, we go even further, and also analyze random sequences of $N$–particle configurations forming Markov chains, see Theorems 2.9, 2.10, 2.11 below. This time the answer, which is the covariance for $\lim_{N\to\infty}\left(\mathcal{L}_{f}^{(N)}-\mathbf{E}\mathcal{L}_{f}^{(N)}\right)$, depends only on asymptotics of $S_{\mathfrak{r}_{N}}(x_{1},x_{2},1,1,\dots,1)$, that is, all variables except for two can be set to $1$ prior to the asymptotic analysis. For proving the asymptotic Gaussianity we need more.

Our theorems reduce the LLN and CLT to asymptotic behavior of Schur generating functions, which is known in many cases. This leads to proofs of the LLN and CLT for a variety of stochastic systems of particles, including:

1. (1)

   Lozenge tilings of trapezoid domains, cf. Figure 2 in Section 3.

2. (2)

   Domino tilings of Aztec diamond, cf. Figure 3 in Section 3 (for the application to domino tilings of more complicated domains see [BuK]).

3. (3)

   Ensembles of non-intersecting random walks, cf. Figure 4 in Section 3.

4. (4)

   $2+1$–dimensional random growth models.

5. (5)

   Measures governing the decomposition into irreducible components for tensor products of irreducible representations of the unitary group $U(N)$.

6. (6)

   Measures governing the decomposition of restrictions onto $U(N)$ of extreme characters of the infinite–dimensional unitary group $U(\infty)$.

7. (7)

   Schur–Weyl measures.

A more detailed exposition of the applications of our method is given in Section 3.

One advantage of our approach through Schur generating functions is that it is quite general, and as a result, in each of our applications we can address more general situations than those rigorously known before. However, particular cases of some of our applications were accessible previously by other important techniques. Let us list several of those.

• •

  Determinantal point processes have led to Central Limit Theorems for uniformly random lozenge tilings of certain domains in [Ken], [Pet2], for $2+1$ dimensional random growth in [BF], [Dui1], [Ku1]. Similar results for domino tilings of the Aztec diamond were announced (without technical details) in [CJY].

• •

  Asymptotic analysis of orthogonal polynomials through the recurrence relations has led to Central Limit Theorems for ensembles of non-intersecting paths with specific initial conditions (which also include some tiling models) in [BrDu], [Dui2].

• •

  Discrete loop equations (also known as Nekrasov equations) have led in [BGG] to Central Limit Theorems for discrete log–gases, which has overlaps with specific ensembles of non-intersecting paths and tilings.

• •

  Various representation–theoretic ideas, involving, in particular, computations in the algebra of shifted symmetric functions and universal enveloping algebra of $\mathfrak{gl}_{N}$ have led in [Ker], [IO], [F], [HO], [BBu], [BBO], [DF], [Ku2], [Mel] to several instances of Central Limit Theorem for the probability distributions of asymptotic representation theory.

• •

  Differential operators acting in the algebra of symmetric functions in infinitely-many variables were used in [Mo] for proving the Central Limit Theorem for the Jack measures.

Let us emphasize, that despite the existence of several competing methods, most of our applications were not previously accessible by any of them. Yet our technique is adapted to the study of the global behavior of probabilistic systems, while some of these methods are more suitable for the study of the local behavior.

Replacing $\ell_{i}^{(N)}\in\mathbb{Z}$ by $\ell_{i}^{(N)}\in\mathbb{R}$ in (1.1), we arrive at continuous analogues of the particle configurations under consideration. In this fashion, our results are closely related to the global asymptotics for the eigenvalues of random matrix ensembles.

One precise example is given by the semiclassical limit, which degenerates the decomposition of tensor products of irreducible representations of $U(N)$ (one of our applications) to spectral decomposition of sums of independent Hermitian matrices, see [BuG, Section 1.3] for the details. The Central Limit Theorem for this random matrix problem is well-known, see [PS, Section 10]. It can be put into the context of the second order freeness in the free probability theory, see [MS], [MSS]. In Section 9.4 we explain how the covariance for our Central Limit Theorem for tensor products degenerates to the random matrix one.

Another degeneration is the appearance of the Gaussian Unitary Ensemble (GUE) as a scaling limit of lozenge and domino tilings near the boundary of the tiled domain, see [OR], [JN], [GP], [No]. Recall that GUE is the eigenvalue distribution of $H=\frac{1}{2}(X+X^{*})$, where $X$ is $N\times N$ matrix of i.i.d. mean $0$ complex Gaussian random variables. And again for GUE the Gaussian asymptotics for global fluctuations is well–known and can be generalized in (at least) two directions. The first one is a general Central Limit Theorem for (continous) log–gases of [J1] based on the loop equations. The second generalization is to replace the Gaussian distributions in the definition of GUE by arbitrary ones and to study the resulting Wigner matrix. Then the global fluctuations can be accessed by the moments method, see e.g. [AGZ, Chapter 2] for an exposition. In more details, one computes the moments of the eigenvalues in the following form

(1.4)

$$\mathbf{E}\left(\prod_{k=1}^{n}{\rm Trace}\bigl{(}H^{m_{k}}\bigr{)}\right)=\mathbf{E}\left(\prod_{k=1}^{n}\sum_{i=1}^{N}(h_{i})^{m_{k}}\right),\quad\{h_{i}\}_{i=1}^{N}\text{ are eigenvalues of }H.$$

The independence of matrix elements of $H$ paves a way to find the asymptotic of the left–hand side of (1.4), which then gives the global asymptotic of linear statistics of the form (1.2) with polynomial test functions $f(x)$.

The moments method was never available for the discrete particle configurations as in (1.1) for a very simple reason: there is no underlying random matrix or an analogue thereof. Here we change this situation by providing a way to efficiently compute (a discrete analogue of) the right–hand side in (1.4). Let us briefly state the key idea.

Let $\partial_{i}$ denote the derivative with respect to the variable $x_{i}$ and consider the differential operator

$$\mathcal{D}_{m}=\prod_{1\leq i<j\leq N}\frac{1}{x_{i}-x_{j}}\left(\sum_{i=1}^{N}\left(x_{i}\partial_{i}\right)^{m}\right)\prod_{1\leq i<j\leq N}(x_{i}-x_{j}).$$

A straightforward computation shows that the Schur functions are eigenvectors of $\mathcal{D}_{m}$:

$$\mathcal{D}_{m}s_{\lambda}=\left(\sum_{i=1}^{N}(\lambda_{i}+N-i)^{m}\right)s_{\lambda},\quad\lambda=(\lambda_{1},\dots,\lambda_{N}).$$

Therefore, applying such operators to (1.3) we get

(1.5)

$$\mathbf{E}\left(\prod_{k=1}^{n}\sum_{i=1}^{N}\left(\ell_{i}^{(N)}\right)^{m_{k}}\right)=\left[\left(\prod_{k=1}^{n}\mathcal{D}_{m_{k}}\right)S_{\mathfrak{r}_{N}}\right]_{x_{1}=x_{2}=\dots=x_{N}=1}.$$

The fact that differential (or difference) operators applied to symmetric functions can be used for the analysis of random particle configurations is by no means new, see e.g. [BC], [BCGS] for recent similar statements, and the asymptotic questions boil down to finding a way to analyze the right–hand side of (1.5). This is where the specific and relatively simple definition of $\mathcal{D}_{m}$ shines, as we are able to develop a combinatorial approach (yet based on several analytic lemmas) to the right–hand side of (1.5).

One important observed feature is that the right–hand side of (1.5) depends only on the values of the Schur generating function $S_{\mathfrak{r}_{N}}$ at points $(x_{1},\dots,x_{N})$ such that all but a bounded number of coordinates (i.e. the total number is not growing with $N$) are equal to $1$. First, this reduces a problem in growing (with $N$) dimension to a much more tractable finite–dimensional form. Second, the values of Schur generating functions at such points are very robust and not too sensitive to small perturbations for $\ell^{(N)}$. This is indicated by the results of [GM], [GP] on the asymptotics of Schur functions, on which we elaborate in Section 8. In particular, these results give enough control on the values of Schur generating functions to give the asymptotic expansion for the left–hand side of (1.5) needed for the Central Limit theorem. In contrast to our method, previous results and related approaches in the area, such as those of [BBu], [BC], [BCGS], [BBO], [Mo] relied on the exact form of the Schur generating function or its analogue; in particular, it was necessary to assume its factorization into a product of $1$–variable functions.

From the technical point of view, even after all these observations are made, the asymptotic analysis still needs many efforts and is much more complicated than that of [BuG] where the Law of Large Numbers was addressed through the same technique.

Let us end this section with a speculation. We believe that it should be possible to reverse the theorems of the present article: the knowledge of the Law of Large Numbers and Central Limit Theorem should give (perhaps, subject to technical conditions) exhaustive information about asymptotics of the Schur generating functions for all but finitely many values of coordinates $x_{i}$ equal to $1$. We plan to develop this direction in a separate publication¹¹1This was subsequently proven to be true, see [BuG2]..

The rest of the text is organized as follows. In Section 2 we formulate our main results linking the Central Limit Theorem for global fluctuations to the asymptotic of Schur generating functions. Numerous applications of these results are presented in Section 3. Section 4 gives a generalization of (1.5) which underlies all our developments. The remaining sections present a step-by-step proof for the statements of Sections 2 and 3.

We would like to thank Alisa Knizel for help with preparing the picture of a domino tiling. We thank Alexei Borodin for useful comments. We are grateful to an anonymous referee for suggestions on improving the text. V.G. was partially supported by the NSF grant DMS-1407562, by the NEC Corporation Fund for Research in Computers and Communications and by the Sloan Research Fellowship. Both authors were partially supported by the NSF grant DMS-1664619.

Here we collect some notations that we use throughout this paper. Note that some of these notations are slightly unconventional.

By $\vec{x}$ we denote the variables $(x_{1},\dots,x_{N})$.

We denote by $(1^{N})$ the sequence $\underbrace{(1,1,\dots,1)}_{N}$.

By $\partial_{i}$ we denote the partial derivative $\frac{\partial}{\partial x_{i}}$. We use $\partial_{z}$ instead of $\frac{\partial}{\partial z}$. For a function of one variable $f(x)$ we sometimes denote the derivative by the conventional notation $f^{\prime}(x)$. By $\partial_{i}^{0}f$ we mean the function $f$ itself.

For a differential operator $\mathcal{D}$ by $\mathcal{D}[F(x)]G(x)$ we mean that the differential operator is applied to $F(x)$ only. Let $S_{r}$ be the group of all permutations of $r$ elements; then

$$Sym_{x_{1},\dots,x_{r}}f(x_{1},\dots,x_{r}):=\frac{1}{r!}\sum_{\sigma\in S_{r}}f(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(r)}),$$

denotes the symmetrization of a function.

Let $V_{N}(\vec{x}):=\prod_{1\leq i<j\leq N}(x_{i}-x_{j})$ be the Vandermond determinant in variables $x_{1},\dotsm x_{N}$.

Sometimes we omit the arguments of functions in formulas. For example, we can use the symbol $V_{N}$ instead of $V_{N}(\vec{x})$.

We use notations $[N]:=\{1,2,\dots,N\}$, $[2;N]:=\{2,3,\dots,N\}$.

$\sum_{\{a_{1},\dots,a_{r}\}\subset[N]}$ denotes the summation over all subsets of $[N]$ consisting of $r$ elements.

All contours of integration in this paper are counter-clockwise.

An $N$-tuple of non-increasing integers $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{N})$ is called a signature of length $N$. We denote by $\mathbb{GT}_{N}$ the set of all signatures of length $N$. The Schur function $s_{\lambda}$, $\lambda\in\mathbb{GT}_{N}$, is a symmetric Laurent polynomial defined by

$$s_{\lambda}(x_{1},\dots,x_{N})=\frac{\det\left[x_{i}^{\lambda_{j}+N-j}\right]}{\det\left[x_{i}^{N-j}\right]}=\frac{\det\left[x_{i}^{\lambda_{j}+N-j}\right]}{\prod\limits_{i<j}(x_{i}-x_{j})}.$$

Let $\mathfrak{r}$ be a probability measure on the set $\mathbb{GT}_{N}$. A Schur generating function $S_{\mathfrak{r}}(x_{1},\dots,x_{N})$ is a symmetric Laurent power series in $x_{1},\dots,x_{N}$ given by

$$S_{\mathfrak{r}}(x_{1},\dots,x_{N})=\sum_{\lambda\in\mathbb{GT}_{N}}\mathfrak{r}(\lambda)\frac{s_{\lambda}(x_{1},\dots,x_{N})}{s_{\lambda}(1^{N})}.$$

In what follows we always assume that the measure $\mathfrak{r}$ is such that this (in principle, formal) sum is uniformly convergent in an open neighborhood of $(1^{N})$. Note that the uniform convergence of such a series in a neighborhood of $(1^{N})$ implies the uniform convergence in an open neighborhood of the $N$-dimensional torus $\{(x_{1},\dots,x_{N}):|x_{i}|=1,i=1,\dots,N\}$. Indeed, it follows from the estimate $|s_{\lambda}(x_{1},\dots,x_{n})|\leq s_{\lambda}(|x_{1}|,\dots,|x_{n}|)$ (which is an immediate corollary of the combinatorial formula for Schur functions as a positive sum of monomials, see [Ma, Chapter I, Section 5, (5.12)]).

The goal of this paper is to show how to extract information about $\mathfrak{r}$ with the help of $S_{\mathfrak{r}}(x_{1},\dots,x_{N})$.

Indeed, for a uniform limit of holomorphic functions the order of taking derivatives and limit can be interchanged, which shows that the example above is correct. In applications studied in this paper all LLN-appropriate measures will come from the construction of Example 2.3. However, we prefer to prove general theorems in a slightly more general setting of Definition 2.2.

For a signature $\lambda\in\mathbb{GT}_{N}$ consider the measure on $\mathbb{R}$

(2.1)

$$m[\lambda]:=\frac{1}{N}\sum_{i=1}^{N}\delta\left(\frac{\lambda_{i}+N-i}{N}\right).$$

The pushforward of a measure $\mathfrak{r}$ on $\mathbb{GT}_{N}$ with respect to the map $\lambda\to m[\lambda]$ defines a random probability measure on $\mathbb{R}$ which we denote by $m[\mathfrak{r}]$.

The following theorem is essentially [BuG, Theorem 5.1]. In Section 10 we comment on the slight difference between this formulation and the one given in [BuG].

Indeed, for a uniform limit of holomorphic functions the order of taking derivatives and limit can be interchanged, which shows that the example above is correct. In applications studied in this paper all CLT-appropriate measures will come from the construction of Example 2.7. However, we prefer to prove theorems for a slightly more general setting of Definition 2.6.

Let $\rho_{N}$ be a probability measure on $\mathbb{GT}_{N}$. Set

$$p_{k}^{(N)}:=\sum_{i=1}^{N}\left(\lambda_{i}+N-i\right)^{k},\qquad k=1,2,\dots,\quad\mbox{$\lambda=(\lambda_{1},\dots,\lambda_{N})$ is $\rho_{N}$-distributed.}$$

The following theorem is the main result of this paper.

This theorem serves as a model example of our approach. However, for applications it is often required to study the joint distributions of several random particle systems. Our approach can be applied to (some of) these cases as well: We deal with them in the next sections.

Let us introduce a general construction of Markov chains which are analyzable by our methods.

For a positive integer $m$ and $\varepsilon>0$ let $\Lambda^{m}_{\varepsilon}$ be the space of analytic symmetric functions in the region

$$\{(z_{1},\dots,z_{m})\in\mathbb{C}^{m}:1+\varepsilon>|z_{1}|>1-\varepsilon,1+\varepsilon>|z_{2}|>1-\varepsilon,\dots,1+\varepsilon>|z_{m}|>1-\varepsilon\}.$$

We consider $\Lambda^{m}_{\varepsilon}$ as a topological space with topology of uniform convergence in this region.

Consider $\Lambda^{m}:=\cup_{\varepsilon>0}\Lambda^{m}_{\varepsilon}$ endowed with the topology of the inductive limit. Note that for $\mathfrak{f}\in\Lambda^{m}$ the function $\mathfrak{f}(x_{1},x_{2},\dots,x_{m})\prod_{1\leq i<j\leq m}(x_{i}-x_{j})$ is an (antisymmetric) analytic function. Therefore, it can be written as an absolutely convergent sum of monomials $x_{1}^{l_{1}}\dots x_{m}^{l_{m}}$, where $l_{i}\in\mathbb{Z}$, $i=1,2,\dots,m$. Dividing both sides of such a sum by $\prod_{1\leq i<j\leq m}(x_{i}-x_{j})$, we obtain that each element of $\Lambda^{m}$ can be written in a unique way as an absolutely convergent sum

$$\sum_{\lambda\in\mathbb{GT}_{m}}\mathfrak{c}_{\lambda}s_{\lambda}(x_{1},\dots,x_{m}),\qquad\mathfrak{c}_{\lambda}\in\mathbb{C},$$

in some neighborhood of the $m$-dimensional torus.

We consider a map $\mathfrak{p}_{m,n}:\Lambda^{m}\to\Lambda^{n}$ with the following properties:

1) $\mathfrak{p}_{m,n}$ is a linear continuous map.

2) For every $\lambda\in\mathbb{GT}_{m}$ we have

$$\mathfrak{p}_{m,n}\left(\frac{s_{\lambda}(x_{1},\dots,x_{m})}{s_{\lambda}(1^{m})}\right)=\sum_{\mu\in\mathbb{GT}_{n}}\mathfrak{c}_{\lambda,\mu}^{\mathfrak{p}_{m,n}}\frac{s_{\mu}(x_{1},\dots,x_{n})}{s_{\mu}(1^{n})},\qquad\mathfrak{c}_{\lambda,\mu}^{\mathfrak{p}_{m,n}}\in\mathbb{R}_{\geq 0}.$$

This property says that the coefficients $\mathfrak{c}_{\lambda,\mu}^{\mathfrak{p_{m,n}}}$ must be nonnegative reals (rather than arbitrary complex numbers). Note that the sum in the right-hand side is absolutely convergent due to the definition of $\mathfrak{p_{m,n}}$.

3) For any $\mathfrak{f}\in\Lambda^{m}$ we have

$$\mathfrak{f}(1^{m})=\mathfrak{p}_{m,n}(\mathfrak{f})(1^{n}).$$

In words, this property asserts that our map should preserve the value at unity.

It follows from conditions 2) and 3) that

$$\sum_{\mu\in\mathbb{GT}_{n}}\mathfrak{c}_{\lambda,\mu}^{\mathfrak{p}_{m,n}}=1.$$

Since these coefficients are nonnegative reals one can consider them as transitional probabilities of a Markov chain. In more details, let $n_{1},\dots,n_{s}$ be positive integers, and let $\mathfrak{p}_{n_{2},n_{1}}$, …, $\mathfrak{p}_{n_{s},n_{s-1}}$ be maps satisfying conditions above. Let $\rho$ be a probability measure on $\mathbb{GT}_{n_{s}}$. Define the probability measure on the set

$$\mathbb{GT}_{n_{1}}\times\mathbb{GT}_{n_{2}}\times\dots\times\mathbb{GT}_{n_{s}}$$

via

(2.3)

$$\mathrm{Prob}(\lambda^{(1)},\lambda^{(2)},\dots,\lambda^{(s)})=\rho(\lambda^{(s)})\prod_{i=2}^{k}\mathfrak{c}_{\lambda^{(i)},\lambda^{({i-1})}}^{\mathfrak{p_{n_{i},n_{i-1}}}}.$$

In Section 4 we prove a formula for the expectation of joint moments of signatures $\{\lambda^{(i)}\}_{i=1,\dots,s}$ distributed according to this measure.

In this section we state three multi-level generalisations of Theorem 2.8. They are mainly shaped to the applications studied in the present paper. With the use of the construction from Section 2.3 it is possible to produce many other similar multi-level generalisations of Theorem 2.8; this should be regulated by applications that one has in mind.

We consider the following particular examples of the map $\mathfrak{p_{m,n}}$ from Section 2.3. In the first case, it is given by $f(x_{1},\dots,x_{m})\to f(x_{1},\dots,x_{n},1^{m-n})$, for $m>n$. In the second case, it is given by $s_{\lambda}(x_{1},\dots,x_{m})\to g(x_{1},\dots,x_{m})s_{\lambda}(x_{1},\dots,x_{m})$, for $m=n$ and a function $g(x_{1},\dots,x_{m})$ which is a Schur generating function of a probability measure on $\mathbb{GT}_{m}$. Finally, in the third case we combine the two previous ones.

In an attempt to make the exposition more explicit, we repeat the construction of Section 2.3 in all three cases below.

Example 1) For $\lambda\in\mathbb{GT}_{k_{1}},\mu\in\mathbb{GT}_{k_{2}}$, with $k_{1}\geq k_{2}$, let us introduce the coefficients $\mathrm{pr}_{k_{1}\to k_{2}}(\lambda\to\mu)$ via

(2.4)

$$\frac{s_{\lambda}(x_{1},\dots,x_{k_{2}},1^{k_{1}-k_{2}})}{s_{\lambda}(1^{k_{1}})}=\sum_{\mu\in\mathbb{GT}_{k_{2}}}\mathrm{pr}_{k_{1}\to k_{2}}(\lambda\to\mu)\frac{s_{\mu}(x_{1},\dots,x_{k_{2}})}{s_{\mu}(1^{k_{2}})}.$$

The branching rule for Schur functions asserts that the coefficients $\mathrm{pr}_{k_{1}\to k_{2}}(\lambda\to\mu)$ are non-negative for all $\lambda$, $\mu$ (see [Ma, Chapter 1.5]). Plugging in $x_{1}=\dots=x_{k}=1$, we see that $\sum_{\mu\in\mathbb{GT}_{k_{2}}}\mathrm{pr}_{k_{1}\to k_{2}}(\lambda\to\mu)=1$.

Let $0<a_{1}\leq a_{2}\leq\dots\leq a_{n}=1$ be fixed positive reals, and let $\rho_{N}$ be a probability measure on $\mathbb{GT}_{N}$.

Let us introduce the probability measure on the set $\mathbb{GT}_{[a_{1}N]}\times\mathbb{GT}_{[a_{2}N]}\times\dots\times\mathbb{GT}_{[a_{n}N]}$ via

(2.5)

$$\mathrm{Prob}(\lambda^{(1)},\lambda^{(2)},\dots,\lambda^{(n)}):=\rho_{N}(\lambda^{(n)})\prod_{i=1}^{n-1}\mathrm{pr}_{[a_{i+1}N]\to[a_{i}N]}(\lambda^{(i+1)}\to\lambda^{(i)}),\qquad\lambda^{(i)}\in\mathbb{GT}_{[a_{i}N]},$$

(the fact that all these weights are summed up to $1$ can be straightforwardly checked by induction over $n$.)

We are interested in the joint distributions of random signatures of this random array. For $t=1,2,\dots,n$, let $p_{k}^{[a_{t}N]}$ be a (shifted) power sum of coordinates of signatures defined by the formula

$$p_{k}^{[a_{t}N]}:=\sum_{i=1}^{[a_{t}N]}\left(\lambda_{i}^{(t)}+[a_{t}N]-i\right)^{k},\qquad k=1,2,\dots,$$

where $\lambda^{(t)}$ is distributed according to the measure (2.5).

Example 2) Let us start with the following classical fact. For $\lambda,\mu\in\mathbb{GT}_{N}$ there is a decomposition of the product of two Schur functions into a linear combination of Schur functions:

(2.6)

$$s_{\lambda}(x_{1},\dots,x_{N})s_{\mu}(x_{1},\dots,x_{N})=\sum_{\eta\in\mathbb{GT}_{N}}c^{\eta}_{\lambda\mu}s_{\eta}(x_{1},\dots,x_{N}).$$

The coefficients $c_{\lambda\mu}^{\eta}$ are well-known under the name of Littlewood-Richardson coefficients. It is known that for arbitrary $\lambda,\mu,\eta$ they are nonnegative (see, e.g., [Ma, Chapter 1.9]).

Let $\rho=\{\rho_{N}\}$, $\{\mathfrak{r}^{(1)}_{N}\}$, $\{\mathfrak{r}^{(2)}_{N}\}$, …, $\{\mathfrak{r}^{(n-1)}_{N}\}$ be sequences of appropriate measures, where $\rho_{N}$, $\mathfrak{r}^{(1)}_{N}$, …, $\mathfrak{r}^{(n-1)}_{N}$ are probability measures on $\mathbb{GT}_{N}$. Let $g_{1}^{(N)}(\vec{x}),\dots,g_{n-1}^{(N)}(\vec{x})$ be the Schur generating functions of $\mathfrak{r}^{(1)}_{N}$, $\mathfrak{r}^{(2)}_{N}$, …, $\mathfrak{r}^{(n-1)}_{N}$, respectively.

Define the coefficients $\mathrm{st}_{(g_{r})}^{(N)}(\lambda\to\mu)$, for $\lambda\in\mathbb{GT}_{N}$, $\mu\in\mathbb{GT}_{N}$, $1\leq r\leq(n-1)$, via

(2.7)

$$g_{r}^{(N)}(\vec{x})\frac{s_{\lambda}(\vec{x})}{s_{\lambda}(1^{N})}=\sum_{\mu\in\mathbb{GT}_{N}}\mathrm{st}_{(g_{r})}^{(N)}(\lambda\to\mu)\frac{s_{\mu}(\vec{x})}{s_{\mu}(1^{N})}.$$

Note that the series in the right-hand side is absolutely convergent and the coefficients $\mathrm{st}_{(g_{r})}^{(N)}$ are nonnegative. Using (2.6), one can write an explicit formula for them:

$$\mathrm{st}_{(g_{r})}^{(N)}(\lambda\to\mu)=\sum_{\eta\in\mathbb{GT}_{N}}\frac{s_{\mu}(1^{N})}{s_{\lambda}(1^{N})s_{\eta}(1^{N})}c^{\mu}_{\lambda\eta}\mathfrak{r}^{(r)}_{N}(\eta).$$

Let us define a probability measure on the set

(2.8)

$$\underbrace{\mathbb{GT}_{N}\times\mathbb{GT}_{N}\times\dots\times\mathbb{GT}_{N}}_{\mbox{$n$ factors}}.$$

We define the probability of the configuration

$$(\lambda^{(1)},\lambda^{(2)},\dots,\lambda^{(n)})\in\mathbb{GT}_{N}\times\mathbb{GT}_{N}\times\dots\times\mathbb{GT}_{N}$$

via

(2.9)

$$\mathrm{Prob}(\lambda^{(1)},\lambda^{(2)},\dots,\lambda^{(n)}):=\rho_{N}(\lambda^{(n)})\prod_{i=1}^{n-1}\mathrm{st}_{g_{i}}^{(N)}(\lambda^{(i+1)}\to\lambda^{(i)}).$$

Let $p_{k;s}^{(N)}$ be the $k$th shifted power sum of $\lambda^{(s)}$:

$$p_{k;s}^{(N)}:=\sum_{i=1}^{N}\left(\lambda_{i}^{(s)}+N-i\right)^{k},\qquad k=1,2,\dots,\qquad\mbox{$(\lambda^{(1)},\dots,\lambda^{(n)})$ is \eqref{eq:gener-def-multiplications}-distributed.}$$

Let

(2.10)

$$H_{n}^{(N)}(\vec{x}):=S_{\rho_{N}}(\vec{x}),\ \ H_{n-1}^{(N)}(\vec{x}):=H_{n}^{(N)}(\vec{x})g_{n-1}^{(N)}(\vec{x}),\ ...\ ,\ H_{1}^{(N)}(\vec{x}):=H_{2}^{(N)}(\vec{x})g_{1}^{(N)}(\vec{x}).$$

It can be directly shown by induction that the functions $H_{s}^{(N)}(\vec{x})$, are Schur generating functions of $\lambda^{(s)}$, $s=1,\dots,n$. Moreover, they are appropriate (in the sense of Definition 2.5) because $g_{i}^{(N)}$ and $S_{\rho_{N}}$ are appropriate sequences of functions.