Fluctuations of Interlacing Sequences
In a series of works published in the 1990s, Kerov put forth various applications of the circle of ideas centered at the Markov moment problem to the limiting shape of random continual diagrams arising in representation theory and spectral theory. We demonstrate on several examples that his approach...
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irk-123456789-1405822018-07-11T01:23:40Z Fluctuations of Interlacing Sequences Sodin, S. In a series of works published in the 1990s, Kerov put forth various applications of the circle of ideas centered at the Markov moment problem to the limiting shape of random continual diagrams arising in representation theory and spectral theory. We demonstrate on several examples that his approach is also adequate to study the fluctuations about the limiting shape. In the random matrix setting, we compare two continual diagrams: one is constructed from the eigenvalues of the matrix and the critical points of its characteristic polynomial, whereas the second one is constructed from the eigenvalues of the matrix and those of its principal submatrix. The fluctuations of the latter diagram were recently studied by Erd}os and Schröder; we discuss the uctuations of the former, and compare the two limiting processes. For Plancherel random partitions, the Markov correspondence establishes the equivalence between Kerov's central limit theorem for the Young diagram and the Ivanov-Olshanski central limit theorem for the transition measure. We outline a combinatorial proof of the latter, and compare the limiting process with the ones of random matrices. 2017 Article Fluctuations of Interlacing Sequences / S. Sodin // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 63. — Бібліогр.: 63 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag13.04.364 Mathematics Subject Classification 2000: 60B20, 34L20, 05E10, 60F05, 44A60 http://dspace.nbuv.gov.ua/handle/123456789/140582 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In a series of works published in the 1990s, Kerov put forth various applications of the circle of ideas centered at the Markov moment problem to the limiting shape of random continual diagrams arising in representation theory and spectral theory. We demonstrate on several examples that his approach is also adequate to study the fluctuations about the limiting shape. In the random matrix setting, we compare two continual diagrams: one is constructed from the eigenvalues of the matrix and the critical points of its characteristic polynomial, whereas the second one is constructed from the eigenvalues of the matrix and those of its principal submatrix. The fluctuations of the latter diagram were recently studied by Erd}os and Schröder; we discuss the uctuations of the former, and compare the two limiting processes. For Plancherel random partitions, the Markov correspondence establishes the equivalence between Kerov's central limit theorem for the Young diagram and the Ivanov-Olshanski central limit theorem for the transition measure. We outline a combinatorial proof of the latter, and compare the limiting process with the ones of random matrices. |
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Sodin, S. Fluctuations of Interlacing Sequences Журнал математической физики, анализа, геометрии |
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Sodin, S. |
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Fluctuations of Interlacing Sequences |
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Fluctuations of Interlacing Sequences |
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Fluctuations of Interlacing Sequences |
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Fluctuations of Interlacing Sequences |
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Fluctuations of Interlacing Sequences |
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fluctuations of interlacing sequences |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Fluctuations of Interlacing Sequences / S. Sodin // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 63. — Бібліогр.: 63 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT sodins fluctuationsofinterlacingsequences |
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2025-07-10T10:47:50Z |
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2025-07-10T10:47:50Z |
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1837256647684128768 |
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Journal of Mathematical Physics, Analysis, Geometry
2017, vol. 13, No. 4, pp. 364–401
doi:10.15407/mag13.04.364
To Uzy Smilansky on his 752
3 -th birthday
Fluctuations of Interlacing Sequences
Sasha Sodin
Tel Aviv University, School of Mathematical Sciences
Tel Aviv 69978, Israel
Queen Mary University of London, School of Mathematical Sciences
London E1 4NS, United Kingdom
E-mail: sashas1@post.tau.ac.il
Received November 07, 2016
In a series of works published in the 1990s, Kerov put forth various
applications of the circle of ideas centered at the Markov moment problem
to the limiting shape of random continual diagrams arising in representation
theory and spectral theory. We demonstrate on several examples that his
approach is also adequate to study the fluctuations about the limiting shape.
In the random matrix setting, we compare two continual diagrams: one
is constructed from the eigenvalues of the matrix and the critical points of
its characteristic polynomial, whereas the second one is constructed from
the eigenvalues of the matrix and those of its principal submatrix. The fluc-
tuations of the latter diagram were recently studied by Erdős and Schröder;
we discuss the fluctuations of the former, and compare the two limiting
processes.
For Plancherel random partitions, the Markov correspondence establishes
the equivalence between Kerov’s central limit theorem for the Young diagram
and the Ivanov–Olshanski central limit theorem for the transition measure.
We outline a combinatorial proof of the latter, and compare the limiting
process with the ones of random matrices.
Key words: interlacing sequences, Markov moment problem, continual
diagrams, random matrices, central limit theorem.
Mathematical Subject Classification 2010: 60B20, 34L20, 05E10, 60F05,
44A60.
1. Overview
1.1. Markov correspondence. Two sequences of real numbers A =
(a1 < · · · < an) and B = (b1 < · · · < bn−1) are called interlacing if
a1 < b1 < a2 < · · · < bn−1 < an.
c© Sasha Sodin, 2017
Fluctuations of Interlacing Sequences
To a pair of interlacing sequences (A,B) one associates the probability measure
µ =
n∑
j=1
pjδaj ,
where pj are defined by the simple fraction decomposition∏n−1
j=1 (z − bj)∏n
j=1(z − aj)
=
n∑
j=1
pj
z − aj
. (1)
Vice versa, a probability measure supported on a finite set of atoms gives rise to
a pair of interlacing sequences.
This construction admits numerous generalizations. The relation
exp
{
−
∫
log(z − x)d(τ+(x)− τ−(x))
}
=
∫
dµ(x)
z − x
, (2)
obtained from (1) by replacing sums with integrals, forms the basis of the solution
of the Markov moment problem (see [5, 37, 43]), and is one of the forms of the
Markov correspondence (which we further discuss below and in 2.2.2). In the
terminology of Kerov [33,34], τ+ and τ− corresponding to a probability measure
µ form a pair of interlacing measures; such pairs are intrinsically characterized
by the inequalities
τ+[x,∞) ≥ τ−[x,∞) , τ+(−∞, x] ≥ τ−(−∞, x] (x ∈ R). (3)
The equality (2) may be viewed as a connection between an additive and a
multiplicative representations of a function from the Nevanlinna class. In this
form it admits further generalizations, extensively used starting from the works
of Akhiezer and Krein (see, e.g., the appendix to [37]); we also refer to the work
of Yuditskii [63] for some recent developments.
In the 1990s, Kerov discovered a number of applications of the Markov corre-
spondence to problems in representation theory and analysis. These applications
form a central theme in the monograph [34], see further the survey [33].
Here we follow Kerov and switch to the language of continual diagrams, which
are 1-Lipschitz functions coinciding with |x − a| for large values of |x| (such as
in Figure 1c and Figure 2; see 2.2.1 for a formal definition). The mapping from
pairs of interlacing sequences to continual diagrams is given by (A,B) 7→ ω,
ω(x) =
n∑
j=1
|x− aj | −
n−1∑
j=1
|x− bj |; (4)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 365
Sasha Sodin
whereas for interlacing measures (τ+, τ−) as in (2) or (3), one sets
ω(x) =
∫
|x− a| dτ+(a)−
∫
|x− b| dτ−(b).
The Markov correspondence (2) induces a bijection between continual diagrams
ω and probability measures µ, which is called the Markov transform and denoted
µ = Mω. Some of its properties are listed in 2.2.2.
Continual diagrams appear naturally as scaling limits of Young diagrams.
Indeed, a Young diagram rotated by 135◦ (with respect to the English convention)
is a continual diagram, see Figure 1 for an illustration and 4.1.1, 4.1.2 for a formal
construction.
(a) (b) (c)
Fig. 1: A Young diagram in English convention (a) and in Russian convention
(b), and the corresponding continual diagram (c).
Kerov showed that the Markov transform µn = Mωn of a continual diagram
ωn obtained in this way encodes the transition probabilities of the Young dia-
gram in a stochastic process called the Plancherel growth (see 4.1.3), and called
µn the transition measure of the Young diagram. If ωn is a random continual di-
agram associated with a Young diagram sampled at random from the Plancherel
measure, the Logan–Shepp–Vershik–Kerov limit law [40,60,61] asserts that (uni-
formly almost surely)
1√
n
ωn(
√
nx) −→ ΩLSVK(x) =
{
2
π
(
x arcsin x
2 +
√
4− x2
)
, |x| ≤ 2,
|x|, |x| > 2.
(5)
Using the Markov correspondence, Kerov deduced that µ̃n (here and further in
the introduction, tildes indicate unspecified scaling) obey the semicircle law [30]
dµ̃n(x)→ dρs.c.(x) =
1
2π
√
(4− x2)+ dx; (6)
vice versa, (6) implies (5).
Kerov also discovered [31] a random matrix counterpart of these statements;
it comes in two flavors. Let P̃n(x) be the characteristic polynomial of a Wigner
366 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
matrix H̃n of dimension n×n (see 2.3.4); let λ̃j be the zeros of P̃n (which are the
eigenvalues of H̃n). Also let λ̃∗j be the zeros of P̃ ′n, and let λ̃∗∗j be the eigenvalues
of the top-left (n− 1)× (n− 1) principal submatrix of H̃n. Then (see 2.3.4)
n∑
j=1
|x− λ̃j | −
n−1∑
j=1
|x− λ̃∗j | → ΩLSVK(x) (7)
and also (see 2.3.5)
n∑
j=1
|x− λ̃j | −
n−1∑
j=1
|x− λ̃∗∗j | → ΩLSVK(x). (8)
The former was proved by Kerov, the latter was proved by Alexey Bufetov [14]
who strengthened the result in [31].
As put forth by Kerov, (7) is equivalent (by the Markov correspondence) to
Wigner’s semicircle law
ρ̃n =
1
n
n∑
j=1
δλ̃j → ρs.c.
for the normalized eigenvalue counting measure, see further 2.3.5. On the other
hand, (8) is equivalent to Wigner’s law for the spectral measure associated with
a fixed vector in Cn, see 2.3.4.
Another similar looking result was found by Kerov [31] in the setting of Jacobi
matrices with regularly varying coefficients. There (see 2.1.1), the counterparts
of λ̃j and λ̃∗∗j are the (properly rescaled) eigenvalues of an n×n and an (n−1)×
(n− 1) principal submatrix, respectively; then (8) holds, as a consequence of the
semicircle limit law for what could be colloquially called the spectral measure at
infinity (see 2.1.1 and 2.1.2). On the other hand, the counterpart of (7) is, in
general, false (see 2.1.5).
In Section 2, mostly following [33, 34] in substance if not in terminology, we
give an overview of these results with some proofs.
1.2. Fluctuations about the limiting shape. In the main part of the
paper (Sections 3 and 4), we use the Markov correspondence to study the de-
viations of diagrams and of interlacing sequences from the limiting shape. We
observe that, although (2) is highly nonlinear, it can be linearized about the lim-
iting shape. Therefore one can study the fluctuations of the left-hand side via
the right-hand side, and vice versa. Several forms of this assertion are proved in
Section 3.1, for example, in the case of the limiting shape ΩLSVK we have:
Proposition 3.1.4. Let µn be probability measures, and let ωn = Mµn be
the corresponding continual diagrams. Let (εn)n and (αk)k≥2 be two sequences
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 367
Sasha Sodin
such that εn → 0 and 1
k log |αk| → 0. Then
lim
n→∞
∫ 2
−2
φ(x)
dµn(x)− dρs.c.(x)
εn
=
∑
k≥2
αk
∫ 2
−2
φ(x) 2Tk(x/2) dρarcsin(x) (9)
for all test functions φ which are analytic in [−2, 2] if and only if
lim
n→∞
∫ 2
−2
φ(x)
ωn(x)− ΩLSVK(x)
εn
dx =
∑
k≥1
4αk+2
k + 1
∫ 2
−2
φ(x)Uk(x/2)dρs.c.(x) (10)
for all such φ.
Here
dρarcsin(x) =
1[−2,2](x)dx
π
√
4− x2
is the arcsine distribution, Tk and Uk are the Chebyshev polynomials (43), and
the Markov transform M is formally defined in 2.2.2. A marginally more general
formulation is given in 3.1.4, and the stochastic setting (as opposed to determin-
istic deviations from the limit shape) is commented upon in 3.2.1. Note that the
integrals are exactly the coefficients of φ in the Chebyshev expansions:
φ(x) =
∑
k≥0
Tk(x/2)
∫ 2
−2
φ(y) (2− δk0)Tk(y/2) dρarcsin(y)
=
∑
k≥0
Uk(x/2)
∫ 2
−2
φ(y)Uk(y/2)dρs.c.(y).
In particular, the assumption (9) implies that (and is almost equivalent to)∫
Tk(x/2)(dµn(x)− dρs.c.(x)) = εnαk + o(εn) , n→∞. (11)
We use Proposition 3.1.4 to study the fluctuations of random diagrams appear-
ing in the theory of random matrices and in the representation theory of the
symmetric group.
The study of fluctuations of diagrams was initiated by a theorem, proved by
Kerov [32] in the 1990s, which describes the fluctuations of a (Plancherel) random
Young diagram about the limiting shape. Informally,
1√
n
ωn(
√
nx) ≈ ΩLSVK(x) +
1√
n
∑
k≥1
2gk+2√
k + 1
Uk(x/2)
√
4− x2
2π
, (12)
where gk are independent, identically distributed standard Gaussian variables (see
4.1.5). Another proof, based on Kerov’s unpublished notes, was given by Ivanov
368 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
and Olshanski in [25]. We refer to the works [15,44,45] for various generalizations,
not discussed here.
Ivanov and Olshanski also described the fluctuations of the transition measure
µ̃n associated with ωn (see 4.1.3):
dµ̃n(x) ≈ dρs.c.(x) +
1√
n
∑
k≥3
√
k − 1
2
gk
2Tk(x/2)dx
π
√
4− x2
. (13)
As observed in [25], (13) bears a similarity to Johansson’s central limit theo-
rem [27] for the Gaussian Unitary Ensemble:
dρ̃n(x) ≈ dρs.c.(x) +
1
n
∑
k≥1
√
k
2
gk
2Tk(x/2)dx
π
√
4− x2
(14)
(see 4.1.5). This raises the question what is the analogue of Kerov’s central limit
theorem (12) in random matrix context.
Recently, this question was studied by Erdős and Schröder [19]. Their result is
both general (generalized Wigner matrices are considered), and strong (the ≈ sign
in (15) below can be understood pointwise, with an explicit power-law estimate
on the error term). In the special case of the Gaussian Unitary Ensemble, the
result of [19] asserts that the fluctuations of the diagram corresponding to the
eigenvalues of H̃n and of its principal submatrix are described by
n+1∑
j=1
|x− λ̃j | −
n∑
j=1
|x− λ̃∗∗j | ≈ ΩLSV K(x) +
1√
n
∆M
1 (x), (15)
∆M
1 (x) =
2g1
π
arcsin
x
2
+
∑
k≥1
2(gk − gk+2)
k + 1
Uk(x/2)
√
4− x2
2π
, (16)
where the argument of the arcsine is truncated at ±1 (see further 3.2.3 and 3.3.3).
Parallel results for Jacobi β-ensembles were obtained by Gorin and Zhang in [22].
The right-hand side of (15) does not look similar to (12) for two reasons. First,
a typical random partition of n has �
√
n rows, therefore the normalization 1/
√
n
in (15) would correspond to 1/ 4
√
n, rather than 1/
√
n in (12). Second, the Gaus-
sian process in (16) has a continuous modification (it is roughly a reparametrized
Brownian motion), whereas the sum in (12) does not even converge in L2.
These differences are natural in view of the Markov correspondence. As men-
tioned in Section 1.1 above, the Markov transform takes the left-hand side of
(15) to the spectral measure µ̃n (see 2.3.4). The fluctuations of the latter were
studied, in the context of Gaussian ensembles, by Lytova and Pastur [41]. For
the GUE, their result (see 3.2.3) asserts that
dµ̃n(x) ≈ dρs.c.(x) +
1√
n
∆1(x)dx , (17)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 369
Sasha Sodin
∆1(x) ∼
∑
k≥1
gkUk(x/2)
√
4− x2
2π
. (18)
As noted in [41], the larger scale 1/
√
n reflects the Gaussian fluctuations of the
eigenvectors of the random matrix. The result (17) was extended to other Wigner
matrices by the same authors [42] and by Pizzo–Renfrew–Soshnikov [53]; in this
more general setting, the limiting process may change and it is not necessarily
Gaussian (see 3.3.3).
In 3.2.3, we feed (17) into the Markov correspondence and obtain a version
of (15), albeit in much weaker topology than [19]. More general Wigner matrices
are considered in 3.3.3. (Vice versa, it may be possible to deduce a strengthened
version of the central limit theorems in [42,53] from the result of Erdős–Schröder
[19] in its full strength and generality; we do not pursue this direction here (See
further 3.3.5).
The discussion above raises the expectation that a version of (15) with the
critical points λ̃∗j in place of the submatrix eigenvalues λ̃∗∗j may bear more sim-
ilarity to Kerov’s theorem (12). Indeed, appealing to Johansson’s central limit
theorem (14), we show that
Corollary 3.2.2. For the Gaussian Unitary Ensemble,
n∑
j=1
|x− λ̃j | −
n−1∑
j=1
|x− λ̃∗j | ≈ ΩLSV K(x) +
1
n
∆M
tr (x), (19)
∆M
tr (x) ∼ 2
π
g1 arcsin(x/2) +
∑
k≥0
2
√
k + 2
k + 1
gk+2
Uk(x/2)
√
4− x2
2π
. (20)
A precise formulation is given in 3.2.2, a comparison between (19) and (15) —
in Figure 2, and a generalization to other Wigner matrices, in 3.3.1.
Recently, Fyodorov asked what are the properties of the critical points λ̃∗j
and, in particular, how to distinguish between them and λ̃∗∗j . Differentiating
the relations (19)–(20) and comparing to (15)–(16), we obtain an answer in the
following form (stated here for GUE; the topology is as in Proposition 3.1.4):
n∑
j=1
δλ̃j −
n−1∑
j=1
δλ̃∗j
≈ dx
π
√
4− x2
+
1
2n
[
d2
dx2
∆M
tr (x)
]
dx, (21)
n∑
j=1
δλ̃j −
n−1∑
j=1
δλ̃∗∗j
≈ dx
π
√
4− x2
+
1
2
√
n
[
d2
dx2
∆M
1 (x)
]
dx. (22)
Note the difference in scaling, and that in the former, the Gaussian process is
a derivative of a log-correlated process, while in the latter, it is a derivative of
370 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
Fig. 2: The random continual diagrams (19) in blue (starts at (−3, 3)) and (15) in
green (starts at (−3, 3.1)), for GUEn=50. The former fluctuates on scale ∼ 1/n,
while the latter — on scale 1/
√
n.
(reparametrized) white noise (cf. (49)). The relation (22) was proved by Erdős
and Schröder, in greater generality (Wigner matrices) and stronger topology (cor-
responding to test functions in the Sobolev space H2[−10, 10]); they used it to
prove (15). The relation (21) seems not to have been observed before. The for-
mulæ (21) and (22) appear as (48) and (52), respectively, in the body of the
paper.
The comparison between the statistical properties of the critical points and
the eigenvalues in the local regime (i.e., on scales comparable to the mean spacing)
is discussed in the companion paper [57].
In Section 4, we return to random partitions and to the theorems of Kerov
and Ivanov–Olshanski stated above as (12) and (13). We describe the setting and
use Proposition 3.1.4 to derive one from the other.
In Section 4.2, we outline a proof of (13) (and thus also of (12)) based on the
combinatorial approach of Biane [10] and Okounkov [46] in the version of [26]. Our
goal is to emphasize the similarity between the transition measure of a random
diagram and the normalized eigenvalue counting measure of a random matrix.
To this end, we compare the Jucys–Murphy elements (see 4.1.4) acting on a
random representation of the symmetric group with Wigner random matrices
chosen from an ensemble (see 4.1.6) for which a particularly clean version of
the moment method is available (cf. [21, 56] and references therein). For this
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 371
Sasha Sodin
ensemble,
dρ̃n(x) ≈ dρs.c.(x) +
1
n
∑
k≥3
√
k
2
gk
2Tk(x/2)dx
π
√
4− x2
, (23)
and accordingly (Proposition 4.1.6),
n∑
j=1
|x− λ̃j | −
n−1∑
j=1
|x− λ̃∗j |
≈ ΩLSVK(x) +
1
n
∑
k≥1
2
√
k + 2
k + 1
gk+2
Uk(x/2)
√
4− x2
2π
. (24)
In the combinatorial approach, the coefficient k (the square root of which appears
in (23)) acquires the interpretation as the number of ways to align two cycles of
length k. The coefficient k−1 in (13) has exactly the same combinatorial meaning
(with cycles of length k − 1).
Many of the results are probably familiar to experts. However, we hope to
find a reader that would enjoy seeing them under a single cover, rephrased in the
peculiar argot of spectral theory. We made no attempt to pursue the limits of the
approach, and instead chose to illustrate the main ideas in the simplest setting.
2. Limit Shape
2.1. Jacobi matrices.
2.1.1. Let J be a Jacobi matrix and let Jn be its top-left n × n principal
submatrix
J =
a1 b1 0 0 · · ·
b1 a2 b2 0 · · ·
0 b2 a3 b3 · · ·
· · · · · · · · · · · · · · ·
, Jn =
a1 b1 0 0 · · ·
b1 a2 b2 0 · · ·
· · · · · · · · · · · · · · ·
0 0 · · · bn−1 an
. (25)
Define a probability measure µn by∫
xkdµn(x) = (Jkn)nn = 〈Jknδn, δn〉, k = 0, 1, 2, . . . (26)
It may be called the spectral measure of Jn at δn or, following Kerov, the n-th
transition measure of J . It is supported on the eigenvalues of Jn; the mass
at an eigenvalue is equal to the squared n-th coordinate of the corresponding
eigenvector.
Also define the normalized eigenvalue counting measure ρn by∫
xkdρn(x) =
1
n
tr Jkn , k = 0, 1, 2, . . . (27)
which has equal masses 1/n at the eigenvalues of Jn.
372 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
2.1.2. The asymptotics of µn (colloquially, the spectral measure at infinity)
is determined by the asymptotics of an and bn. The following proposition is
folklore (cf. Kerov [31]):
Proposition. Let J be a Jacobi matrix (25) such that
lim
n→∞
bn−1
bn
= 1, lim
n→∞
an−1 − an
bn
= 0. (28)
Then the sequence of measures µ̃n defined by
µ̃n(B) = µn(bn−1B + an), B ∈ B(R),
converges weakly to the semicircle measure ρs.c.,
ρs.c.(B) =
∫
B∩(−2,2)
1
2π
√
4− x2 dx.
Proof. The measure µ̃n is the spectral measure of J̃n at δ1, where
(J̃n)i j = b−1
n−1 ((Jn)n+1−i n+1−j − an) .
As n→∞,
b−1
n−1(J̃n − an)→ T
in strong operator topology, where Tij = δi,j+1 + δi+1,j is the adjacency matrix of
Z>0. The spectral measure of T at δ1 is exactly ρs.c., as one can see, for example,
from the relation
((T − z)−1)11 = (−z − ((T − z)−1)11)−1, z ∈ C \ R,
which follows from the formula for the top-left matrix element of a matrix inverse.
Therefore µ̃n → ρs.c..
2.1.3. The previous discussion remains valid for a sequence of Jacobi ma-
trices which are not necessarily the sub-matrices of one infinite matrix. By the
same argument as above, we have:
Proposition. Let (Jn)n≥0 be a sequence of finite Jacobi matrices
Jn =
a1,n b1,n 0 0 · · ·
b1,n a2,n b2,n 0 · · ·
· · · · · · · · · · · · · · ·
0 0 · · · bn−1,n an,n
such that for any k ≥ 1,
lim
n→∞
bn−k−1,n
bn−1,n
= 1, lim
n→∞
an−k,n − an,n
bn−1,n
= 0. (29)
Then the sequence of measures µ̃n(B) = µn(bn−1,nB + an,n) converges weakly to
the semicircle measure ρs.c..
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Sasha Sodin
2.1.4. The asymptotics of ρn is also determined by the asymptotics of an
and bn. However, µn is insensitive to multiplication of the coefficients by a se-
quence cn such that lim cn
cn−1
= 1, whereas ρn depends on the growth of the
coefficients. For example, we have
Proposition. Suppose
an = o(
√
n), bn =
√
n(1 + o(1)). (30)
Then the rescaled measures ρ̃n(B) = ρn(
√
nB) converge weakly to ρs.c..
Proof. Let aH
n = 0, bHn =
√
n. Here H stands for Hermite, and ρH
n is the
normalized zero counting measure of Hermite polynomials. It is known [59] that
ρ̃H
n → ρs.c.. This can also be proved directly from the Jacobi matrix as in 2.1.2.
To justify the approximation ρn ≈ ρH
n , let ε > 0. Choose n0 such that for
n > n0
|an| < ε
√
n, |bn −
√
n| < ε
√
n. (31)
Let
a1
n =
{
0, n ≤ n0
an , n > n0
,
{
b1n =
√
n, n ≤ n0
bn , n > n0
.
Then, for any segment [a, b],∣∣ρ̃n[a, b]− ρ̃1
n[a, b]
∣∣ ≤ n0
n
by the interlacing property of rank-one perturbation, and
ρ̃H
n [a+ ε, b− ε] ≤ ρ̃1
n[a, b] ≤ ρ̃H
n [a− ε, b+ ε]
by (31). It remains to let n→∞ and then ε→ +0.
2.1.5. In the case of Proposition 2.1.4, the sequences ρn and µn share the
same asymptotics. As emphasized by Kerov, this is an exception rather than a
rule. The limit of the former is a kind of integrated density of states, while the
latter describes the spectral properties at infinity. Neither is directly related to
the usual spectral measure at δ1.
Example. For J = T , ρn → ρarcsin, where
ρarcsin(B) =
∫
B∩(−2,2)
1
π
dx√
4− x2
.
The same conclusion holds for any Jacobi matrix with
an = o(1), bn = 1 + o(1).
374 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
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2.2. Continual diagrams.
2.2.1. A continual diagram is a function ω : R → R such that for some
a ∈ R
|ω(x)− ω(y)| ≤ |x− y|, (32)
ω(x) = |x− a| for sufficiently large x. (33)
The collection of diagrams is equipped with the topology of uniform conver-
gence. A diagram ω is said to be supported in a (closed) segment I (denoted:
ω ∈ D(I)) if (33) holds for all x /∈ I.
2.2.2. Denote by M(I) the collection of Borel probability measures on I,
equipped with weak topology.
Theorem (Markov [43]; Akhiezer–Krein [3]; Kerov [30]). For any segment
I, the relation
exp
{
−1
2
∫
log(1− zx)dω′(x)
}
=
∫
dµ(x)
1− zx
defines a homeomorphism M : D(I)←→M(I).
The homeomorphism M is called the Markov transform. In the language of
what is now called the Markov moment problem, the construction of the bijection
M for the case of a segment goes back to Markov [43]. It was developed and
generalized by Akhiezer and Krein in the 1930s, who published a series of papers
[1–4] and a book [5] on this subject. The formulation in the language of continual
diagrams is due to Kerov [30], who also observed that M is a homeomorphism;
see further [34].
Example. The Logan–Shepp–Vershik–Kerov diagram
ΩLSVK(x) =
{
2
π
(
x arcsin x
2 +
√
4− x2
)
, |x| ≤ 2,
|x|, |x| > 2,
corresponds to the semicircle: MΩLSVK = ρs.c..
2.2.3. Let J be a Jacobi matrix. Denote by λ
(n)
j the eigenvalues of Jn,
and define the diagram ωn corresponding to the interlacing sequences (λ
(n)
j ) and
(λ
(n−1)
j ) via (4), i.e., as a continuous function such thatω
′
n(x) = −1 + 2 #
{
j, λ
(n)
j ≤ x
}
− 2 #
{
j , λ
(n−1)
j ≤ x
}
,
ωn(x) =
∣∣∣x−∑n
j=1 λ
(n)
j +
∑n−1
j=1 λ
(n−1)
j
∣∣∣ for sufficiently large x.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 375
Sasha Sodin
Equivalently, if Pn(z) = det(z − Jn),
ω′n(x) = sign
Pn−1(x)
Pn(x)
.
Also define a diagram $n such that
$′n(x) = sign
P ′n(x)
Pn(x)
= sign
1
n
P ′n(x)
Pn(x)
.
Lemma. Mωn = µn and M$n = ρn.
Proof. Representing Pn−1/Pn and P ′n/Pn as a sum of simple fractions, we
obtain:
Pn−1(z)
Pn(z)
=
∫
dµn(x)
z − x
,
1
n
P ′n(z)
Pn(z)
=
∫
dρn(x)
z − x
and then use the bijection M from Theorem 2.2.2.
As noted by Kerov (e.g., [33, Section 6]), this lemma is a finite-dimensional
trace formula (the study of trace formulæ goes back to the works of Lifshits [39]
and Krein [36], see further the survey of Birman and Yafaev [11]).
2.2.4.
Corollary (Kerov [31]). Let J be a Jacobi matrix (25) satisfying (28). Then
b−1
n−1ωn(bn−1x+ an)→ ΩLSVK(x)
uniformly in x.
Proof. Follows from Proposition 2.1.2, Lemma 2.2.3 and Theorem 2.2.2.
2.2.5. The corresponding statement for $n = M−1 ρn is much less gen-
eral.
Corollary (Kerov [31]). Let J be a Jacobi matrix (25) satisfying (30). Then
1√
n
$n(
√
nx)→ ΩLSVK(x)
uniformly in x.
Proof. Follows from Proposition 2.1.4, Lemma 2.2.3 and Theorem 2.2.2.
376 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
2.2.6. Introduce the rescaling operators
(RL µ)(B) = µ(LB) (B ∈ B(R)), (RL ω)(x) =
1
L
ω(Lx).
Consider the following random Jacobi matrix, constructed by Dumitriu and
Edelman [16]. Fix β > 0, and let Jβ be such that
aβn ∼ N(0, 2/β), β bβn ∼ χnβ,
and these random variables are jointly independent. Recall that the χa distribu-
tion is defined by the density
pa(x) =
21−a/2xa−1e−x
2/2
Γ(a/2)
.
We have almost surely:
aβn = O(
√
log n), bβn =
√
n(1 + o(1)).
Therefore, by Propositions 2.1.2 and 2.1.4,
R√n µ
β
n → ρs.c. , R√n ρ
β
n → ρs.c. weakly,
and by Corollaries 2.2.4 and 2.2.5,
R√n ω
β
n → ΩLSVK(x) , R√n$
β
n → ΩLSVK(x) uniformly,
almost surely.
Remark. These statements are not directly related to the spectral properties
of Jβ, see [12, 13] for the properties of the latter and a discussion.
2.3. Random matrices.
2.3.1. The Gaussian Unitary Ensemble (GUE) is the ensemble of semi-
infinite Hermitian matrices H = (H(i, j))i,j≥1 such that for any n the top-left
n× n submatrix Hn has the probability density
Z−1
n exp
{
−1
2
trH2
n
}
with respect to the Lebesgue measure on the space of n× n Hermitian matrices.
Let λ
(n)
j be the eigenvalues of Hn, and let Pn(z) = det(z −Hn). Define the
spectral measure µGUE
n and the normalized eigenvalue distribution ρGUE
n by the
formulæ∫
xkdµGUE
n = (Hk
n)nn ,
∫
xkdρGUE
n =
1
n
trHk
n , k = 0, 1, 2, . . .
Also define the diagrams ωGUE
n and $GUE
n by
d
dx
ωGUE
n (x) = sign
Pn−1(x)
Pn(x)
,
d
dx
$GUE
n (x) = sign
P ′n(x)
Pn(x)
.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 377
Sasha Sodin
2.3.2.
Theorem (Dumitriu–Edelman [16]). The joint distribution of
(λ
(n)
j )nj=1 and (λ
(n−1)
j )n−1
j=1
for the GUE coincides with the joint distribution of the same quantities for Jβ=2
from 2.2.6.
In other words, Jβ=2 is obtained from the GUE by tridiagonalization.
Corollary. The distribution of µGUE
n , ρGUE
n , ωGUE
n , $GUE
n from 2.3.1 coin-
cides with the distribution of µβ=2
n , ρβ=2
n , ωβ=2
n , $β=2
n associated with the random
Jacobi matrix Jβ=2 of 2.2.6.
2.3.3. Combining Corollary 2.3.2 with the conclusion of 2.2.6, we obtain:
Corollary (Kerov [31], special case). Almost surely
R√n µ
GUE
n ,R√n ρ
GUE
n → ρs.c. weakly,
R√n ω
GUE
n ,R√n$
GUE
n → ΩLSVK uniformly.
2.3.4. Corollary 2.3.3 can be extended to the class of Wigner matrices. Let
H = (H(i, j))i,j≥1 be an arbitrary semi-infinite Hermitian random matrix such
that
{H(i, j), i ≤ j}
are jointly independent with EH(i, j) = 0, (H(i, i)) are identically distributed,
and (H(i, j))i<j are identically distributed with E|H(i, j)|2 = 1. Let Hn be the
top-left principal submatrix of H, and let µn, ρn, ωn, $n be defined as in 2.3.1.
Let us quote Wigner’s law [62]. In the current generality it was proved by
Pastur [51].
Proposition (Wigner; Pastur). Almost surely R√n ρn → ρs.c..
Corollary (Kerov [31]). Almost surely R√n$n → ΩLSVK.
2.3.5. Proposition 2.3.4 and Corollary 2.3.4 have a counterpart for µn and
ωn. The former is the following version of Wigner’s law (proved similarly to [51]):
Proposition. In the setting of 2.3.4, R√n µn → ρs.c. almost surely.
Corollary (Kerov [31]; Bufetov [14] /general case/). R√n ωn → ΩLSVK al-
most surely.
378 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
3. Corrections to the Limit Shape
3.1. Deterministic corrections.
3.1.1. Let ωn be a sequence of continual diagrams such that ωn → Ω in
uniform topology. Then
µn = Mωn → ρ = MΩ
in weak topology. Our goal is to relate the corrections ωn−Ω to µn−ρ. We start
with a lemma.
Lemma. Let εn → +0. Suppose ωn and Ω are continual diagrams such that
the corresponding measures µn = Mωn and ρ = MΩ satisfy∫
(x− z)−1 dµn(x) =
∫
(x− z)−1dρ(x) + εnR(z) + o(εn), z ∈ C \ R. (34)
Then ∫
(x− z)−1 d(ωn(x)− Ω(x)) = −2εn
R(z)
wρ(z)
+ o(εn), z ∈ C \ R, (35)
where we defined the Stieltjes transform
wρ(z) =
∫
(x− z)−1 dρ(x). (36)
As one can see from the proof below, if the convergence in the assumption is
uniform on compact sets, it is so also in the conclusion.
Proof. By Theorem 2.2.2 applied with z−1 in place of z,∫
log(1− z−1x) dω′n(x) = −2 log
∫
(1− z−1x)−1 dµn(x)
= −2 log z + 2 log
∫
(x− z)−1 dµn(x).
Using the assumption (34), we deduce∫
log(1− z−1x) dω′n(x)
= −2 log z + 2 log
∫
(x− z)−1 dρ(x) + 2εn
R(z)
wρ(x)
+ o(εn).
Similarly, Ω and ρ are related by∫
log(1− z−1x) dΩ′(x) = −2 log z + 2 log
∫
(x− z)−1 dρ(x),
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 379
Sasha Sodin
hence ∫
log(1− z−1x) d(ω′n(x)− Ω′(x)) =
2εnR(z)
wρ(x)
+ o(εn). (37)
Integrating by parts, we rewrite the left-hand side of (37) as∫
log(1− z−1x) d(ω′n(x)− Ω′(x)) = −
∫
(x− z)−1 d(ωn(x)− Ω(x))
and this concludes the proof.
3.1.2. The conclusion of Lemma 3.1.1 can be reformulated in the following
form, from which one can see that the asymptotics of ωn−Ω (and not just of the
derivative) is determined.
Corollary. In the setting of Lemma 3.1.1, suppose that the convergence is
uniform on compact subsets of C \ R. Then∫
I
(x− z)−1(ωn(x)− Ω(x)) dx = 2εn
∫ z R(ζ) dζ
wρ(ζ)
+ o(εn), z ∈ C \ R, (38)
where I is an interval in which Ω is supported, the integral is from ±i∞ along
a path avoiding the real axis, and the convergence is also uniform on compact
subsets of C \ R.
Proof. Let I be an interval in which Ω is supported. The assumption of
uniform convergence on compact subsets implies that, for real x outside I,
ωn(x)− Ω(x) = o(εn). (39)
Therefore we can choose δn = o(εn) such that the shifted diagram
ω→n (x) = ωn(x− δn)
and Ω(x) have the same center, i.e., coincide for sufficiently large |x|. Note that
ω→n (x)− ωn(x) = o(εn) uniformly in x. By Lemma 3.1.1,∫
(x− z)−1 d(ω→n (x)− Ω(x)) = −2εn
R(z)
wρ(z)
+ o(εn), z ∈ C \ R,
whence, integrating by parts and replacing z with ζ,
d
dζ
∫
(x− ζ)−1(ω→n (x)− Ω(x)) dx = 2εn
R(ζ)
wρ(ζ)
+ o(εn), ζ ∈ C \ R. (40)
Integrating (40), we obtain∫
(x− z)−1(ω→n (x)− Ω(x)) dx = 2εn
∫ z R(ζ) dζ
wρ(ζ)
+ o(εn), z ∈ C \ R,
where the integral can be taken over any interval containing the support of Ω,
and this implies (38).
380 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
3.1.3. Corollary 3.1.2 allows us to drag the corrections to the limiting shape
through the Markov correspondence. Denote by B[a,b] the space of analytic test
functions φ : [a, b]→ C. The space B[a,b] is topologized as the projective limit of
the spaces of analytic functions in shrinking neighborhoods of [a, b]. Also consider
the space of continuous functionals B′[a,b], and observe that the topology on this
space coincides with the minimal topology in which the functionals
F 7→ 〈F, (· − z)−1〉, z /∈ [a, b]
are continuous. In the setting of 3.1.2, define F, FM ∈ B′[a,b] by
〈F, φ〉 =
∮
Γ
φ(z)R(z)
dz
2πi
, 〈FM, φ〉 = −
∮
Γ
Φ(z)
R(z)
wρ(z)
dz
πi
,
where Φ(z) = 1
2(
∫ z
−2−
∫ 2
z )φ(x) dx, and Γ encircles I counterclockwise within the
domain of analyticity of φ.
Proposition. Let µn, ρ be probability measures such that supp ρ ⊂ [a, b], and
let ωn = M−1 µn and Ω = M−1 ρ. If, for some εn → +0,
ε−1
n (dµn(x)− dρ(x))→ F in B′[a,b], (41)
then
ε−1
n (ωn(x) dx− Ω(x) dx)→ FM in B′[a,b]. (42)
Proof. Use Corollary 3.1.2 and the Cauchy theorem.
By the construction of B′[−2,2], the left-hand side of (42) is implicitly multi-
plied by the indicator 1[a,b](x). The indicator can be dropped if∫
x dρn(x) =
∫
x dρ(x) + o(εn).
Otherwise, it is necessary, as observed in [19] and as one can see from Figure 2.
Second, the implication in the proposition is in fact an equivalence: (42) implies
(41), as one can see by tracing the arguments.
3.1.4. Consider the following example. Assume that ρ = ρs.c. is the semi-
circle measure, the Stieltjes transform of which is given by
wρ(z) =
∫ 2
−2
1
2π
√
4− x2
dx
x− z
=
−z +
√
z2 − 4
2
.
Recall the definition of Chebyshev polynomials
Tn(cos θ) = cos(nθ), Un(cos θ) =
sin((n+ 1)θ)
sin θ
, (43)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 381
Sasha Sodin
or explicitly,
T0(x/2) = 1, T1(x/2) =
x
2
, T2(x/2) =
x2
2
− 1, · · ·
U0(x/2) = 1, U1(x/2) = x, U2(x/2) = x2 − 1, · · ·
They satisfy the orthogonality relations∫
Tk(x/2)Tl(x/2) dρarcsin(x) =
1 + δk0
2
δkl,
∫
Uk(x/2)Ul(x/2) dρs.c.(x) = δkl.
Assume that
1
εn
d(µn(x)− ρs.c.(x))→
∑
k≥1
2ckTk(x/2) dx
π
√
4− x2
in B′[−2,2], (44)
i.e., for any z ∈ C \ [−2, 2],∫
d(µn(x)− ρs.c.(x))
x− z
= εn
∑
k≥1
ck
∫ 2
−2
2Tk(x/2) dx
π
√
4− x2(x− z)
+ o(εn). (45)
Observing that
wρarcsin,k(z) =
∫ 2
−2
Tk(x/2) dx
π
√
4− x2(x− z)
=
−1√
z2 − 4
Tk(z/2) +
1
2
Uk−1(z/2)
= − 1√
z2 − 4
{
z −
√
z2 − 4
2
}k
,
we deduce that for k ≥ 2,
−2
wρarcsin,k(z)
wρs.c.(z)
= 2wρarcsin,k−1(z) =
2
k − 1
∫ 2
−2
d
dx
[
Uk−2(x/2)
√
4−x2
2π
]
x− z
dx,
whereas
−2
wρarcsin,1(z)
wρs.c.(z)
= 2wρarcsin(z) = − 2
π
∫ 2
−2
d
dx arcsin(x/2)
x− z
dx,
and finally
ε−1
n (ωn(x)− ΩLSVK(x))dx −→−4c1
π
arcsin(x/2) +
∑
k≥2
4ck
k − 1
Uk−2(x/2)
√
4− x2
2π
dx in B′[−2,2]. (46)
We proved:
382 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
Proposition. Let {ck} be a sequence with lim sup |ck|1/k = 1. If µn = Mωn
satisfy (44), then (46) holds.
As we noted above, the arguments go in both directions, and (44) is actually
equivalent to (46).
3.2. Fluctuations about the limit shape.
3.2.1. Although Proposition 3.1.4 was proved in the deterministic setting,
standard arguments allow us to apply it in the stochastic setting, i.e., for ran-
dom measures µn, after proper modifications. For example, if the assumptions
holds almost surely, then so does the conclusion; if the assumption holds in dis-
tribution, then so does the conclusion. The former version follows directly from
the deterministic statement, while the latter version follows from the former one
using Skorokhod’s representation theorem. In the sequel we will work with the
convergence in distribution.
3.2.2. Let gk be independent, identically distributed standard Gaussian
random variables, and let
∆tr(x) =
∑
k≥1
√
k
2
gk
2Tk(x/2)
π
√
4− x2
,
∆M
tr (x) = − 2
π
g1 arcsin(x/2)−
∑
k≥0
2
√
k + 2
k + 1
gk+2
Uk(x/2)
√
4− x2
2π
,
where the series are understood in B′[−2,2]. Following [25], we note the similarity
between ∆tr(x) and
d
dx
∆M
tr (x) ∼ − g1
π
√
4− x2
−
∑
k≥1
√
kgk+1
2Tk(x/2)
π
√
4− x2
.
Theorem (Johansson [27]). For the Gaussian Unitary Ensemble,
nd(R√n[ρGUE
n ](x)− ρs.c.(x))→ ∆tr(x) dx
as random functionals on B[−2,2].
In the original work [27], the convergence was established in slightly weaker
topology; now the result is available in the topology corresponding to B[−2,2] and
even a much stronger one (cf. 3.3.2 below). Appealing to (a stochastic version
of) Proposition 3.1.4, we obtain:
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 383
Sasha Sodin
Corollary. For the GUE,
n (R√n[$GUE
n ](x)− ΩLSVK(x))dx→ ∆M
tr (x) dx, (47)
ndR√n[ρGUE
n − ρGUE,∗
n−1 ](x)→ 1
2
d2
dx2
∆M
tr (x) dx, (48)
where ρ∗n−1 is the normalized counting measure of the zeros of P ′n.
3.2.3. Now let
∆1(x) =
∑
k≥1
gkUk(x/2)
√
4− x2
2π
=
∑
k≥1
gk
2
2Tk(x)− 2Tk+2(x)
π
√
4− x2
,
∆M
1 (x) = −2g1
π
arcsin
x
2
−
∑
k≥1
2(gk − gk+2)
k + 1
Uk(x/2)
√
4− x2
2π
.
We mention that ∆1 can also be described as follows:
∆1(x) dx =
√
ρ′s.c.(x)dB(x)−
{∫ 2
−2
√
ρ′s.c.(y)dB(y)
}
ρ′s.c.(x) dx, (49)
where ρ′s.c.(x) = 1
2π
√
(4− x2)+, and B(x) is the Brownian motion. While
∆1(x)dx is a generalized Gaussian process, its integral
∆
∫
1 (x) =
∫ min(x,2)
−2
∆1(y)dy
has a continuous modification, and so does ∆M
1 (x).
The fluctuations of the measure µn are described by the following result of
Lytova–Pastur [41] (where a stronger topology defined by test functions in C1 is
used, see further 3.3.4).
Theorem (Lytova–Pastur [41] /weak form/). For the GUE,
√
nd(R√n[µGUE
n ](x)− ρs.c.(x))
distr−→ ∆1(x) dx (50)
as random functionals on B[−2,2].
This theorem and Proposition 3.1.4 imply:
Corollary (Erdős and Schröder [19] /special case, weak form/).
√
n(R√n[ωGUE
n ](x)− ΩLSVK(x)) dx
distr−→ ∆M
1 (x) dx, (51)
√
ndR√n[ρGUE
n − ρGUE
n−1 ](x)
distr−→ 1
2
d2
dx2
∆M
1 (x) dx. (52)
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3.3. On generalizations.
3.3.1. Johansson’s Theorem 3.2.2 has been extended beyond the Gaussian
Unitary Ensemble, see [52, Section 18.4], [6, Section 2.1.7] and references therein.
We quote (53) below from the work of Bai and Yao [8]; related results were proved
earlier by Khorunzhy, Khoruzhenko, and Pastur [35].
Let H be a Wigner matrix as in 2.3.4; let us assume that the matrix entries
have finite fourth moments, and that EH(1, 2)2 = 0 (the second condition can be
omitted at the expense of making the formulæ more cumbersome). Then
nd(R√n[ρGUE
n ](x)− ρs.c.(x))
distr−→
∑
k≥1
akgk + bk
2
2Tk(x/2)
π
√
4− x2
(53)
as random functionals on B′[−2,2], where
a1 =
√
EH(1, 1)2, a2 =
√
2E(|H(1, 2)|2 − 1)2, a3 =
√
3, a4 =
√
4, . . .
b2 = EH(1, 1)2 − 1, b4 = E(|H(1, 2)|2 − 1)2 − 1, b1 = b3 = b5 = b6 = · · · = 0.
Using (53) in place of Theorem 3.2.2, Corollary 3.2.2 is extended as follows:
n (R√n[$GUE
n ](x)− ΩLSVK(x)) dx
distr−→ − 2
π
a1g1 arcsin(x/2)dx
−
∑
k≥0
2
√
k + 2
k + 1
(ak+2gk+2 + bk+2)
Uk(x/2)
√
4− x2
2π
dx. (54)
3.3.2. It may also be possible to strengthen the topology in Corollary 3.2.2,
using as an input a topologically stronger central limit theorem for linear statistics
such as the one proved by Shcherbina [55] (see further Sosoe and Wong [58]). To
follow this strategy, one needs a version of Proposition 3.1.4 for non-analytic test
functions; this can be obtained, for example, by the method of pseudoanalytic
extension [17,18,23].
3.3.3. As for the fluctuations of the spectral measure, Theorem 3.2.3 of
Lytova–Pastur was extended beyond the Gaussian ensembles by Lytova–Pastur
[42] and Pizzo–Renfrew–Soshnikov [53]. We quote one of the results in [53]: for
Wigner matrices as in 2.3.4 having finite fourth moments,
√
nd(R√n[µGUE
n ](x)− ρs.c.(x))
distr−→
∑
k≥1
hkUk(x/2) dρs.c.(x), (55)
where hk are independent random variables: h1 = −H(n, n), h2 is centered
Gaussian with variance depending on the moments of H(1, 2), and (hk)k≥3 are
centered Gaussian with variance depending on the symmetry class of H (cf. (58)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 385
Sasha Sodin
below for explicit formulæ). The topology is stronger than that of B′[−2,2], see
3.3.4 below.
Applying Proposition 3.1.4, we deduce that for H as above,
√
n(R√n[ωGUE
n ](x)− ΩLSVK(x)) dx
distr−→ −2h1
π
arcsin
x
2
dx−
∑
k≥1
2(hk − hk+1)
k + 1
Uk(x/2) dρs.c.(x) (56)
in the topology of random functionals on B[−2,2].
3.3.4. The result of Erdős and Schröder [19, Theorem 3] is topologically
much stronger than our statements in (51) and (56), and is applied, for exam-
ple, to test functions which are indicators of intervals. This raises the question
whether the topology can be strengthened in the central limit theorems (50), (55)
for the spectral measure.
In the case of the GUE, one can use the convergence to the semicircle on
scale n−1/2 and a functional central limit theorem for sums of independent ex-
ponentially distributed random variables to obtain the following version of The-
orem 3.2.3: √
n
(
µGUE
n (x
√
n)− ρs.c.(x)
) distr−→ ∆
∫
1 (x) (57)
as random continuous functions (here we identify measures with their cumulative
distribution functions).
In the case of general Wigner matrices with finite fourth moments, the results
of [53] imply that (55) holds in the topology of functionals on test functions φ ∈
C7[−2− δ, 2 + δ]. It is also proved in [53] that for Wigner matrices the entries of
which satisfy the Poincaré inequality the limit theorem (55) holds for Lipschitz
test functions. This topology is still (most probably) insufficient to recover the
results of [19] in their full strength.
3.3.5. In a remark in Section 1.2 (following (17)), we suggested that the
results of [19] could be pulled through the Markov correspondence to obtain a
version of (55) in stronger topology. However, Erdős and Schröder found [20] a
direct approach to the latter problem.
For the special case of Wigner matrices as in 2.3.4, their result asserts the
following (the more general setting of [20] applies to matrices the entries are
not necessarily identically distributed): if H(1, 1) and H(1, 2) have moments of
arbitrary order, then
√
n(dR√n µ
H
n (x)− dρs.c.(x))
distr−→ dρs.c.(x)
[
−H(n, n)U1(x/2)
386 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
+
√
E|H(1, 2)|4 − 1 g2U2(x/2) +
∑
k≥3
√
1 + (EH(1, 2)2)kgkUk(x/2)
]
, (58)
where gk are independent standard Gaussian, and the topology is defined by test
functions φ : R → R of bounded variation in [−3, 3]. For the convenience of the
reader, we have not modified the phrasing of the remark in Section 1.2.
4. Random Partitions and Random Matrices
4.1. Plancherel growth.
4.1.1. A partition λn of n ≥ 0 (denoted λn ` n) is a non-increasing se-
quence
λn = (λn1 ≥ λn2 ≥ λn3 ≥ · · · )
of non-negative integers adding up to n. We identify a partition λn with the
Young diagram with rows of lengths λn1 , λ
n
2 , · · · , i.e., the union of unit squares
(boxes) with a corner at {
(j, k) | 1 ≤ j, 1 ≤ k ≤ λnj
}
.
For example, the partition 19 = 7 + 4 + 4 + 3 + 1 of 19 corresponds to the Young
diagram in Figure 1a. The content of a box � = (j, k) is by definition ct(�) =
k− j. For two diagrams λn ` n and λn+1 ` n+ 1, we write λn ↗ λn+1 if λn+1 is
obtained by adding a single box λn+1 \λn to λn. A box � ∈ λn is called an inner
corner of λn if λn \� is a partition (Young diagram). A box � /∈ λn is called an
outer corner if λn ∪� is a partition. The continual diagram ω[λn] associated to
a partition λn is defined by
ω′(x) = −1 + 2 # {outer corners � with ct(�) < x}
− 2 # {inner corners � with ct(�) < x}
ω(x) = |x| for sufficiently large x.
See Figure 1c.
4.1.2. A standard Young tableau is a chain
T = (λ0 = ∅↗ λ1 ↗ λ2 ↗ · · · ↗ λn). (59)
The dimension dim λ̃n of a Young diagram λ̃n is the number of chains (59) with
λn = λ̃n. In the representation theory of the symmetric group, the irreducible
representations are in one-to-one correspondence with the partitions of n, and
dimλ equals the dimension of the irreducible representation corresponding to λ.
One has (see 4.1.4 below):
dimλn =
∑
λn−1↗λn
dimλn−1,
∑
λn`n
dim2 λn = n!. (60)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 387
Sasha Sodin
4.1.3. Consider the space Tab∞ of infinite sequences
λ0 = ∅↗ λ1 ↗ λ2 ↗ · · · ↗ λn ↗ · · · (61)
equipped with product topology. Define a probability distribution on Tab∞ by
P
{
λ1 = λ̃1 , · · · , λn = λ̃n
}
=
dim λ̃n
n!
for any n and any chain λ̃0 ↗ · · · ↗ λ̃n. The corresponding process is called
Plancherel growth, and the distribution
P
{
λn = λ̃n
}
=
dim2 λ̃n
n!
of λn — the Plancherel measure. The transition measure of a diagram λ̃n ` n is
the probability measure
µ[λ̃n] =
∑
outer corners �
P
{
λn+1 = λ̃n ∪�, λn = λ̃n
}
P
{
λn = λ̃n
} δct(�). (62)
This measure was introduced by Kerov [30], who proved that Mω[λn] = µ[λn].
Theorem (Logan–Shepp [40], Vershik–Kerov [60,61]; Kerov [31]).
R√n {ω[λn]} → ΩLSVK and R√n µ[λn]→ ρs.c.
(in distribution and almost surely).
The first part was proved by Logan–Shepp [40] and Vershik–Kerov [60, 61],
while the second part was deduced by Kerov [31] using the Markov correspon-
dence. Vice versa, one can first prove the second part (see Biane [10] and 4.2.4,
4.2.5 below), and then deduce the first part.
4.1.4. The construction above arises in the representation theory of the
symmetric group. Let us explain the minimum that we need below, and refer
to [34] for more details. The left regular representation of the symmetric group
Sn is the space C[Sn] of linear combinations
∑
π∈Sn aππ, equipped with the action
of Sn by left multiplication. It can be decomposed into a sum of irreducible
representations so that the multiplicity of the representation corresponding to λn
is equal to dimλn. This implies the second equality in (60).
The decomposition of an irreducible representation corresponding to λn ` n
into irreducible representations of Sn−1 contains exactly the representations cor-
responding to λn−1 ↗ λn, and the multiplicity of each of these is exactly one.
388 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
This implies the first equality in (60). Furthermore, it follows that an irreducible
representation corresponding to λn ` n is decomposed into one-dimensional sub-
spaces corresponding to chains T of the form (59).
The arguments below also rely on the properties of the Jucys–Murphy ele-
ments Xm ∈ C[Sn], which are defined as sums of transpositions
Xm = (1m) + (2m) + · · ·+ (m− 1m).
Every T is invariant under all Xm, and Xm |T= ct(λm \ λm−1) (see [47], where
the Jucys–Murphy elements are used to reconstruct the representation theory of
the symmetric group, and [10,46]). This implies
1
n!
trP (Xn) = E
∫
P dµ[λn−1]
for any polynomial P . The use of Jucys–Murphy elements to compute moments
of the transition measure goes back to the work of Biane [10].
4.1.5. Define two Gaussian processes
∆part(x) =
∑
k≥3
√
k − 1
2
gk
2Tk(x/2)
π
√
4− x2
, (63)
(−)∆M
part(x) =
∑
k≥1
2gk+2√
k + 1
Uk(x/2)
√
4− x2
2π
. (64)
Theorem (Kerov [32]; Ivanov–Olshanski [25]).
√
n
{
R√n ω[λn]− ΩLSVK
}
dx→ ∆M
part(x)dx
and √
n
{
dR√n µ[λn](x)− dρs.c.(x)
}
→ ∆part(x)dx
in distribution, as random functionals on B[−2,2].
The first part was proved by Kerov [32] in 1993. A simplified proof, based on
Kerov’s notes, was published by Ivanov and Olshanski [25]. The second part was
proved in [25], independently of Theorem 4.1.5 (although by a similar method).
In view of Proposition 3.2.1, the second part implies the first part (and vice
versa). In Section 4.2, we sketch a proof of the second part using a combinatorial
approach which was used by Biane [10] and by Okounkov [46], and recently
developed in [26] As pointed out in [25], ∆part is similar to ∆tr from 3.2.2. It
is even more similar to the process from Proposition 4.1.6. The argument will
highlight this similarity, and also explain the appearance of the factor
√
k − 1 in
(63) in place of
√
k in (14).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 389
Sasha Sodin
4.1.6. To emphasize the similarity between random matrices and random
partitions, we consider the following special Wigner matrix: H = Hunif =
(H(i, j)) with
H(i, i) = 0 , H(i, j) ∼ Unif(S1) (i < j) .
This ensemble is particularly convenient due to the identities (66), see [56] for
a survey of applications of relations of this kind and historical remarks. (For
other ensembles of Wigner matrices (66) is no longer an identity, however, the
difference between the left-hand side and the right-hand side is a small quantity,
cf. [21].) For the ensemble Hunif, the central limit theorem is stated as follows.
Proposition. For Hunif,
nd(R√n[ρunif
n ](x)− ρs.c.(x))→
∑
k≥3
√
k
2
gk
2Tk(x/2) dx
π
√
4− x2
,
n (R√n[$unif
n ](x)− ΩLSVK(x))dx→
∑
k≥1
2
√
k + 2
k + 1
gk+2
Uk(x/2)
√
4− x2 dx
2π
.
In the next section, we prove this proposition in parallel with Theorem 4.1.5.
4.2. Asymptotics of moments.
4.2.1. Consider the following sequence of polynomials:
Pl,m(x) = (m− 1)
l
2Ul
(
x
2
√
m− 1
)
− (m− 1)
l−2
2 Ul−2
(
x
2
√
m− 1
)
(65)
with the convention U−2 ≡ U−1 ≡ 0. For p = (u0, u1, . . . , ul), set
π̂(p;H) = H(u0, u1)H(u1, u2) . . . H(ul−1, ul).
Also let
P̂l,n(u, v) =
{
(u0, u1, . . . , ul−1, ul) ∈ {1, . . . , n}l |
u0 = u, ul = v, ur 6= ur−1, ur−2
}
.
Graphically, we may represent a tuple in P̂l,n(u, v) as a path of length l from u
to v which does not backtrack.
Lemma (e.g. [56]). For any Hermitian H with |H(u, v)| = 1 − δuv and any
u, v,
Pl,n−1(Hn)(u, v) =
∑
p∈P̂l,n(u,v)
π̂(p;H), (66)
390 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
and consequently
E
∏
r
Plr,n−1(H)(ur, vr) =
{
p ∈
∏
r
P̂lr,n |
∏
r
π̂(pr;H) ≡ 1
}
.
Graphically, the product is identically one if every edge is traversed forward
the same number of times as backward.
4.2.2. Now we state (following [26]) the counterpart of (65) for random
partitions. Let p = (j1, j2, . . . , jl) be a sequence of numbers. Denote:
πm(p) = (j1m)(j2m) · · · (jlm) .
Also let
Pl,m =
{
p ∈ {1, · · · ,m− 1}l | ∀1 ≤ r ≤ l − 1 jr 6= jr+1
}
. (67)
The following lemma can be checked by induction:
Lemma ([26, Lemma 4.1]).
Pl,m−1(Xm) =
∑
p∈Pl,m
πm(p),
and consequently
tr
∏
r
Plr,mr−1(Xmr) = #
{
p ∈
∏
r
Plr,mr |
∏
r
πmr(pr) = 1(= idSn)
}
. (68)
4.2.3. Let us prove that Wigner’s law holds in the mean:
ER√n ρ
unif
n = ER√n µ
unif
n → ρs.c. . (69)
We omit many details which can be found, for example, in [56, 2.4.1].
Proof of Wigner’s law in the mean. The polynomials Uk(x/2) are orthogonal
with respect to ρs.c., therefore to prove (69) it suffices to show that
lim
n→∞
E
∫
Uk(x/2)dR√n ρ
unif
n (x)
= lim
n→∞
E
∫
Uk(x/2)dR√n µ
unif
n (x) = 0, k = 1, 2, · · · ,
which, by (65), is equivalent to
lim
n→∞
n−
k
2 EPk,n−1(H)(n, n) = 0, k ≥ 1, (70)
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 391
Sasha Sodin
and we do this using Wigner’s power-counting argument, as follows.
The quantity EPk,n(H)(n, n) counts the number of paths in Pk,n−1(n, n)
which satisfy the parity condition
∀a 6= b # {j, uj = a, uj+1 = b} = # {j, uj = b, uj+1 = a} ,
where we set u0 = uk = n. First, we divide Pk,n−1(n, n) into isomorphism classes:
p is isomorphic to p′ if p = σ ◦p′ for some permutation σ ∈ Sn. The number of
isomorphism classes is bounded as n→∞, for every k. Second, to every class we
associate the characteristic χ = V −E+1, where V is the number of vertices and
E is the number of edges, and observe that E ≤ k/2 and hence the contribution
of a class is bounded by Ckn
k
2
+χ−1. Finally, a path can not be tree-like (such
as (u0, · · · , u10) = (j1, j2, j3, j4, j3, j5, j6, j5, j3, j2, j1)), since trees have leaves at
which the path backtracks (in the example in the previous parentheses, u2 = u4 =
j3, et cet.), hence χ ≤ 1. A somewhat more careful argument shows that χ ≤ 0
with equality for paths isomorphic to the one on [26, Figure 4, right], [56, Figure
3.1, middle], but this is not needed at the moment. Finally,
0 ≤ n−
k
2 EPk,n−1(H)(n, n) ≤ C ′kn−1 .
This proves (69) and (70).
4.2.4. In the same way, 4.2.2 can be used to prove that Theorem 4.1.3
holds in the mean:
R√n Eµ[λn]→ ρc.
Proof of Kerov’s law in the mean. It suffices to show that
lim
n→∞
1
nk/2
1
n!
trPk,n−1(Xn) = 0, k = 1, 2, 3, · · · (71)
The argument, essentially due Biane (see [10]), is similar to 4.2.3. We write
trPk,n−1(Xn) = # {p ∈ Pk,n | πn(p) = 1}
and divide the solutions to the equation πn(p) = 1 into isomorphism classes (by
Sn−1-conjugation). Then we assign to every class the characteristic
χ = #{distinct indices in p} − k
2
+ 2.
See [26, 4.2.1] for a graphical interpretation. The irreducibility condition (67)
rules out tree-like solutions such as
(j1 n) (j2 n) (j3 n) (j3 n) (j2 n) (j4 n) (j4 n) (j1 n) = 1
with χ = 2, therefore χ ≤ 1 (in fact, χ ≤ 0, but this is not required at the
moment), and this implies (71).
392 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
4.2.5. To prove that Wigner’s law and Kerov’s law hold in distribution, we
need to show that also the variance of∫
Uk(x/2) dR√nρ
unif
n and
∫
Uk(x/2) dR√nµ[λn]
tends to zero. This is accomplished via a combinatorial argument which is similar
to that in 4.2.3 and 4.2.4: one shows that, for any isomorphism class of pairs
contributing to the variance, the number of distinct vertices / indices is at most
k/2. We omit the details.
4.2.6. In order to consider fluctuations, we introduce another family of
polynomials
Qk,m(x) = 2(m− 1)
k
2Tk
(
x
2
√
m− 1
)
− 2(m− 1)
k−2
2 Tk−2
(
x
2
√
m− 1
)
. (72)
Then we have, for k ≥ 3,
trQk,n−1(H) =
∑
p∈P̂◦k,n
π̂(p, H)−
∑
p∈P̂◦k−2,n
π̂(p, H), (73)
where
P̂◦k,n =
{
p ∈ P̂k,n | uk−1 6= u1
}
is the collection of cyclically non-backtracking paths. The relation (72) is derived
from (65) using the identity
Uk − Uk−2 = 2Tk.
For k = 1, 2, we have:
tr 2
√
n− 2T1
(
H
2
√
n− 2
)
= 0, tr 2(n− 2)T2
(
H
2
√
n− 2
)
= −n(n− 3).
Consequently, we obtain that
tr 2(n− 2)
k
2Tk
(
H
2
√
n− 2
)
=
∑
p∈P̂◦k,n
π̂(p, H)− n(n− 3)1n is even. (74)
The relation (74) was first derived by Oren, Godel and Smilansky [48], using
a trace formula (see further [49, 50] and 4.3.3 below). Related combinatorial
interpretations of the traces of Chebyshev polynomials of the first kind appeared
in the works of Anderson and Zeitouni [7] (who used generating functions), and
of Kusalik, Mingo and Speicher [38] and Schenker and Schulz-Baldes [54].
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 393
Sasha Sodin
Proof of Proposition 4.1.6. The proposition is derived from (74) using an
approximation argument and the following two claims.
Claim A. Let Ĉk,n be the sub-collection of paths in P̂◦k,n in which all the
vertices are distinct. Then
(n− 2)−
k
2
∑
p∈P̂◦k,n\ Ĉk,n
π̂(p, H)→ 0 (75)
in distribution.
Claim B. Let g3, g4, · · · be independent standard Gaussian variables. Then(n− 2)−
k
2
∑
p∈ Ĉk,n
π̂(p, H)
k≥3
→
{√
kgk
}
k≥3
(76)
in distribution.
Claim A is proved by power-counting as in the 4.2.3 above: the elements of
Ĉk,n have V = E, whereas for all the other paths in P̂◦k,n V < E. This implies
that E[LHS of (75)]2 → 0.
To prove Claim B, denote the expression inside the braces on the left-hand
side of (76) by Sk,n; then (and this is the main combinatorial step in the proof)
lim
n→∞
E
r∏
j=1
Skj ,n
=
∑
pairings of {1, · · · , r}
∏
pair (j, j′)
#
{
alignments of a kj-cycle with a kj′-cycle
}
=
∑
pairings of {1, · · · , r}
∏
pair (j, j′)
kj δkj ,kj′ ,
where the first equality is a consequence of power-counting. Claim B follows by
the Wick rule.
4.2.7.
Proof of Proposition 4.1.5. Let us denote by movem any linear combination
of permutations for which m is not a fixed point. Then (for k ≥ 3)
Qk,m−1(Xm) =
∑
p∈P◦k,m
πm(p) + movem,
394 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
where
P◦k,m =
{
p ∈ Pk,m
∣∣uk = u1 , uk−1 6= u2
}
is the collection of cyclically non-backtracking paths. In particular,
tr
r∏
j=1
Qkj ,mj−1(Xmj ) = tr
r∏
j=1
∑
p∈P◦kj,mj
πmj (p),
if m1 < · · · < mr. Consider the sub-sub-collection
Ck,m = {(j1, · · · , jk < m | jk = j1 and j1, · · · , jk−1 are pairwise distinct} .
(Note that the cycles have k−1 vertices, as opposed to the k-long cycles of 4.2.6.)
Then Claim 4.2.6.A takes the form
n−
k
2
Qk,n−1(Xn)|T −
∑
p∈Ck,n
πn(p)|T
−→ 0,
where T is sampled from the Plancherel growth process, whereas in place of Claim
4.2.6.B one has ∑
p∈Ck,n
πn(p)|T
k≥3
−→
{√
k − 1 gk
}
k≥3
.
Both claims are proved by evaluating moments, where we take mj = n − j + 1.
Then Proposition 4.1.5 follows from the claims by an approximation argument.
4.3. Some comments.
4.3.1. The counterpart of Theorem 3.2.3 (i.e., the special case of (55)) for
the ensemble Hunif is stated as follows:
√
nd(R√n[µunif
n ](x)− ρs.c.(x))
distr−→
∑
k≥3
gkUk(x/2)
√
4− x2
2π
dx , (77)
while (56) takes the form
√
n(R√n[ωunif
n ](x)− ΩLSVK(x)) dx→∑
k≥3
2gk
{
Uk−2(x/2)
k − 1
− Uk(x/2)
k + 1
} √
4− x2
2π
dx.
The coefficient “1” in front of gk in (77) is combinatorially interpreted as the
number of ways to align two cycles of length k with a marked vertex.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 395
Sasha Sodin
4.3.2. The parallelism between random matrices and random Young dia-
grams is somewhat more transparent if described via the matrix of transpositions
Γn =
0 (12) (13) · · · (1n)
(12) 0 (23) · · · (2n)
. . .
(1n) (2n) (3n) · · · 0
introduced (in a different gauge) by Biane [10]. The spectral properties of the
restriction Γn|λn to λn ⊗ Cn, where λn is an irreducible representation, are sim-
ilar to those of the Jucys–Murphy elements; if λn is randomly sampled from
the Plancherel measure, the corresponding matrix Γn|λn can be considered as a
counterpart to the random matrix Hn.
In particular, the use of Γn in place of Xn makes the argument in 4.2.7 even
more similar to that in 4.2.6, and also introduces a conceptual simplification by
allowing to work with central elements only.
4.3.3. The combinatorial constructions of Section 4 can be recast in terms
of Ihara-type zeta-functions [9, 24]. For a Hermitian matrix H such that
|H(u, v)| = 1 − δuv (or, more suggestively, H(u, v)H(v, u) = 1 − δuv), consider
the zeta functions
ζHn(u) =
∏
l≥0
∏
p∈P̂◦l,n
(1− π̂(p;H)ul)−1.
A variant of the Bass determinantal formula [9] asserts that
ζHn(u)−1 = (1− u2)(
n−1
2 )−1 det(1− uHn + (n− 1)u2
1). (78)
Similarly, consider the C[Sn]-valued function
ζn(u) =
∏
l≥0
∏
p∈P◦l,n
(1− πn(p)ul)−1,
for which one has
ζn(u)−1 = (1− u2)(
n−1
2 )−1 detn×n(1− uΓn + (n− 1)u2
1).
Both sides of this identity are central, and hence act as scalars on each irreducible
representation λn of Sn; thus we obtain:
[ζn(u)|λn ]−1 = (1− u2)(
n−1
2 )−1 detn×n(1− uΓn|λn + (n− 1)u2
1). (79)
396 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
Fluctuations of Interlacing Sequences
The logarithms of the relations (78) and (79) are trace formulæ relating the
spectra of Hn and Γn with the quantities π̂n and πn, respectively.
The trace formulæ thus obtained do not explicitly separate between the semi-
circle and the corrections to it. Improved trace formulæ, in which such a separa-
tion is explicit, were obtained in the setting of d-regular graphs by Oren, Godel
and Smilansky, see [48, 49] and further [50]. Their approach can be applied to
the problems discussed here.
4.3.4. Chebyshev polynomials appear naturally in the combinatorial ap-
proach described in Section 4.2, and also in the trace formulæ mentioned in
4.3.3. Another approach in which their rôle is apparent was developed by Joyner
and Smilansky [28, 29]; it relies on the study of the Fokker–Planck equation de-
scribing the evolution of the ensemble under random walk. We refer in particular
to [29], where Gaussian fluctuations are analyzed.
Acknowledgement. It is a pleasure to express my gratitude to Yan Fy-
odorov, for emphasizing the importance of critical points; to Vadim Gorin, for
many helpful explanations, for sending me a copy of Kerov’s habilitation thesis,
and for sharing the preprint [22] before publication; to Grigori Olshanski, for
discussions of the central limit theorems for Young diagrams and of many other
interesting topics, and for patiently answering some questions which should be
categorized as philosophical only if philosophy is understood in the colloquial
meaning listed third in the dictionary of Ushakov; and to Peter Yuditskii, for
correspondence on the Markov moment problem and its continual analogues. It
is equally pleasant to acknowledge my recognition to Alexey Bufetov, for show-
ing me the matrix Γn used in 4.3.2 and 4.3.3, to László Erdős and Dominik
Schröder, for pointing at several mistakes and irresponsible assertions in a pre-
liminary version of this text; and to Christopher Joyner and Uzy Smilansky, for
spotting lapses of various kinds and suggesting numerous improvements, and for
explaining in detail some results from the work [29].
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