Gaussian fluctuations of Jackdeformed random Young diagrams
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Abstract
We introduce a large class of random Young diagrams which can be regarded as a natural oneparameter deformation of some classical Young diagram ensembles; a deformation which is related to Jack polynomials and Jack characters. We show that each such a random Young diagram converges asymptotically to some limit shape and that the fluctuations around the limit are asymptotically Gaussian.
Keywords
Jack polynomials Jack characters Random Young diagrams Random matrices \(\beta \)ensembleMathematics Subject Classification
Primary 05E05 Secondary 20C30 60K35 60B201 Introduction
1.1 Random partitions...
An integer partition, called also a Young diagram, is a weakly decreasing finite sequence \(\lambda =(\lambda _1,\ldots ,\lambda _l)\) of positive integers \(\lambda _1\ge \cdots \ge \lambda _l>0\). We also say that \(\lambda \) is a partition of \(\lambda :=\lambda _1+\cdots +\lambda _l\).
Random partitions occur in mathematics and physics in a wide variety of contexts, in particular in the Gromov–Witten and Seiberg–Witten theories, see the overview articles of Okounkov [24] and Vershik [30].
1.2 ...And random matrices
1.3 Random Young diagrams related to loggases
Opposite to the special cases \(\beta \in \{1,2,4\}\), in the generic case of \(\beta \)ensembles there seems to be no obvious unique way of defining their discrete counterparts and several alternative approaches are available, see the work of Moll [23] as well as the work of Borodin et al. [2]. In the current paper we took another approach based on a deformation of the character theory of the symmetric groups.
The class of random Young diagrams considered in the current paper as well as the classes from [2, 23] are of quite distinct flavors and it is not obvious why they should contain any elements in common, except for the trivial example given by the Jack–Plancherel measure. The problem of understanding the relations between these three classes does not seem to be easy and is out of the scope of the current paper.
1.4 Random Young diagrams and characters
The names integer partitions and Young diagrams are equivalent, but they are used in different contexts; for this reason we will use two symbols \(\mathcal {P}_{n}\) and \(\mathbb {Y}_{n}\) to denote the same object: the set of integer partitions of n, also known as the set of Young diagrams with n boxes. Any function on the set of partitions (or its some subset) will be referred to as character.
1.5 Irreducible characters of symmetric groups: Plancherel measure
1.6 Irreducible Jack characters
In the current paper we will use another, more general, family \((\chi ^{(\alpha )}_\lambda )\) of irreducible characters in (1.1). Our starting point is the family of Jack polynomials \(J^{(\alpha )}_\lambda \) [11] which can be regarded as a deformation of the family of Schur polynomials; a deformation that depends on the parameter \(\alpha >0\). We use the normalization of Jack polynomials from [21, Section VI.10].
It is worth pointing out that in the special case \(\alpha =1\) the corresponding Jack character \(\chi ^{(1)}_\lambda (\pi )=\chi _\lambda (\pi )\) coincides with the irreducible character (1.2) of the symmetric group \(\mathfrak {S}(n)\), see [5, 19].
1.7 Probability measures \(\mathbb {P}^{(\alpha )}_\chi \)
In the simplest example of \(\chi :=\chi _{{{\mathrm{reg}}}}\) given by (1.3) the corresponding probability measure \(\mathbb {P}^{(\alpha )}_{\chi _{{{\mathrm{reg}}}}}\) turns out to be the celebrated Jack–Plancherel measure [2, 5, 9, 22, 23, 28] which is a oneparameter deformation of the Plancherel measure.
1.8 Random Young diagrams related to Thoma’s characters
An additional motivation for considering this particular class of random Young diagrams stems from the research related to the problem of finding extremal characters of the infinite symmetric group \(\mathfrak {S}(\infty )\), solved by Thoma [29]. Vershik and Kerov [32] found an alternative, more conceptual proof of Thoma’s result, which was based on the observation that characters of \(\mathfrak {S}(\infty )\) are in a natural bijective correspondence with certain sequences \((\lambda _1\nearrow \lambda _2\nearrow \cdots )\) of growing random Young diagrams.
The original Thoma’s problem can be equivalently formulated as finding all homomorphisms from the ring of symmetric functions to real numbers which are Schurpositive, i.e. which take nonnegative values on all Schur polynomials. In this formulation the problem naturally asks for generalizations in which Schur polynomials are replaced by another interesting family of symmetric functions. Kerov et al. [17] considered the particular case of Jack polynomials and they proved that a direct analogue of Thoma’s result holds true also in this case. The main idea behind their proof was that the probabilistic viewpoint from the abovementioned work of Vershik and Kerov [32] can be adapted to the new setting of Jack polynomials. Thus, a side product of the work of Kerov et al. is an interesting, natural class of random Young diagrams which fits into the framework which we consider in the current paper, see Sect. 1.16 and the forthcoming paper [7] for more details.
1.9 Cumulants
For partitions \(\pi _1,\ldots ,\pi _k\) we define their product \(\pi _1 \cdots \pi _k\) as their concatenation, for example \((4,3) \cdot (5,3,1)=(5,4,3,3,1)\). In this way the set of all partitions \(\mathcal {P}=\bigsqcup _{n\ge 0} \mathcal {P}_{n}\) becomes a unital semigroup with the unit \(1=\emptyset \) corresponding to the empty partition; we denote by \(\mathbb {R}[\mathcal {P}]\) the corresponding semigroup algebra, the elements of which are formal linear combinations of partitions. Any character \(\chi :\mathcal {P}\rightarrow \mathbb {R}\) with \(\chi (\emptyset )=1\) can be canonically extended to a linear map \(\chi :\mathbb {R}[\mathcal {P}]\rightarrow \mathbb {R}\) (such that \(\chi (1)=\chi (\emptyset )=1\)) which will be denoted by the same symbol.
1.10 Asymptotics
In the current paper we consider asymptotic problems which correspond to the limit when the number of boxes \(n\rightarrow \infty \) of the random Young diagrams tends to infinity. This corresponds to considering a sequence \((\chi _n)\) of reducible characters \(\chi _n:\mathcal {P}_{n}\rightarrow \mathbb {R}\) and the resulting sequence \((\mathbb {P}^{(\alpha )}_{\chi _n})\) of probability measures on \(\mathbb {Y}_{n}\).
We also allow that the deformation parameter \(\alpha (n)\) depends on n; in order to make the notation light we will make this dependence implicit and write shortly \(\alpha =\alpha (n)\).
1.11 Hypothesis: asymptotics of \(\alpha \)
Note that the most important case when \(\alpha \) is constant fits into this framework with \(g=0\). The generic case \(g\ne 0\) will be referred to as double scaling limit.
1.12 Hypothesis: approximate factorization of characters
In this way the cumulant \(k^{\chi _n}_\ell (l_1,\ldots ,l_\ell )\) is well defined if \(n\ge l_1+\cdots +l_\ell \) is large enough.
Definition 1.1
We say that the sequence \((\chi _n)\) has enhanced approximate factorization property if, additionally, in the case \(\ell =1\) the rate of convergence in (1.11) takes the following explicit form:
Example 1.2
1.13 Drawing Young diagrams
1.13.1 Anisotropic Young diagrams
1.13.2 Russian convention: profile of a Young diagram
1.14 The first main result: law of Large Numbers
The following theorem is a generalization of the results of Biane [4] who considered the special case \(\alpha =1\) and the corresponding representations of the symmetric groups.
Theorem 1.3
(Law of large numbers) Assume that \(\alpha =\alpha (n)\) is such that (1.9) holds true. Assume that \(\chi _n:\mathcal {P}_{n}\rightarrow \mathbb {R}\) is a reducible Jack character; we denote by \(\lambda _n\) the corresponding random Young diagram with n boxes distributed according to \(\mathbb {P}^{(\alpha )}_{\chi _n}\). We assume also that the sequence \((\chi _n)\) of characters fulfills the enhanced approximate factorization property.
Remark 1.4
The concrete formula for the profile \(\omega _{\Lambda _\infty }\) may be obtained by computing the corresponding Rtransform and Cauchy transform, see for example [4, Theorem 3].
The proof is postponed to Sect. 5.
1.15 The second main result: Central Limit Theorem
The following result is a generalization of Kerov’s CLT [10, 12] which concerned Plancherel measure in the special case \(\alpha =1\) as well as a generalization of its extension by the firstnamed author and Féray [5] for the generic fixed value of \(\alpha >0\). Indeed, Example 1.2 shows that the assumptions of the following theorem are fulfilled for \(\chi _n:=\chi _{{{\mathrm{reg}}}}\), thus CLT holds for Jack–Plancherel measure in a wider generality, when \(\alpha =\alpha (n)\) may vary with n.
On the other hand the following result is also a generalization of the results of the secondnamed author [26] who considered a setup similar to the one below in the special case \(\alpha =1\).
Theorem 1.5
(Central Limit Theorem) We keep the assumptions and the notations from Theorem 1.3.
Then for \(n\rightarrow \infty \) the random vector \({\Delta }_n\) converges in distribution to some (noncentered) Gaussian random vector \(\Delta _\infty \) valued in the space \((\mathbb {R}[x])'\) of distributions, the dual space to polynomials.
Remark 1.6
The proof is postponed to Sect. 6.
1.16 Example
The convergence in (1.19) is illustrated in Figs. 6, 7 and 8: the function \(\mathbb {E}\Delta _n\) is the difference between the red solid curve (i.e. the plot of \(\sqrt{n} \ \mathbb {E}\omega _{\Lambda _n}\)) and the blue dashed curve (i.e. the plot of \(\sqrt{n}\ \omega _{\Lambda _\infty }\)). As one can see on these examples, the function \(\mathbb {E}\Delta _n\) has oscillations of period and amplitude related to the grid of the boxes of the Young diagrams. As \(n\rightarrow \infty \), the amplitude of these oscillations does not converge to zero (so that the convergence in the supremum norm does not hold) but their frequency tends to infinity (which is sufficient for convergence in the weak topology).
The central limit theorem in Theorem 1.5 is somewhat reminiscent to CLT for random walks. A significant difference lies in the nature of the limit object: in the case of the random walks it is the Brownian motion which has continuous trajectories while in the case considered in Theorem 1.5 it is a random Schwartz distribution \(\Delta _\infty \) for which computer simulations (such as the one shown in Fig. 9) suggest that it has quite singular ‘trajectories’, reminiscent to that of the white noise. A systematic investigation of such trajectorywise properties of \(\Delta _\infty \) via shortdistance asymptotics of the covariance of the corresponding Gaussian field is out of the scope of the current paper.
1.17 Content of the paper
In Sect. 2 we introduce the main algebraic tool for our considerations, namely Theorem 2.3 which gives several convenient characterizations of the approximate factorization property. In Sect. 3 we prove this result. Section 4 is devoted to some technical results, mostly related to probability measures which are uniquely determined by their moments. In Sect. 5 we give the proof of Law of Large Numbers (Theorem 1.3). Finally, in Sect. 6 we give the proof of Central Limit Theorem (Theorem 1.5).
2 Approximate factorization of characters
The purpose of this section is to give a number of conditions which are equivalent to the approximate factorization property (Definition 1.1). These conditions often turn out to be more convenient in applications, such as the ones from [7].
2.1 Conditional cumulants
Let \(\mathcal {A}\) and \(\mathcal {B}\) be commutative unital algebras and let \(\mathbb {E}:\mathcal {A}\rightarrow \mathcal {B}\) be a unital linear map. We will say that \(\mathbb {E}\) is a conditional expectation value; in the literature one usually imposes some additional constraints on the structure of \(\mathcal {A}\), \(\mathcal {B}\) and \(\mathbb {E}\), but for the purposes of the current paper such additional assumptions will not be necessary.
Note that the cumulants for partitions which we introduced in Sect. 1.9 fit into this general framework: for \(\mathcal {A}:=\mathbb {R}[\mathcal {P}]\) one should take the semigroup algebra of partitions, for \(\mathcal {B}:=\mathbb {R}\) the real numbers and for \(\mathbb {E}:=\chi :\mathbb {R}[\mathcal {P}]\rightarrow \mathbb {R}\) the character.
2.2 Normalized Jack characters
The usual way of viewing the characters of the symmetric groups is to fix the irreducible representation \(\lambda \) and to consider the character as a function of the conjugacy class \(\pi \). However, there is also another very successful viewpoint due to Kerov and Olshanski [16], called dual approach, which suggests to do roughly the opposite. Lassalle [18, 19] adapted this idea to the framework of Jack characters. In order for this dual approach to be successful one has to choose the most convenient normalization constants. In the current paper we will use the normalization introduced by Dołęga and Féray [5] which offers some advantages over the original normalization of Lassalle. Thus, with the right choice of the multiplicative constant, the unnormalized Jack character \(\chi ^{(\alpha )}_\lambda \) from (1.5) becomes the normalized Jack character \({{\mathrm{Ch}}}^{(\alpha )}_\pi (\lambda )\), defined as follows.
Definition 2.1
Each Jack character depends on the deformation parameter \(\alpha \); in order to keep the notation light we will make this dependence implicit and we will simply write \({{\mathrm{Ch}}}_{\pi }(\lambda )\).
2.3 The deformation parameters
In order to avoid dealing with the square root of the variable \(\alpha \) which is ubiquitous in the subject of Jack deformation, we introduce an indeterminate \(A := \sqrt{\alpha }\). The algebra of Laurent polynomials in the indeterminate A will be denoted by \(\mathbb {Q}\left[ A,A^{1}\right] \).
2.4 The linear space of \(\alpha \)polynomial functions
In a paper [27] the secondnamed author has defined a certain filtered linear space of \(\alpha \)polynomial functions. This linear space consists of certain functions in the set \(\mathbb {Y}\) of Young diagrams with values in the ring \(\mathbb {Q}\left[ A,A^{1}\right] \) of Laurent polynomials and, among many equivalent definitions, one can define it using normalized Jack characters.
Definition 2.2
2.5 Algebras \(\mathscr {P}\) and \(\mathscr {P}_\bullet \) of \(\alpha \)polynomial functions
The vector space of \(\alpha \)polynomial functions can be equipped with a product in two distinct natural ways (which will be reviewed in the following). With each of these two products it becomes a commutative, unital filtered algebra.
Firstly, as a product we may take the pointwise product of functions on \(\mathbb {Y}\). The resulting algebra will be denoted by \(\mathscr {P}\) (the fact that the \(\mathscr {P}\) is closed under such product was proved by Dołęga and Féray [5, Theorem 1.4]).
2.6 Two probabilistic structures on \(\alpha \)polynomial functions
Assume that \(\chi :\mathcal {P}_{n}\rightarrow \mathbb {R}\) is a reducible Jack character and let \(\mathbb {P}_\chi \) be the corresponding probability measure (1.7) on the set \(\mathbb {Y}_{n}\) of Young diagrams with n boxes. With this setup, functions on \(\mathbb {Y}_{n}\) can be viewed as random variables; we denote by \(\mathbb {E}_\chi \) the corresponding expectation.
On the other hand, by considering the disjoint product, we get a conditional expectation \(\mathbb {E}_\chi :\mathscr {P}_\bullet \rightarrow \mathbb {R}\); the corresponding cumulants will be denoted by \(\kappa _{\bullet \ell }^{\chi }\).
2.7 Equivalent characterizations of approximate factorization of characters

The family (2.8) (and its subset, the family (2.7)) has a direct probabilistic meaning. It contains information about the cumulants of some random variables which might be handy while proving probabilistic statements such as Central Limit Theorem or Law of Large Numbers.

On the other hand, the cumulants appearing in the families (2.6) and (2.9) are purely algebraic and do not have any direct probabilistic meaning. However, their merit lies in the fact that in many concrete applications (such as the ones from [7]) it is much simpler to verify algebraic conditions (A) and (D) than their probabilistic counterparts (B) and (C).
Theorem 2.3
 (a)
Equivalent characterization of approximate factorization property.
Then the following four conditions are equivalent: (A)for each integer \(\ell \ge 1\) and all integers \(l_1,\ldots ,l_\ell \ge 2\) the limitexists and is finite;$$\begin{aligned} \lim _{n\rightarrow \infty } k^{\chi _n}_\ell (l_1,\ldots , l_\ell ) \ n^{\frac{l_1+\cdots +l_\ell +\ell 2}{2}} \end{aligned}$$(2.6)
 (B)for each integer \(\ell \ge 1\) and all \(x_1,\ldots ,x_\ell \in \{{{\mathrm{Ch}}}_1,{{\mathrm{Ch}}}_2,\ldots \}\) the limitexists and is finite;$$\begin{aligned} \lim _{n\rightarrow \infty } \kappa ^{\chi _n}_{\ell }(x_{1},\ldots ,x_{\ell })\ n^{ \frac{\deg x_1+ \cdots + \deg x_\ell  2(\ell 1)}{2}} \end{aligned}$$(2.7)
 (C)for each integer \(\ell \ge 1\) and all \(x_1,\ldots ,x_\ell \in \mathscr {P}\) the limitexists and is finite;$$\begin{aligned} \lim _{n\rightarrow \infty } \kappa ^{\chi _n}_{\ell }(x_{1},\ldots ,x_{\ell })\ n^{ \frac{\deg x_1+ \cdots + \deg x_\ell  2(\ell 1)}{2}} \end{aligned}$$(2.8)
 (D)for each integer \(\ell \ge 1\) and all \(x_1,\ldots ,x_\ell \in \mathscr {P}_\bullet \) the limitexists and is finite.$$\begin{aligned} \lim _{n\rightarrow \infty } \kappa ^{\chi _n}_{\bullet \ell }(x_{1},\ldots ,x_{\ell })\ n^{ \frac{\deg x_1+ \cdots + \deg x_\ell  2(\ell 1)}{2}} \end{aligned}$$(2.9)
 (A)
 (b)Assume that the conditions from part (a) hold true. Furthermore, assume that for \(\ell =1\) the rate of the convergence of any of the four expressions under the limit symbol in (2.6)–(2.9) is of the formin the limit \(n \rightarrow \infty \) and all choices of \(l_1\) (respectively, for all choices of \(x_1\)); the constants depend on the choice of \(l_1\) (respectively, \(x_1\)).$$\begin{aligned} {\text {const}}_1+ \frac{{\text {const}}_2+o(1)}{\sqrt{n}} \end{aligned}$$(2.10)
When \(\alpha =1\), part (a) of the above result corresponds to [26, Theorem and Definition 1]. The proof is postponed to Sect. 3.
3 Proof of Theorem 2.3
In the current section we shall prove the key tool, Theorem 2.3.
Additionally, concerning part (a) of Theorem 2.3 we shall discuss the exact relationship between the limits of the quantities (2.6)–(2.9) in the case \(\ell \in \{1,2\}\). This relationship provides the information about the limit shape of random Young diagrams in Theorem 1.3 as well as about the covariance of the limit Gaussian process describing the fluctuations in Theorem 1.5.
Concerning part (b) of Theorem 2.3 we shall discuss the exact relationship between the constants which describe the fine asymptotics (2.10) of the quantities (2.6)–(2.9) in the case \(\ell =1\). This relationship provides the information about the mean value \(\mathbb {E}\Delta _\infty \) of the limit Gaussian process from Eq. 1.19.
3.1 Approximate factorization property for \(\alpha \)polynomial functions
Definition 3.1
We are now ready to state the main auxiliary result, proved very recently by the secondnamed author, which will be necessary for the proof of the key tool, Theorem 2.3.
3.2 Approximate factorization for \(\alpha =1\)
We denote by \(\mathscr {P}^{(1)}\) a version of the filtered algebra of \(\alpha \)polynomial functions \(\mathscr {P}\) obtained by the specialization \(\alpha :=1\), \(\gamma :=0\); analogously we denote by \(\mathscr {P}^{(1)}_\bullet \) the algebra \(\mathscr {P}^{(1)}\) equipped with the multiplication given by the disjoint product.
The following result has been proved earlier by the secondnamed author.
Theorem 3.3
Theorems 3.2 and 3.3 are of the same flavor. There are two major differences between them: firstly, the arrows in (3.2) and (3.3) point in the opposite directions; secondly, the algebra \(\mathscr {P}\) is more rich than its specialized version \(\mathscr {P}^{(1)}\), in particular the variable \(\gamma \in \mathscr {P}\) is not treated like a scalar since \(\deg \gamma =1 >0\).
3.3 Proof of Theorem 2.3, part (a)
This proof follows closely its counterpart from the work of the secondnamed author [26, Theorem and Definition 1] with the references to Theorem 3.3 replaced by Theorem 3.2 and, occasionally, the roles of \(\mathscr {P}\) and \(\mathscr {P}_\bullet \) reversed. We present the details below.
3.3.1 Proof of the equivalence (A)\(\iff \)(D)
The quantities (2.6) and (2.9) coincide with their counterparts from the work of the secondnamed author [26, Eqs. (12) and (13)]. The equivalence of the conditions (A) and (D) was proved in [26, Section 4.7].
3.3.2 Proof of the equivalence (B)\(\iff \)(C)
The implication (C)\(\implies \)(B) is immediate since \({{\mathrm{Ch}}}_1,{{\mathrm{Ch}}}_2,\ldots \in \mathscr {P}\).
The following result was proved by the secondnamed author [26, Corollary 19] (note that the original paper does not contain the assumption (3.5) without which it is not true). The proof did not use any specific properties of the filtered algebra \(\mathscr {P}\) and thus it remains valid also in our context when the original algebra of polynomial functions is replaced by the algebra of \(\alpha \)polynomial functions.
Lemma 3.4
Under the above assumption, if condition (B) holds true for all \(x_1,\ldots ,x_\ell \in X\) then more general condition (C) holds true for arbitrary \(x_1,\ldots ,x_\ell \in \mathscr {P}\).
In this way we verified that the assumptions of Lemma 3.4 are fulfilled. Condition (C) follows immediately.
3.3.3 Proof of the equivalence (C)\(\iff \)(D)
The proof will follow closely the ideas from [26, Section 4.7] with the roles of the cumulants \(\kappa ^{\chi _n}\) and \(\kappa ^{\chi _n}_{\bullet }\) interchanged. The original proof was based on the observation that the conditional cumulants (denoted in the original work [26] by the symbol \(k^{{\text {id}}}=\kappa _{\mathscr {P}^{(1)}}^{\mathscr {P}_\bullet ^{(1)}}\)) related to the map (3.3) fulfill the degree bounds (3.1) given by the approximate factorization property. By changing the meaning of the symbol \(k^{{\text {id}}}\) and setting \(k^{{\text {id}}} :=\kappa ^{\mathscr {P}}_{\mathscr {P}_\bullet } =\kappa _{\bullet }\) to be the conditional cumulants related to the map (3.2) and by applying Theorem 3.2 we still have in our more general context that the cumulants \(k^{{\text {id}}}\) fulfill the degree bounds (3.1). The reasoning from [26, Section 4.7] is still valid in our context.
3.4 Functionals \(\mathcal {S}_k\)
In [27, Proposition 4.6] the secondnamed author proved that \(\mathcal {S}_k\in \mathscr {P}\) is an \(\alpha \)polynomial function of degree k.
3.5 Free cumulants \(\mathcal {R}_k\)
In many calculations related to the asymptotic representation theory it is convenient to parametrize the shape of the Young diagram \(\lambda \) by free cumulants \(\mathcal {R}_k(\lambda )\) (which depend on the parameter \(\alpha \) in our settings). In the context of the representation theory of the symmetric groups these quantities have been introduced by Biane [3].
3.6 The case \(\alpha =1\)

Each free cumulant \(\mathcal {R}_k\) of a given Young diagram \(\lambda \) can be efficiently calculated [3] and its dependence on the shape of \(\lambda \) takes a particularly simple form (more specifically, “\(\mathcal {R}_k\) is a homogeneous function”).
 The family \((\gamma ,\mathcal {R}_2,\mathcal {R}_3,\ldots )\) forms a convenient algebraic basis of the algebra \(\mathscr {P}\) with \(\deg \gamma =1\) and \(\deg \mathcal {R}_k=k\). In the special case \(\alpha =1\) (which corresponds to \(\gamma =0\)) the expansion of \({{\mathrm{Ch}}}_l\) in this basis takes the following, particularly simple form:$$\begin{aligned} {{\mathrm{Ch}}}_l= \mathcal {R}_{l+1} + (\text {terms of degree at most }l1). \end{aligned}$$(3.8)
3.7 Details of Theorem 2.3 part (a) in the generic case \(\alpha \ne 1\)
The original proof of [26, Theorem and Definition 1] was based on the idea of expressing various elements F of the algebra of \(\alpha \)polynomial functions \(\mathscr {P}^{(1)}\) (such as the characters \({{\mathrm{Ch}}}_\pi \) or the conditional cumulants \(\kappa _{\bullet }\) of such characters) as polynomials in the basis \(\mathcal {R}_2,\mathcal {R}_3,\ldots \) of free cumulants and studying the topdegree of such polynomials, as we did in Sect. 3.6. In our more general context of \(\alpha \ne 1\) (or, in other words, \(\gamma \ne 0\)) the corresponding polynomial for F might have some extra terms which depend additionally on the variable \(\gamma \). These extra terms might influence the asymptotic behavior of random Young diagrams. We shall discuss this issue in more detail in the remaining of this section as well as in Sect. 3.9.
3.7.1 The firstorder asymptotics when \(\alpha \) is constant
Anticipating the proof of Theorem 1.5, the above considerations imply the following explicit description of the covariance of the limiting Gaussian process \(\Delta _\infty \).
3.7.2 The firstorder asymptotics in the double scaling limit
In the asymptotics when \(\alpha =\alpha (n)\) depends on n in a way described in Sect. 1.11 with \(g\ne 0\), the extra terms in both examples (3.15), (3.16) considered above are of the same order as the original terms. It follows that the relationship between the quantities \((a_l)\), \((a'_l)\) and \((a''_l)\) is altered and depends on the constant \(g\) from (1.9), see Sect. 3.9.2 below. Also the covariance of the Gaussian process describing the fluctuations of random Young diagrams is altered; finding an explicit form for this covariance is currently beyond our reach because no closed formula for the topdegree part of the conditional cumulant \(\kappa _{\bullet }({{\mathrm{Ch}}}_{l_1},{{\mathrm{Ch}}}_{l_2})\) (an analogue of the results of [27, Section 1] for \({{\mathrm{Ch}}}_n\)) is available.
3.8 Refined asymptotics of characters
In order to find more subtle relationships between the asymptotics of various quantities appearing in Theorem 2.3 we need an analogue of Equation (3.8) between the character \({{\mathrm{Ch}}}_l\) and the free cumulants in the generic case \(\alpha \ne 1\). We present below two such formulas: the one from Sect. 3.8.1 is conceptually simpler and will be sufficient for the scaling when \(\alpha \) is fixed; in the case of the double scaling limit we will need a more involved formula from Sect. 3.8.2.
3.8.1 The rough estimate
3.8.2 Closed formula for topdegree part of Jack characters
For a permutation \(\pi \) we denote by \(C(\pi )\) the set of its cycles.
Lemma 3.6
3.9 Proof of part (b) of Theorem 2.3
3.9.1 The scaling when \(\alpha \) is constant
Part (b) of Theorem 2.3 concerns the trivial case \(\ell =1\) which one can easily prove from scratch, based on (3.17). We shall present a detailed proof only for a specific case which will be useful in applications (more specifically, for the proof of Theorem 1.5 from Sect. 6) and we shall assume that the refined asymptotics of characters specified in part (b) of Theorem 2.3 holds true for the quantity (2.6). The other implications are analogous.
Remark 3.7
One can see that generically for \(\alpha \ne 1\) and \(\gamma \ne 0\) (even if the initial characters \(\chi _n(l)\) have small subleading terms which corresponds to \(b_l\equiv 0\)) the subleading terms in (3.23) are much bigger than their counterparts for \(\alpha =1\) from (3.12), namely they are of order \(\frac{1}{\sqrt{n}}\) times the leading asymptotic term. As we shall see in Sect. 6, this leads to noncenteredness of the limiting Gaussian process \(\Delta _\infty \).
3.9.2 The double scaling limit
4 Technical results
This section is devoted to some technical results necessary for the proof of Theorem 1.3.
4.1 Slowly growing sequence of moments determines the measure
Lemma 4.1
Then the measure \(\mu \) is uniquely determined by its moments.
Proof
In the case when \(\mu \) is supported on the interval \([0,\infty )\) this is exactly Stieltjes moment problem, while in the case when \(\mu \) is supported on the real line \(\mathbb {R}\) this is exactly the Hamburger moment problem. It is easy to check that the assumptions (4.1), and (4.2) imply that the Carleman’s conditions in both Stieltjes and Hamburger, respectively, problems are satisified and it follows that the measure \(\mu \) is uniquely determined by its moments.
The case, when \(\mu \) is supported on the interval \((\infty ,x_0]\) is analogous, and we leave it as a simple exercise. \(\square \)
4.2 Slow growth of \((\mathcal {R}_n)\) implies slow growth of \((\mathcal {S}_n)\)
Lemma 4.2
Proof
4.3 Estimates on some classes of permutations
Lemma 4.3
Proof
4.4 Growth of free cumulants
Proposition 4.4
Proof
5 Law of large numbers: Proof of Theorem 1.3
For Reader’s convenience the proof of Theorem 1.3 was split into several subsections which consitute the current section.
5.1 Measure associated with a Young diagram
5.2 Random variables \(S^{[n]}_k\) and their convergence in probability
5.3 The limiting probability measure \(P_{\Lambda _\infty }\)
5.4 The measure \(P_{\Lambda _\infty }\) is determined by its moments
We consider first the case \(g=0\). Lemma 4.1 implies immediately that the measure \(P_{\Lambda _\infty }\) is uniquely determined by its moments which concludes the proof.
In the case \(g>0\) the height of each box constituting the anisotropic Young diagram \(\Lambda _n\) is equal to \(g+o(1)>c\) for some constant \(c>0\), uniformly over n, cf. Sect. 1.11. By comparison of the areas it follows that the length l of the bottom rectangle constituting \(\Lambda _n\) fulfills \( l c \le 1 \); in particular it follows that the support of the measure \(P_{\Lambda _n}\) is contained in the interval \(\left( \infty , \frac{1}{c} \right] \). It follows that an analogous inclusion holds true for the support of the mean value \(\mathbb {E}P_{\Lambda _n}\); by passing to the limit the same is true for \(P_{\Lambda _\infty }\). It follows that Lemma 4.1 can be applied which concludes the proof.
The case \(g<0\) is fully analogous.
5.5 Weak convergence of probability measures implies uniform convergence of densities
To conclude, we proved the following theorem which might be of independent interest.
Theorem 5.1
Let \((\Lambda _n)\) be a sequence of random anisotropic Young diagrams, each with the unit area. Let \((P_{\Lambda _n})\) be the corresponding sequence of random probability measures on \(\mathbb {R}\) with densities \((f_{\Lambda _n})\) as in Sect. 5.1. Assume that the sequence of probability measures \((\mathbb {E}P_{\Lambda _n})\) converges to some limit in the weak topology.
Then there exists a function \(f_{\Lambda _\infty }\) such that \(\mathbb {E}f_{\Lambda _n} \rightarrow f_{\Lambda _\infty }\) uniformly on \(\mathbb {R}\).
5.6 Convergence of densities, in probability
5.7 Back to Young diagrams
6 Central Limit Theorem: Proof of Theorem 1.5
Proof of Theorem 1.5
Notes
Acknowledgements
Research supported by Narodowe Centrum Nauki, Grant No. 2014/15/B/ST1/00064.
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