NORMAL CONVERGENCE FOR RANDOM PARTITIONS WITH
MULTIPLICATIVE MEASURES
ZHONGGEN SU∗
Abstract. Let Pn be the space of partitions of integer n ≥ 0, P the space of
all partitions, and define a class of multiplicative measures induced by Fβ (z) =
Q
k kβ with β > −1. Based on limit shapes and other asymptotic
k (1 − z )
properties studied by Vershik, we establish normal convergence for the size
and parts of random partitions.
1. Introduction
The problems considered in this paper concern the fluctuations around its limit
shape of random partitions of a large integer with respect to multiplicative statistics.
Such a study is stimulated by a wide range of applications of random partitions to
combinatorics, statistical mechanics, stochastic processes, analytic number theory,
etc. The interested reader is referred to Okounkov [18] and Vershik [24] for excellent
introduction at this aspect.
Let us start with the precise description of multiplicative statistics introduced
first by Vershik [23]. Let Pn be the set of all partitions of integer n ≥ 0 and
P = ∪n Pn the set of all partitions. For λ = (λ1 , λ2 , · · · , λl ) ∈ P, λ1 ≥ λ2 ≥ · · · ≥
λl > 0, we write rk (λ) = #{i : λi = k} for the number of summands equal to k in
the partition λ, namely rk (λ) is the k-th occupation number. Obviously,
P {rk (λ)}
fully determinePthe partition λ; in particular, the size |λ| is equal to k krk and
the length l = k rk . Now we introduce a class of measures on Pn and P that are
said to be multiplicative. Consider a sequence of functions fk (z), k ≥ 1, analytic in
the open disk D = {z ∈ C : |z| < %}, % = 1 or % = ∞, such that fk (0) = 1. And
assume that
(i) the Taylor series
fk (z) =
∞
X
sk (j)z j
j=0
have all coefficients sk (j) ≥ 0 and
(ii) the infinite product
F(z) =
∞
Y
fk (z k )
k=1
Date: April 5, 2011.
2010 Mathematics Subject Classification. 60F05, 60F17.
Key words and phrases. Central limit theorems, Multiplicative measures, Random partitions .
* Supported partly by NSFC grant 11071213, ZJNSF grant R6090034 and Doctoral Fund of
Ministry of Education of China grant 20100101110001.
1
ZHONGGEN SU∗
2
converges in D. We can define a family of probability measures Pq , q ∈ (0, %), on P
in the following way: put
sk (j)q kj
, j ≥ 0, k ≥ 1
fk (q k )
and assume that different occupation numbers are independent. Thus
Q∞
sk (rk ) |λ|
Pq (λ) = k=1
q , λ ∈ P.
F(q)
Now define the measure Pn on Pn by
sk (j)
Pn (λ ∈ Pn : rk (λ) = j) =
Qn
and
Q∞
sk (rk )
(1.1)
, λ ∈ Pn ,
Pn (λ) = k=1
Qn
where
∞
XY
Qn =
sk (rk ).
Pq (λ ∈ P : rk (λ) = j) =
λ`n k=1
We remark a basic realtion
F(q) =
∞
X
Qn q n .
n=0
The following lemma due to Vershik [23] describes a key feature of the above measures.
Lemma 1.1. For any q ∈ (0, %) and n ≥ 0, we have
(1)
Pq |Pn = Pn
i.e., Pn is the conditional probability measure induced on Pn by Pq ;
(2)
∞
1 X
Pq =
Qn q n Pn
F(q) n=0
i.e., Pq is a convex combination of measures Pn .
We call the family (Pn , Pn ) a small canonical ensemble of partitions and the
(P, Pq ) a grand canonical ensemble of partitions in view of similarities to statistical
The generating function F(z), along with its decomposition F(z) =
Q∞physics.
k
f
(z
),
completely determines such a family. The above multiplicative meak=1 k
sures contain many important examples as discussed in Vershik [23], Vershik and
Yakubovich [26].
For clarity and simplicity, we shall in the sequel restrict our attention to the
β
special case in which the F(z) is generated by fk (z) = 1/(1 − z)k , β > −1. To emphasize the dependence of the ensembles upon the parameter β, we write Pβ,q , Pn,β
for the probabilities, and Eβ,q , En,β for the mathematical expectations. Also, set
Fβ (z) =
∞
Y
1
.
(1 − z k )kβ
k=1
Normal convergence for random partitions
3
In such a special case, the convergence radius of fk and F is % = 1. Vershik
[23], Vershik and Yakubovich [26] treat Pβ,q and Pn,β as generalized Bose-Einstein
models of ideal gas; while in combinatorics and number theory they are well known
for a long time as weighted partitions.
Remark 1. Pn,0 corresponds to the uniform measure on Pn , and Qn is Euler’s
function p(n): the number of partitions of n. In the case of β = 1, the Fβ (z) is the
generating function for the numbers p3 (n) of 3-dimensional plane partitions of n:
X
Y
1
,
p3 (n)z n =
(1 − z k )k
n≥0
k≥1
see, e.g. Andrews [1]. However, the Pn,1 is completely different from the uniform
measure on 3-dimensional plane partitions of n
To a partition λ ∈ Pn we assign a function ϕλ on [0, ∞) by the following rule:
ϕλ (t) = ϕλ (1),
0≤t<1
and
ϕλ (t) =
∞
X
rk ,
1 ≤ t < ∞,
k=[t]
where [x] denotes the integer part of x. Clearly, by definition,
ϕλ (·) is a monotone
R∞
decreasing, piecewise constant function of t, and n = 0 ϕλ (t)dt. We refer to such
a graphical description as a Young diagram of the partition λ.
Certain asymptotic properties of Pn,β and Pβ,q have already been well studied
in the literature. Vershik [23], in an attempt to capture various limiting results
concerning particular functionals in a unified framework, posed the question of
evaluating the limit shape for ϕλ (t) under Pn,β . For later use, we restate a part of
his limit shape results as follows.
)1/(β+2) . Consider the scaled
Lemma 1.2. Assume β ≥ 0, let hn = ( Γ(β+2)ζ(β+2)
n
function
t , t ≥ 0.
ϕ̃n (t) = hβ+1
ϕλ
n
hn
Then we have
ϕ̃n → Ψβ
in the sense of uniform convergence on compact sets, where Ψβ is the function
defined by
Z ∞ β −u
u e
Ψβ (t) =
du.
1 − e−u
t
More precisely, for any ε > 0 and 0 < a < b < ∞, there exists an n0 such that for
n > n0 we have
Pn,β λ ∈ Pn : sup |ϕ̃n (t) − Ψβ (t)| > ε < ε.
a≤t≤b
Remark 2. The value of hn is in essence determined so that Eβ,q |λ|
√ ∼ n, where
q = e−hn . For β = 0, the scaling constants along both axes are πn/ 6. Moreover
Ψ0 (t) can be written in a more symmetric form
e
− √π6 x
+e
− √π6 y
= 1.
ZHONGGEN SU∗
4
Note also that Ψ0 (t) is symmetric about the line x = y, and x = 0 and y = 0 are
its two asymptotic lines respectively.
For β > 0, two distinct scaling constants must be adapted. In fact, the value
on the y axis is more compressed than the indices on the x axis. Also, it is worth
noting
Z ∞ β −u
u e
du < ∞
Ψβ (0) =
1
− e−u
0
by virtue of β > 0.
Having the limit shape, essentially the law of large numbers, of random partitions, it is natural to ask the question about the asymptotic distribution of fluctuations from the limit curve. Under the probability model (P, Pβ,q ) with β > −1,
direct calculations can be made to show the limiting distribution of a maximal summand is the Gumbel distribution and analogous results hold for the first d largest
summands, see Vershik and Yakubovich [26]. More specifically, let
A(q) = (β + 1)| log(1 − q)| + β| log(1 − q)| + β log(β + 1)
and
Wi = (1 − q)λi − A(q).
Then for any m ≥ 1,
d
(W1 , W2 , · · · , Wm ) −→ (Y1 , Y2 , · · · , Ym )
as
q → 1,
where Y1 , Y2 , · · · , Ym is a Markov chain such that the density of Y1 is exp(−x−e−x )
and the density of Yi conditioned on Yi−1 = x is exp(−y − e−y + e−x ), y ≤ x.
The corresponding results for (Pn , Pn,β ) can be obtained by taking q = 1 − e−hn
where hn is as in Lemma 1.2 and noting the asymptotic equivalence between grand
and small canonical ensembles. In particular, we have
−t
t An
(1.2)
≤
= e−e ,
lim Pn,β λ ∈ Pn : λ1 −
n→∞
hn
hn
where
β+1
β+1 β+1
An =
log n + β log log n + β log
−
log Γ(β + 2)ζ(β + 2).
β+2
β+2 β+2
The Gumbel limit in (1.2) first appeared in the pioneering work of Erdös and Lehner
[7] on uniform random partitions. However, the proof was completely different
from that of Vershik and Yakubovich [26]. In fact, using the Hardy-Ramanujan
asymptotic formula for p(n), Erdös and Lehner found the limiting distribution of
the number l of summands (parts) and so obtained by a classic duality the limiting
distribution of the largest summand λ1 .
Remarkably, the classic duality plays a very important role in the asymptotic
study of uniform random partitions. Recall that the partition λ0 dual to λ is
obtained by transposition of λ. Thus if we write (λ01 , · · · , λ0l0 ) for λ0 , then clearly
λ01 = l, l0 = λ1 and there is a simple and useful connection between λ0j and the
sums of rk (λ), namely
λ0k =
∞
X
rj (λ),
k ≥ 1.
j=k
d
Since λ is uniformly random, so is λ0 . Therefore (λk , k ≥ 1) = (λ0k , k ≥ 1).
Normal convergence for random partitions
5
On the basis of such an elegant duality, Fristedt [10], Pittel [20, 21] undertook a
systematic study of the distribution of fluctuations in the bulk of summands for the
uniform random partitions that very fruitfully combined analytic and probabilistic
tools. The conditioning devices used in their argument is conceptually analogous
to the method of equivalence of great and small canonical ensembles above. One
of their fundamental results in this aspect can be read as follows: under (Pn , Pn,0 ),
we have for any 0 ≤ t < ∞
1
d
(1.3)
h1/2
λ[ ht ] −
Ψ0 (t) −→ N (0, σ02 (t)),
n
n
hn
where
3
e−t
− 2
[(1 − e−t ) log(1 − e−t ) + te−t ].
σ02 (t) =
−t
1−e
π (1 − e−t )2
Pittel [20, 21] also discussed a deeper functional central limit theorem and its applications to the character ratio in symmetric group representation and the total
number of standard Young tableaux.
Our goal in this paper is to study the distribution of fluctuations from the limit
shape under (Pn , Pn,β ) with β > 0. We shall prove the following analogue of (1.3):
for any 0 < t < ∞
(1.4)
d
h−(β+1)/2
(ϕ̃n (t) − Eβ,q ϕ̃n (t)) −→ N (0, κ2β (t)).
n
See Theorem 3.3 in Section 3 for precise statement.
In particular, each λ0k , k ≥ 1, after properly scaled, is approximately normally
distributed when β ≥ 1 . It is at this point that the fluctuations of the diagram λ
substantially differ from those in the case of β = 0.
We also remark that the Pn,β is no longer a uniform measure on Pn when β > 0;
so λ and λ0 is not necessarily identically distributed. Thus, unfortunately, we
cannot use the classic duality to obtain any distribution of fluctuations in the bulk
of summands, i.e., λ[ ht ] . Still, we used Eβ,q ϕ̃n (t) in (1.4) as the centering constant
n
instead of Ψβ (t). The reason for this is that λ0 oscillates around its mean more
wildly than around its limit curve.
The proof of (1.4) will be given following the line of Pittel [20] in Section 3. In
fact, by virtue of Lemma 1.1, the probability generating function of λ0k under Pβ,q
is
∞
X
0
0
Eβ,q ezλk =
q n Qn En,β ezλk , k ≥ 1
n=0
from which and using the Cauchy formula, we can express the probability generating
0
0
function En,β ezλk as an integral of Eβ,q ezλk along a contour around zero. Note that
Pn,β is Pβ,q conditioned on Pn irrespective of the value of q ∈ (0, 1) by virtue of
Lemma 1.1 again, so we can choose q = e−hn . It remains to calculate the complex
integral, in which the saddle point method (equivalently the local limit theorem) is
needed. A functional central limit theorem is also proved in Section 3.
Since Pβ,q is a Poissonization of measures Pn,β , then the grand ensembles must
give preponderance to partitions of larger and larger sizes as q grows to 1. Although needed further confirmation, the asymptotics of the grand ensembles as q
approaches 1 will often provide a first hint as to what occurs in the small ensembles. As a warm-up, we discuss the distribution of fluctuations for partitions under
ZHONGGEN SU∗
6
(P, Pβ,q ) in Section 2. In this setting, we are in a position to deal with the infinite
sum of independent and non-identically distributed random variables with negative
binomial distributions. Our main result is as follows. Let q = e−h and
Vh (t) = h(β+1)/2 λ0[ t ] − Eβ,q λ0[ t ] , t ≥ 0,
h
h
then under (P, Pβ,q ) with β > 1, we have as h → 0
Vh ⇒ V
in the sense of weak convergence in D[0, ∞), where V = (V (t), t ≥ 0) is a Gaussian
process with independent increments.
In Section 4 we shall study the distribution of the total number of standard
Young tableaux under (Pn , Pn,β ). To associated with a partition λ ∈ Pn , a standard tableau is obtained by labeling the n cells of the diagram representing λ by
1, 2, · · · , n, so that the labels strictly increase both in columns and rows, in the direction leading away from the diagram’s corner. Let dλ denote the total number of
standard tableaux. Remarkably, the dλ is also equal to the degree of the irreducible
representation of the symmetric group Sn of permutations on {1, 2, · · · , n}. The
Frenbenius formula and the hook formula ( see (4.1) and (4.2)) can often be used
to effectively compute the dλ . We shall use the functional central limit theorem obtained in Section 3 to prove that log(n!)−1/(β+2) dλ suitably scaled is approximately
normal distribution when β > 1.
To conclude the Introduction, we remark that the multiplicative measures considered in this paper concern only measures induced by Euler type generating functions. A number of important measures on partitions do not belong to this class.
One example is the Plancherel measure much studied and still active in the literature. The so-called Plancherel measure assigns a probability d2λ /n! to a partition
λ ∈ Pn . In 1977 Vershik and Kerov [25], and independently, Logan and Shepp [16]
found the limit shape of a Young diagram with respect to the Plancherel measure.
Around 2000 several groups of researchers, Borodin, Okounkov, and Olshanski [5],
Johannson [14], Okounkov[17], derived the Tracy-Widom distribution of fluctuations at the end of a Young diagram, while Kerov[15], Ivanov and Olshanski [13]
obtained the global Gaussian fluctuation around the limit shape. Bogachev and Su
[4] proved a central limit theorem in the bulk of partitions.
Another interesting example is the multiplicative measure given by the exponential generating P
function. Let a = (ak , k ≥ 1) be a parameter function determined
by g(x) = exp( k≥1 ak xk ). Define a probability µn on Pn by
µn (λ) =
1 Y arkk
,
Cn
rk !
λ ∈ Pn ,
k=1
where Cn is the partition function.
In terms of the form of parameter function, the measure µn substantially differ
from the Pn defined in (1.1). The reader is referred to Erlihson and Granovsky [8]
and the reference therein for the limit shape and functional central limit theorem
for the fluctuation.
Throughout the paper we denote by cβ and Cβ numerical constants possibly
depending on the parameter β. They may take different values from line to line.
Normal convergence for random partitions
7
2. Grand Canonical Ensembles
Recall that rk (λ) = #{i : λi = k} for λ = (λ1 , · · · , λl ) ∈ P. Under Pβ,q , the
rk ’s are independent random variables with negative binomial distributions. In
particular,
β
Pβ,q (rk = j) = (1 − q k )k sk,β (j)q jk ,
j ≥ 0,
where sk,β (j) is such that
∞
X
1
sk,β (j)q j .
β =
k
k
(1 − q )
j=0
In this section we shall use the standard argument for sums of independent random
variables to obtain the limiting distributions of |λ| and λ0k of a random partition as
q → 1.
We need the following basic facts on the rk ’s:
(2.1)
Eβ,q (rk ) = k β
qk
,
1 − qk
V arβ,q (rk ) = k β
qk
(1 − q k )2
and
β
Eβ,q exrk =
(2.2)
(1 − q k )k
.
(1 − ex q k )kβ
Our first result is
Theorem 2.1. Let q = e−h .
(1) Under Pβ,q with β > −1, we have as h → 0
(2.3)
d
2
h(β+3)/2 (|λ| − Eβ,q |λ|) −→ N (0, σβ+2
),
where
2
σβ+2
∞
Z
uβ+2 e−u
du.
(1 − e−u )2
=
0
(2) Under P−1,q , we have as h → 0
(2.4)
h
p
d
| log h|
(|λ| − E−1,q |λ|) −→ N (0, 1).
Proof. We shall first give the proofPof (2.3), and (2.4) can be similarly proved with
∞
a minor modification. Since |λ| = k=1 krk , then by virtue of (2.1) and (2.2), it is
easy to obtain
µh =: Eβ,q |λ| =
∞
X
k β+1
k=1
σh2 =: V arβ,q (|λ|) =
∞
X
k=1
e−hk
,
1 − e−hk
k β+2
e−hk
(1 − e−hk )2
and
(2.5)
Eβ,q (e
x|λ|
∞
Y
β
(1 − e−hk )k
)=
.
(1 − exk e−hk )kβ
k=1
ZHONGGEN SU∗
8
We shall prove for each x ∈ C,
Eβ,q e
|λ|−µh
σh
x
=e
x2
2
+o(1)
,
from which the desired result (2.3) immediately follows. Here and in the sequel the
o(·), O(·) and ∼ refers to h → 0+ (equivalently q → 1−).
To this end, it is sufficient to show
|λ|
log Eβ,q e
xσ
h
=
x2
xµh
+
+ o(1).
σh
2
It follows from (2.5) that
∞
X
|λ|
(2.6)
log Eβ,q e
xσ
h
=
k β log
1 − e−hk
xk
1 − e σh e−hk
xk
∞
X
(e σh − 1)e−hk β
.
= −
k log 1 −
1 − e−hk
k=1
k=1
To compute the sum of (2.6), we use the Taylor expansion for the logarithm function
log(1 − x) to yield
(2.7)
xk
(e σh − 1)e−hk log 1 −
1 − e−hk
xk
xk
(e σh − 1)e−hk
(e σh − 1)2 e−2hk
−
−hk
1−e
2(1 − e−hk )2
xk
(e σh − 1)3 e−3hk +O
(1 − e−hk )3
= −
and then for the exponential function ex to yield
xk
(e σh − 1)e−hk
1 − e−hk
(2.8)
x
x2
k 2 e−hk
ke−hk
+ 2 ·
·
−hk
σh 1 − e
2σh 1 − e−hk
x 3 k 3 e−hk +O
·
σh
1 − e−hk
=
and
xk
x 3
(e σh − 1)2 e−2hk
x2
k 2 e−2hk
k 3 e−2hk =
·
+
O
(2.9)
·
.
2(1 − e−hk )2
2σh2 (1 − e−hk )2
σh
(1 − e−hk )2
Substituting (2.7)-(2.9) into (2.6) gives
−
∞
X
k=1
xk
1 − e σh e−hk
k log
1 − e−hk
β
=
∞
1 X
xµh
k β+3 e−kh x2
+O 3
.
+
σh
2
σh
(1 − e−kh )3
k=1
To complete the proof, we need only to verify
∞
1 X k β+3 e−kh
(2.10)
= O h(β+1)/2 = o(1).
3
−kh
3
σh
(1 − e
)
k=1
Note that (2.10) explains the usage of the expansion formulae above was indeed
legitimate. Now use the approximation of Riemann sums by integral to get as
h→0
Z ∞ β+2 −u
∞
X
k β+2 e−kh
1
u
e
2
(2.11)
σh =
∼ β+3
du
−u )2
(1 − e−kh )2
h
(1
−
e
0
k=1
Normal convergence for random partitions
9
and
(2.12)
Z ∞ β+3 −u
∞
X
k β+3 e−kh
1
u
e
∼
du.
−kh
3
β+4
(1 − e
)
h
(1 − e−u )3
0
k=1
Thus (2.10) is valid as desired.
When β = −1, the integrals in the right hand sides of (2.11) and (2.12) do not
exist. However, we can use the Euler-Maclaurin sum formula (see de Bruijn [6]) to
estimate the infinite sums in the left hand sides. In fact, we have
∞
X
ke−kh
(1 − e−kh )2
k=1
1
1
|
log
h|
+
o
=
h2
h2
σh2
=
and
∞
X
k=1
1
k 2 e−kh
1
=
|
log
h|
+
o
,
(1 − e−kh )3
h3
h3
from which (2.10) becomes
∞
1 X k 2 e−kh
1
p
=
O
= o(1).
σh3
(1 − e−kh )3
| log h|
k=1
Now (2.4) easily follows.
Remark 3. In the case β = 0, Pittel [20], Bloch and Okounkov [3] contained the
proof of Theorem 2.1.
Analogously, we can obtain the following central limit theorem for λ0k , k ≥ 1.
Theorem 2.2. (1) Under Pβ,q with β > 1, we have as h → 0
d
h(β+1)/2 (λ0k − Eβ,q λ0k ) −→ N (0, σβ2 ),
where
σβ2
Z
=
0
∞
uβ e−u
du.
(1 − e−u )2
(2) Under P1,q , we have as h → 0
h
d
p
(λ0k − E1,q λ0k ) −→ N (0, 1).
| log h|
Remark 4. When β = 0, the central limit theorem obviously fails for λ0k , k ≥ 1.
Indeed, λ0k asymptotically follows the Gumbel distribution by (1.2) and the duality.
Even when 0 < β < 1, Theorem 2.2 is no longer valid. This is easily understood
from the following simple observation: it follows
2
σh,k
=: V arβ,q (λ0k ) =
∞
X
j=k
∼
1
(1 − β)k 1−β h2
jβ
e−hj
(1 − e−hj )2
ZHONGGEN SU∗
10
and
∞
X
jβ
j=k
1
e−hj
∼
,
(1 − e−hj )3
(2 − β)k 2−β h3
and so
∞
1 X
3
σh,k
j=k
(1 − β)3/2
j β e−jh
∼
> 0.
(1 − e−jh )3
(2 − β)k (β+1)/2
Next we turn to the distribution of fluctuations in the bulk of a partition. Define
(2.13)
Vh (t) = h(β+1)/2 λ0[ t ] − Eβ,q λ0[ t ] , t ≥ 0
h
where by convention
λ00
=
h
λ01 .
Theorem 2.3. Under (P, Pβ,q ) with β > −1, we have as h → 0
(1) for each t > 0
(2.14)
d
Vh (t) −→ N (0, σβ2 (t)),
where
σβ2 (t) =
Z
t
∞
uβ e−u
du.
(1 − e−u )2
(2) for any a > 0
(2.15)
Vh ⇒ G,
in
D[a, ∞)
where G is a continuous Gaussian process with independent increments.
Proof. (2.14) can be proved in a completely similar argument to (2.3). We need
only to prove (2.15) below.
Since Vh is a partial sum process of independent random variables, we can apply
the general theory of weak convergence in D[a, ∞) (see Billingsley [2]). From (2.14)
and the independence of the rk ’s one easily see the convergence of finite dimensional
distribution of Vh . Therefore it is sufficient to establish the uniform tightness for
Vh in D[a, ∞). In turn, it is enough to show that for each positive ε and η, there
exist a δ, 0 < δ < 1, and a h0 > 0 such that for any h < h0 ,
Pβ,q sup |Vh (s) − Vh (t)| > η < ε.
(2.16)
|t−s|≤δ
Let
Sk = λ0k − Eβ,q λ0k .
Then according to (7.12) of Billingsley [2], a sufficient condition for (2.16) to hold
is for every t ≥ a
η
1
(2.17)
Pβ,q
max
|S[ ht ] − Sk | > (β+1)/2 < ε.
δ
h
[ ht ]≤k≤[ t+δ
h ]
To this end, we use Levy’s inequality (see Theorem 12 of Chapter III, Petrov [19])
to get
η
Pβ,q
max
|S[ ht ] − Sk | > (β+1)/2
(2.18)
h
[ ht ]≤k≤[ t+δ
h ]
q
η
≤ 2Pβ,q |S[ ht ] − S[ t+δ ] | > (β+1)/2 − 2V arβ,q (S[ ht ] − S[ t+δ ] ) .
h
h
h
Normal convergence for random partitions
11
Choose δ small enough (not depending on t and h) that
Z t+δ
Cβ
uβ e−u
η2
t
V arβ,q S[ h ] − S[ t+δ ] ≤ β+1
du
<
.
h
h
(1 − e−u )2
8hβ+1
t
Then the probability in the right hand side of (2.18) is bounded by
η
(2.19)
Pβ,q |S[ ht ] − S[ t+δ ] | > (β+1)/2 .
h
2h
Next we shall use Chebyschev’s inequality to further control (2.19) from above.
Note that
Eβ,q rk3
= k β (k β + 1)(k β + 2)
q 3k
+ 3Eβ,q rk2 − 2Eβ,q rk
(1 − q k )3
and
Eβ,q rk4
= k β (k β + 1)(k β + 2)(k β + 3)
q 4k
(1 − q k )4
+6Eβ,q rk3 − 11Eβ,q rk2 + 6Eβ,q rk
These together with (2.1) leads to
Eβ,q (rk − Eβ,q rk )4
= Eβ,q rk4 − 4Eβ,q rk3 (Eβ,q rk ) + 6Eβ,q rk2 (Eβ,q rk )2 − 3(Eβ,q rk )4
h 11e−4u − 4e−3u − 4e−2u
18e−2u
11e−2u i
= k 2β
+
−
(1 − e−u )4
(1 − e−u )3
(1 − e−u )2
h 6e−4u
12e−2u
5e−u
6e−u i
+k β
+
−
+
(1 − e−u )4
(1 − e−u )3
(1 − e−u )2
(1 − e−u )
=: k 2β g1 (u) + k β g2 (u).
where e−u = q k .
Using the independence of rk ’s we have
[ t+δ
h ]
(2.20) Eβ,q |S[ ht ] − S[ t+δ ] |4 ≤
X
h

[ t+δ
h ]
Eβ,q (rk − Erk )4 + 3 
X
k=[ ht ]
2
V arβ,q (rk ) .
k=[ ht ]
Thus by Chebyschev’s inequality and (2.20) we obtain
η
Pβ,q |S[ ht ] − S[ t+δ ] | > (β+1)/2
h
2h
t+δ
t+δ
[ h ]
h ]
[X
2 i
4 2(β+1) h X
2 h
4
E
(r
−
Er
)
+
3
V
ar
(r
)
≤
β,q
k
k
β,q
k
η4
t
t
k=[ h ]
2(β+1) h
Cβ h
η4
1
k=[ h ]
Z
t+δ
h2β+1
t
u2β g1 (u)du +
1
Z
t+δ
uβ g2 (u)du
hβ+1 t
Z
2 i
3 t+δ uβ e−u
+ 2(β+1)
du
−u
2
(1 − e )
h
t
Z t+δ
Z t+δ
h
Z t+δ uβ e−u
2 i
Cβ
2β
β+1
β
= 4 h
u g1 (u)du + h
u g2 (u)du + 3
du
.
η
(1 − e−u )2
t
t
t
≤
ZHONGGEN SU∗
12
Note
sup u2β g1 (u) < ∞,
a≤u<∞
uβ e−u
< ∞.
(1 − e−u )2
sup u2β g2 (u) < ∞,
a≤u<∞
Hence we have
Pβ,q |S[ ht ] − S[ t+δ ] | >
(2.21)
h
η
2h(β+1)/2
≤
Cβ
(h + hβ+1 + δ)δ
η4
which immediately implies (2.17) holds as long as we choose δ and η sufficiently
small. This completes the proof of Theorem 2.3 as desired.
Remark 5. As we have already seen in Lemma 1.2, the curve Ψβ (t) is the limit
shape for random partitions under (P, Pβ,q ). But we cannot in general replace the
centering constant Eβ,q λ0[ t ] by Ψβ (t)/hβ+1 in (2.13). In fact, let
h
xβ e−x
,
1 − e−x
then by the Euler-Maclaurin sum formula, we have for t > 0 and m ≥ 1
∞
∞
X
1 X
k β e−kh
=
f (hk)
Eβ,q λ0[ t ] =
h
1 − e−kh
hβ
t
t
(2.22)
f (x) =
k=[ h ]
=
k=[ h ]
Z
m
1 t X B2l 2l−1 (2l−1) t 1 h ∞
f
(hx)dx
+
f
h[
]
−
h
f
h[ ]
hβ [ ht ]
2
h
(2l)!
h
l=1
Z ∞
i
B2m (x − [x])
−h2m
f (2m) (hx)
dx ,
t
(2m)!
[h]
where Bl (x) denotes the l-th Bernoulli polynomials.
It is easy to see
Z ∞
Z t
1
1 h[ h ]
f (hx)dx = Ψβ (t) −
f (x)dx.
h
h t
[ ht ]
Also, since |B2m (x − [x])| ≤ |B2m | < ∞, then
Z ∞
B2m (x − [x]) f (2m) (hx)
dx ≤
(2m)!
[ ht ]
≤
|B2m |
(2m)!
Z
|B2m |
h(2m)!
∞
|f (2m) (hx)|dx
[t]
Zh ∞
|f (2m) (x)|dx.
t
When β = 0, we have for t > 0
1
Ψ0 (t) + O(1),
h
which implies (2.14) is still valid with h1 Ψ0 (t) in place of E0,q λ0[ t ] in (2.13).
E0,q λ0[ t ] =
h
h
3. Small Canonical Ensembles
Let
(3.1)
σn2
=
−(β+1)
hn
µn,k =
and define for k ≥ 1
∞
X
j=k
jβ
e−hn j
,
1 − e−hn j
where hn is as in Lemma 1.2.
2
σn,k
=
∞
X
j=k
jβ
e−hn j
,
(1 − e−hn j )2
Normal convergence for random partitions
13
Theorem 3.1. (1) Under Pn,β with β > 1, we have as n −→ ∞,
λ0k − µn,k d
−→ N (0, κ2β (0)),
σn
(3.2)
where
κ2β (0) = Γ(β + 1)ζ(β + 1, 0) −
Γ(β + 2)ζ 2 (β + 2, 0)
ζ(β + 2)
and
1
ζ(r + 1, 0) =:
Γ(r + 1)
∞
Z
0
ur e−u
du
(1 − e−u )2
for r > 1.
(2) Under Pn,1 , we have as n → ∞,
λ0k − µn,k
d
p
−→ N (0, 1).
σn | log hn |
(3.3)
Proof. The proof basically follows the line of Theorem 5 of Pittel [20]. We shall
only give the proof of (3.2), and (3.3) can be similarly proved with some minor
modifications.
It suffices to verify for each u ∈ R
1
λ0 − µ n,k
(3.4)
−→ exp κ2β (0)u2 .
En,β exp u k
σn
2
Recall the generating function of λ0k under Pβ,q is
Eβ,q e
xλ0k
β
∞
Y
(1 − q j )j
=
.
(1 − ex q j )j β
j=k
So by Lemma 1.1, we have
(3.5)
∞
X
0
0
q m Qm Em,β exλk
= Fβ (q)Eβ,q exλk
m=0
= Fβ (q)
β
∞
Y
(1 − q j )j
.
(1 − ex q j )j β
j=k
We view the right hand side of (3.5) as an analytic function in {q ∈ C : |q| < 1},
and make use of the Cauchy integral formula to get
Z π
β
∞
Y
0
(1 − (γeiθ )j )j
1
−inθ
iθ
e
F
(γe
)
(3.6) En,β exλk =
dθ.
β
2πQn γ n −π
(1 − ex (γeiθ )j )j β
j=k
We are now in a position to choose a suitable radius γ and to estimate the complex
contour integral in (3.6). For simplicity, define
(3.7)
β
∞
Y
(1 − (γeiθ )j )j
Fn (x, γe ) = Fβ (γe )
.
(1 − ex (γeiθ )j )j β
j=k
iθ
iθ
Let γ = e−τ , τ > 0 is to be determined (see (3.11) below). We need the following
lemma to estimate the generating function Fβ (γeiθ ) and the partition function Qn .
The reader is referred to Lemma 6.1 and Theorem 6.2 of Chapter 6, Andrews [1]
for more detail.
ZHONGGEN SU∗
14
Lemma 3.2. Let Fβ and Qn be as in the Introduction, then
(i) as τ → 0
(3.8)
Fβ (e−τ +iθ )
= exp Γ(β + 1)ζ(β + 2)τ −(β+1) − D(0) log τ + D0 (0) + O(τ A0 )
uniformly in θ provided |θ| ≤ π/4, where A0 (0 < A0 < 1) is a constant and
D(s) =
∞
X
n=1
1
ns−β
,
s = x + iy
possesses an analytic continuation in the region x ≥ −A0 .
(ii) as n → ∞
β + 2
2D(0)−(β+3)
β+1
1
Qn = A1 n 2(β+2) exp
(Γ(β + 2)ζ(β + 2)) β+2 n β+2 (1 + o(1)),
β+1
where
0
A1 = p
eD (0)
2π(β + 2)
(Γ(β + 2)ζ(β + 2))
1−2D(0)
2(β+2)
.
We proceed to prove Theorem 3.1. In view of Lemma 3.2, letting θ = 0 in (3.7)
yields
(3.9)
γ −n Fn (x, γ) = enτ Fβ (e−τ )
β
∞
Y
(1 − e−jτ )j
.
(1 − ex e−jτ )j β
j=k
Define
H(t, x) = nt + Γ(β + 1)ζ(β + 2)t−(β+1) +
∞
X
j=k
j β log
1 − e−jt
.
1 − ex e−jt
Then substituting (3.8) with θ = 0 into (3.9) gives
(3.10) γ −n Fn (x, γ) = exp H(τ, x) − D(0) log τ + D0 (0) + O(τ A0 ) .
To estimate the value of H at τ , we need its first three derivatives:
Ht (t, x)
n − Γ(β + 2)ζ(β + 2)t−(β+2)
∞
h e−jt
X
ex e−jt i
+
j β+1
−
,
1 − e−jt
1 − ex e−jt
=
j=k
Htt (t, x)
=
Γ(β + 3)ζ(β + 2)t−(β+3)
∞
h
i
X
e−jt
ex e−jt
−
j β+2
−
(1 − e−jt )2
(1 − ex e−jt )2
j=k
and
Httt (t, x)
= −Γ(β + 4)ζ(β + 2)t−(β+4)
∞
h e−jt (1 + e−jt ) ex e−jt (1 + ex e−jt ) i
X
−
j β+3
−
.
(1 − e−jt )3
(1 − ex e−jt )3
j=k
Normal convergence for random partitions
15
Now define τ as follows:
1 − ex
τ = τ∗ 1 −
S(τ ∗ , x) ,
(β + 2)n
(3.11)
where
∗
τ = hn ,
S(t, x) =
∞
X
j=k
j β+1 e−jt
.
(1 − e−jt )(1 − ex e−jt )
It is easy to see
∞
H(τ ∗ , x)
=
∗
(β + 2)Γ(β + 2)ζ(β + 2) X β
1 − e−jτ
+
.
j log
∗
β+1
(β + 1)(τ )
1 − ex e−jτ ∗
j=k
Let x
(3.12)
= σun
∞
X
j=k
. Then as in the proof of Theorem 2.1, we have
∗
j β log
1 − e−jτ
µn,k
u2
u + Γ(β + 1)ζ(β + 1, 0) + o(1)
∗ =
x
−jτ
1−e e
σn
2
and
∗
∞
e−jτ
1 X β
j
−→ 0.
σn3
(1 − e−jτ ∗ )3
j=k
To estimate Ht (t, x) at t = τ , we note
−(β+2)
1 − ex
(3.13) Ht (τ, x) = n − n 1 −
S(τ ∗ , x)
+ (1 − ex )S(τ, x).
(β + 2)n
Making use of the Taylor expansion yields
−(β+2)
1 − ex
(3.14)
S(τ ∗ , x)
1−
n(β + 2)
1 1 − ex
(β + 3)(1 − ex )2
∗
2
=1+
S(τ ∗ , x) +
S(τ
,
x)
+
O
n
2n2 (β + 2)
σn2
and
n
τ ∗ (1 − ex )
(3.15)
S(τ ∗ , x) + O 2 .
S(τ, x) = S(τ ∗ , x) − St (τ ∗ , x)
n(β + 2)
σn
Inserting (3.14) and (3.15) into (3.13) we obtain
n
Ht (τ, x) = O 2 = O n1/(β+2) .
σn
Analogously, for τ ∗ ≤ t̃ ≤ τ ,
(3.16)
Htt (t̃, x)
=
Γ(β + 3)ζ(β + 2)(τ ∗ )−(β+3) + O n(β+5)/2(β+2) .
Again, by the Taylor expansion of the function H(t, x) in the point t = τ , we have
1
(3.17) H(τ ∗ , x) = H(τ, x) + Ht (τ, x)(τ ∗ − τ ) + Htt (t̃, x)(τ ∗ − τ )2
2
which in turn, together with (3.16), implies
1
(3.18) H(τ, x) = H(τ ∗ , x) + Ht (τ, x)(τ − τ ∗ ) − Htt (t̃, x)(τ − τ ∗ )2
2
(β + 2)Γ(β + 2)ζ(β + 2) µn,k
u2 2
=
+
u
+
κ (0) + o(1).
(β + 1)(τ ∗ )β+1
σn
2 β
ZHONGGEN SU∗
16
Substituting (3.18) into (3.10) and by the definition (3.11) of τ , we obtain
(3.19)
γ −n Fn (x, γ)
uµ
u2 κ2β (0) nD(0)/(β+2)
n,k
= exp
+
σn
2
(Γ(β + 2)ζ(β + 2))D(0)/(β+2)
(β + 2)Γ(β + 2)ζ(β + 2)
0
A0
+
D
)
(1 + o(1)).
× exp
(0)
+
O(τ
(β + 1)(τ ∗ )β+1
Let us turn to the general case θ ∈ (−π, π) and γ = e−τ . Applying Lemma 3.2 to
(3.7), and noting the definition of H(t, x), we have
γ −n e−inθ Fn (x, γeiθ )
θ .
= exp H(τ − iθ, x) − D(0) log τ + D0 (0) + O(τ A0 ) + O
τ
It similarly follows
1
Httt (τ, x) = O
.
(τ ∗ )β+4
(3.20)
Thus we make use of the Taylor expansion of H(τ − iθ, x) in θ = 0 to get
θ3 θ2
H(τ − iθ, x) = H(τ, x) − iθHt (τ, x) − Htt (τ, x) + O
2
(τ ∗ )β+4
which together with (3.10) and (3.20) yields
γ −n e−inθ Fn (x, γeiθ )
θ3 θ θ2
+
O
= γ −n Fn (x, γ) exp −iθHt (τ, x) − Htt (τ, x) + O
.
2
(τ ∗ )β+4
τ
(β+3)/2
log n.
We are now ready to calculate the contour integral (3.6). Let θn = hn
The integral will be split into two parts: {θ : |θ| ≤ θn } and {θ : |θ| > θn }.
A direct calculation shows
Z
θ2
1
e−iθHt (τ,x)− 2 Htt (τ,x) dθ
2π |θ|≤θn
H 2 (τ, x) 1
1
−1/2
=p
exp − t
+ O Htt
exp − θn2 Htt (τ, x) .
2Htt (τ, x)
2
2πHtt (τ, x)
Since
β+1 Ht2 (τ, x)
= O n− β+2
Htt (τ, x)
and
θn2 Htt (τ, x) ≥ a log2 n,
for some a > 0,
then we obtain
(3.21)
1
2πγ n
Z
e−inθ Fn (x, γeiθ )dθ
|θ|≤θn
γ −n Fn (x, γ)(τ ∗ )(β+3)/2
= p
(1 + o(1))
2πΓ(β + 3)ζ(β + 2)
uµ
u2 κ2β (0) n,k
= Qn exp
+
(1 + o(1)).
σn
2
Normal convergence for random partitions
17
To estimate the integral over the region {θ : |θ| > θn }, observe that it follows by
an elementary inequality due to (1.11) of Pittel [20],
∞
∞
∞
X
Y
Y
1
1 − γj
β j
exp
j
(3.22) |Fn (x, γeiθ )| ≤
γ
(cos
jθ
−
1)
(1 − γ j )j β j=k 1 − ex γ j
j=1
j=1
∞
X
= Fn (x, γ) exp
j β γ j (cos jθ − 1) .
j=1
We need to calculate the infinite sum in the exponential of (3.22). In a similar
argument to (1.9) of Pittel [20], one can prove that there is a positive constant A3
such that if |θ| ≤ A3 hn then
∞
X
cβ θ2
(3.23)
j β γ j (1 − cos jθ) ≥ β+3 .
hn
j=1
On the other hand, if |θ| ≥ A3 hn , then by the Euler-Maclaurin formula
∞
∞
X
X
j β γ j (1 − cos jθ) =
(3.24)
j β e−jτ (1 − cos jθ)
j=1
j=1
=
Z
1
τ β+1
∞
0
1 θ uβ e−u 1 − cos u du + O β
τ
τ
cβ
≥
,
hβ+1
n
where in the last inequality we used (3.11).
(β+3)/2
log n, it easily follows
By (3.23), (3.24) and noting θn = hn
Z
∞
X
(3.25)
j β γ j (cos jθ − 1) dθ
exp
|θ|≥θn
j=1
Z
≤
exp −
θn ≤|θ|≤A3 hn
cβ θ2 hβ+3
n
Z
dθ +
c β
exp − β+1 dθ
hn
|θ|≥A3 hn
≤ 2 exp(−cβ log2 n).
Thus by (3.22) and (3.25) we have
1 Z
(3.26) n
e−inθ Fn (x, γeiθ )dθ ≤ γ −n Fn (x, γ) exp(−cβ log2 n).
2γ |θ|≥θn
Combining (3.6),(3.19), (3.21) and (3.26), we have shown
uµ
u2 κ2β (0) 0
n,k
En,β exλk = exp
+
(1 + o(1)).
σn
2
Therefore (3.4), and so (3.2), holds true as desired.
The proof of (3.3) is completely similar; we only remark that the difference
mainly stems from (3.12). The limit variance κ2β (0) in the case β > 1 (see (3.2))
consists of two parts: one is given by the second term of the right hand side of
(3.12), while the other is induced by the third term of the right hand side of (3.17).
When β = 1,
∞
X
je−jhn
2
σn,k
=
= σn2 | log hn |(1 + o(1))
(1 − e−jhn )2
j=k
ZHONGGEN SU∗
18
p
In this case we choose σn | log hn | as a normalizing constant. Given u ∈ R, let
, then (3.12) becomes
x= √u
σn
| log hn |
∞
X
∗
1 − e−jτ
µn,k u
u2
p
j log
=
+
+ o(1).
∗
1 − ex e−jτ
2
σn | log hn |
j=k
The third term of the right hand side of (3.17) is now negligible.
Theorem 3.1 corresponds to the end of partitions. We consider the fluctuations
in the deep bulk of partitions below. Let
1 0
λ[ t ] − µn,[ ht ] , t ≥ 0,
Vn (t) =
hn
n
σn
0
0
where λ0 = λ1 .
Theorem 3.3. Under Pn,β with β > −1, we have as n → ∞
(1) for each t > 0,
d
Vn (t) −→ V (t),
where V (t) is a normal random variable with zero mean and variance
1
κ2β (t) = σβ2 (t) −
(σ 2 (t))2 .
Γ(β + 3)ζ(β + 2) β+1
(2) for 0 < t1 < t2 < · · · < tm < ∞,
d
(Vn (t1 ), Vn (t2 ), · · · , Vn (tm )) −→ (V (t1 ), V (t2 ), · · · , V (tm )),
where (V (t1 ), V (t2 ), · · · , V (tm )) is a Gaussian vector with covariance structure
Cov(V (s), V (t)) = σβ2 (t) −
2
2
σβ+1
(s)σβ+1
(t)
,
Γ(β + 3)ζ(β + 2)
s < t.
(3) Each separable version of V is continuous in (0, ∞).
The proof is similar to that of Theorem 3.1 and is left to the interest reader. The
R ∞ uβ e−u
hypothesis β > 1 was used to guarantee the integral 0 (1−e
−u )2 du < ∞. However,
for each fixed t > 0 we can relax this requirement.
Having convergence of finite-dimensional distributions, it is natural to expect
the weak convergence of Vn (·) in function space. This requires that Vn (·) satisfies
a certain uniform tightness, namely for every ε > 0
(3.27)
lim lim Pn,β sup |Vn (t) − Vn (s)| ≥ ε = 0.
δ→0 n→∞
|t−s|≤δ
Note a similar uniform tightness was proved true in the independent case (see
Theorem 2.3). Unfortunately, we can only prove a weaker version of (3.27), i.e.,
stochastic equi-continuity holds for Vn (·). However, this will still guarantee convergence in distribution of every integral functional from a broad class; see Section 7,
Chapter IX in Gihman-Skorohod [12].
Let G be the set of all continuous functions g(t, x) in R+ × R such that for some
β−1
β+1
ν > ω if β > 2 and 2(β+2)
ν > ω if 1 < β ≤ 2
ω and ν with 6(β+2)
|g(t, x)| = O(eωt |x|ν )
uniformly over R+ × R.
Normal convergence for random partitions
19
Theorem 3.4. Under Pn,β with β > 1, we have for each g ∈ G
Z ∞
Z ∞
d
(3.28)
g(t, Vn (t))dt −→
g(t, V (t))dt.
0
0
We shall apply the Gihman-Skorohod method to prove (3.28). To this end, we
need the following lemmas to show that Vn (·) satisfies the stochastic equi-continuity
condition.
Lemma 3.5. Let 1 ≤ k1 < k2 and
 (β+1)/3
,
β>2
 hn
−1/3
ηn =
n, β = 2
 hn log
hn ,
1 < β < 2.
Then for each η such that
O(ηn ), β ≥ 2
o(ηn ), 1 < β < 2,
η=
(3.29)
we have
η2
2
2
(σn,k
−
σ
.
En,β exp(η(λ0k1 − λ0k2 − (µn,k1 − µn,k2 ))) ≤ exp
)
n,k
1
2
2
Proof. We can adapt a completely similar argument to Proposition of Pittel [20]
with some modifications given in the proof of Theorem 3.1.
Lemma 3.6. Assume β > 1. Then for each ε > 0
(3.30)
lim lim
sup Pn,β (|Vn (t) − Vn (s)| ≥ ε) = 0
δ→0 n→∞ |t−s|≤δ
and for m ≥ 1
(3.31)
En,β |Vn (t)|m = O(σβm (t) + (σn ηn )−m (log n)m ).
Proof. Assume 0 ≤ t < s. By the Euler-Maclaurin formula, we have
Z s
1 1
uβ e−u
2
2
2
σn;t,s
=: σn,[
du
+
O
−
σ
=
,
s
t
n,[ hn ]
−u )2
hn ]
hβ+1
hβn
t (1 − e
n
and so
2
σn;t,s
−→
σn2
Z
t
s
uβ e−u
du = σβ2 (t) − σβ2 (s).
(1 − e−u )2
By Lemma 3.5, we obtain for each η satisfying condition (3.29)
η2
2
En,β eσn η(Vn (t)−Vn (s)) ≤ exp
σn;t,s
2
σ2 η2
≤ a exp n (σβ2 (t) − σβ2 (s))
2
where a > 0 is a numerical constant. Consequently, given x > 0,
σ2 η2
(3.32) Pn,β (|Vn (t) − Vn (s)| ≥ x) ≤ a exp n (σβ2 (t) − σβ2 (s)) − σn ηx .
2
Note that the quadratic function in η in the above exponent attains its minimum
at
x
ηn (x) =:
.
2
σn (σβ (t) − σβ2 (s))
ZHONGGEN SU∗
20
Put x0 = σn η(σβ2 (t) − σβ2 (s)). We further control the upper bound (3.32) below.
(i) x ≤ x0 . Plugging η = ηn (x) into (3.32) yields
x2
(3.33)
Pn,β (|Vn (t) − Vn (s)| ≥ x) ≤ a exp −
.
2(σβ2 (t) − σβ2 (s))
(ii) x > x0 . Plugging η = ηn (x0 ) into (3.32) yields
σ ηx n
Pn,β (|Vn (t) − Vn (s)| ≥ x) ≤ a exp −
.
2
Now choose η = ηn if β ≥ 2; otherwise η = ηn (log n)−1 , where ηn is given by
Lemma 3.5. Then for every ε > 0 we have
σ η ε
ε2
n n
+
a
exp
−
.
Pn,β (|Vn (t) − Vn (s)| ≥ ε) ≤ a exp −
2(σβ2 (t) − σβ2 (s))
2 log n
(3.34)
n ηn
Note 2σlog
n → ∞ for any β > 1. Letting n → ∞ and then letting δ → 0 gives
(3.30).
Next we turn to prove (3.31). Analogously to (3.33) and (3.34), it follows

 a exp − x22 , x ≤ x0
2σβ (t)
(3.35)
Pn,β (|Vn (t)| ≥ x) ≤
 a exp − σn ηx , x > x0 .
2
where x0 = σn ησβ2 (t).
Choosing η as above and by the integration by part formula
En,β |Vn (t)|m = O σβm (t) + (σn ηn )−m (log n)m .
The proof is complete.
Proof of Theorem 3.4. Fix g ∈ G. First, it follows from (3.30) and the GihmanSkorohod lemma (see Theorem 2 in Section 7, Chapter IX of Gihman-Skorohod
[12]) that for any M > 0
Z M
Z M
d
g(t, Vn (t))dt −→
g(t, V (t))dt.
0
0
Also, we have as M → ∞
Z
∞
P
|g(t, V (t))|dt −→ 0.
M
Indeed, for g ∈ G
Z
∞
Z
∞
E|g(t, V (t))|dt ≤
(3.36)
M
eωt E|V (t)|ν dt.
M
Since V (t) is normal with zero mean and variance V ar(V (t)) = κ2β (t), then
E|V (t)|ν ≤ Aν κνβ (t)) ≤ Aν σβν (t),
where Aν is a constant depending on ν.
In addition, obviously ν/2 > ω by the hypothesis on ν and ω. This gives
Z ∞
Z ∞
ν/2
Z ∞ uβ e−u
eωt σβν (t)dt =
eωt
du
dt
−u
2
(1 − e )
M
M
t
ν
= O M βν/2 e−( 2 −ω)M
−→
0
(M → ∞).
Normal convergence for random partitions
21
Thus (3.36) goes to 0 as M → ∞, as desired.
Next we prove as n → ∞, M → ∞
Z ∞
P
(3.37)
g(t, Vn (t))dt −→ 0.
M
As in (3.36), we have
Z log n
(3.38)
En,β |g(t, Vn (t))|dt ≤
log n
Z
M
eωt En,β |Vn (t)|ν dt.
M
By virtue of (3.31), the right hand side of (3.38) is bounded up to a factor by
Z ∞
eωt σβν (t)dt + (σn ηn )−ν (log n)ν nω ,
M
which in turn goes to 0 as n → ∞, M → ∞ by the definition of ηn and the
hypotheses on ν and ω.R
∞
Turn to the integral log n g(t, Vn (t))dt. It follows from (1.2)
β + 1
log n lim Pn,β λ1 >
+ε
= 0, ∀ ε > 0
n→∞
β+2
hn
In particular, the event {λ1 ≤ log n/hn } occurs with high probability. On the other
hand, on this event λ0[ t ] = 0 whenever t > log n. Thus we have
hn
Vn (t) = −
µn,[ ht
n
]
σn
and so
Z
∞
Z
|g(t, Vn (t))|dt
∞
eωt |Vn (t)|ν dt
≤
log n
log n
≤
→
∞
∞
X
1
σnν
Z
0
(n → ∞).
eωt
log n
k=[ htn ]
k β e−khn ν
dt
1 − e−khn
Combined, we have shown (3.37). The proof of Theorem 3.4 is now complete.
4. Normal convergence of dλ
In this section we are concerned with the number dλ of standard Young tableaux
of a Young diagram λ ∈ Pn . Let us start with two useful expressions for the
computation of dλ .
Assume λ = (λ1 , · · · , λl ) and λ0 = (λ01 , · · · , λ0λ1 ). A so-called hook formula
discovered by Frames, Thrall and Robinson [9] reads
(4.1)
n!
,
(λ
−
j + λ0j − i + 1)
(i,j)∈λ i
dλ = Q
where the product is over all n unit cells in the diagram λ. Trivially, dλ = dλ0 .
An alternative expression, due to Frobenius [11], is given by
(4.2)
dλ =
n!
,
Hλ
ZHONGGEN SU∗
22
where
Ql
Hλ = Q
+ l − i)!
.
(λ
−
i − (λj − j))
1≤i<j≤l i
i=1 (λi
Considerably deeper is the RSK correspondence between the set of partitions of the
integer n and the symmetric group Sn . The RSk correspondence actually gives a
procedure for obtaining Young tableaux with the help of permutations, from which
follows the Burnside identity
X d2
λ
= 1.
n!
λ∈Pn
In the remarkable series of papers Szalay, Turán [22] and Pittel [20, 21] studied
at length the likely magnitude of dλ under (Pn , Pn,0 ). In particular, log(d2λ /n!)
is asymptotically normal with mean −An (A a positive constant) and standard
deviation of order n3/4 .
The aim of this section is to find the limiting distribution of log(dβ+2
/n!) under
λ
(Pn , Pn,β ), β > 0. Our main result is
Theorem 4.1. Under (Pn , Pn,β ) with β > 1, we have as n → ∞
dλ
d
(4.3)
−
b
−→ N (0, Λ2 ),
h(β+3)/2
log
n
n
(n!)1/(β+2)
where q = e−hn ,
(4.4)
bn
∞
∞
k=1
k=1
∞
X
X
β+1
β+1 X
µn,k −
µn,k log µn,k +
µn,k log n −
n
β+2
β+2
=
−
β+1
(log Γ(β + 2)ζ(β + 2))
β+2
k=1
∞
X
µn,k − n
k=1
and
2
Z
∞
Z
∞
Cov(V (x), V (y)) log Ψβ (x) log Ψβ (y)dxdy.
Λ =
0
0
Remark 6. We conjecture that (4.3) is also valid for β ∈ (0, 1). However, the
present technique does not allow to do it. The centering constant bn in (4.4) is of
the order n, which can be seen from the Euler-Maclaurin formula.
In order to prove Theorem 4.1, we need the following lemma.
Lemma 4.2. Assume β > 1. Then
2
σn,k
≤ Cβ ,
n,k≥1 µn,k
(4.5)
max
(4.6)
(4.7)
where µn,k and
lim Pn,β
n→∞
lim Pn,β
n→∞
2
σn,k
λ1
X
Cβ σn,k
≥ (β−1)/2 = 0,
µn,k
hn
k=1
max
1≤k≤λ1
|λ0k − µn,k | p
≥ 3 log n = 0,
σn,k
are as in (3.1).
Normal convergence for random partitions
23
Proof. We start with the proof of (4.5). Recall the definition of f in (2.22). Then
by the Euler-Maclaurin sum formula, we have for k ≥ 1
Z ∞
1
1
(4.8)
µn,k =
f (x)dx + β f (khn ) + Rn,k (f, 1),
β+1
hn
2hn
khn
where Rn,k (f, 1) is an error term satisfying
Z ∞
1 0
00
|Rn,k (f, 1)| ≤
|f
(kh
)|
+
|f
(x)|dx
.
n
12hnβ−1
khn
Similarly, putting
f2 (x) =
xβ e−x
,
(1 − e−x )2
we have for k ≥ 1
(4.9)
2
σn,k
=
Z
1
hβ+1
n
∞
f2 (x)dx +
khn
1
2hβn
f2 (khn ) + Rn,k (f2 , 1),
where Rn,k (f2 , 1) is an error term satisfying
Z ∞
1 0
|Rn,k (f2 , 1)| ≤
|f2 (khn )| +
|f200 (x)|dx .
β−1
12hn
khn
A key observation is
Z ∞
Z ∞
cβ
Cβ
(4.10)
f (x)dx ≤ µn,k ≤ β+1
f (x)dx
hβ+1
hn
khn
khn
n
and
Z ∞
Cβ
2
σn,k ≤ β+1
(4.11)
f2 (x)dx
hn
khn
We only show (4.10), the other is very similar. Fix M > 0 such that eM > 2. We
consider two subsets of integers separately: {k : khn ≤ M } and {k : khn > M }.
First, note
Z ∞
xe−x
(4.12)
< ∞,
f (x)dx < ∞
C =: sup
−x
0≤x<∞ 1 − e
0
Also,
f 0 (x) =
βxβ−1 e−x
xβ e−x
−
−x
1−e
(1 − e−x )2
and
f 00 (x)
xβ−2 e−x
βxβ−1 e−x
−
−x
1−e
1 − e−x
β−1 −x
β −x
βx
e
x e
xβ e−x
−
+
+
2
.
(1 − e−x )2
(1 − e−x )2
(1 − e−x )3
= β(β − 1)
(i) Assume k is such that khn ≤ M . Then we have the following
Z ∞
Z ∞
Z ∞
f (x)dx ≤
f (x)dx ≤
f (x)dx,
M
khn
0
f (khn ) ≤ CM β−1 ,
ZHONGGEN SU∗
24
|f 0 (khn )| ≤ Cβ (khn )β−2 ≤
Cβ M β−2 , β > 2
Cβ hβ−2
,
1<β≤2
n
and
Z
∞
|f 00 (x)|dx ≤
khn
Cβ
.
hn
(ii) Assume k is such that khn > M . Then we have the following
Z ∞
cβ (khn )β e−khn ≤
f (x)dx ≤ Cβ (khn )β e−khn ,
khn
f (khn ) ≤ 2(khn )β e−khn ,
f 0 (khn ) ≤ Cβ (khn )β e−khn ,
Z
∞
f 00 (x)dx ≤ Cβ (khn )β e−khn .
khn
Combining these with (4.8) easily yields (4.10).
In view of (4.10) and (4.11), it follows
R∞
2
f2 (x)dx
σn,k
Cβ
n
max
≤
· max Rkh
,
∞
n,k≥1 µn,k
cβ n,k≥1 kh f (x)dx
n
which implies (4.5).
Let us turn to the sum estimation in (4.6). Fix ε > 0. Let
1 β + 1
(4.13)
Bn =
log n + (β + ε) log log n .
hn β + 2
then by (1.2), it follows
(4.14)
lim Pn,β (λ1 > Bn ) = 0
n→∞
So it suffices to prove
X
(4.15)
1≤k≤Bn
σn,k
Cβ
≤ (β−1)/2 .
µn,k
hn
Again, according to (4.10) and (4.11), it follows
(4.16)
X
1≤k≤Bn
σn,k
≤ Cβ h(β+1)/2
n
µn,k
X
1≤k≤Bn
R∞
( khn f2 (x)dx)1/2
R∞
f (x)dx
khn
As in the proof of (4.10) above, split the sum of (4.16) into two subsums over
{k : khn ≤ M } and {k : khn > M }. It is easy to see
R∞
X ( kh
f2 (x)dx)1/2
Cβ M
R n∞
≤
hn
f (x)dx
khn
k:kh ≤M
n
and
X
M
hn
≤k≤Bn
(
R∞
f (x)dx)1/2
khn 2
R∞
f (x)dx
khn
≤ Cβ (log n)ε/2
These together with (4.16) concludes the proof of (4.6).
Normal convergence for random partitions
25
Finally, we shall prove (4.7). By (4.14), it suffices to prove
|λ0k − µn,k | p
Pn,β max
(4.17)
≥ 3 log n −→ 0.
1≤k≤Bn
σn,k
To this end, let us first prove for each b > 0,
|λ0 − µ | p
b
n,k
k
Pβ,q
(4.18)
≥ b log n ≤ 2 exp − log n ,
σn,k
3
k ≥ 1,
where q = e−hn . As in Theorem 2.1, we have for every x ∈ R
∞
X
0
x2 2
j β e−3jhn .
Eβ,q exλk = exp xµn,k + σn,k
+ O |x|3
2
(1 − e−jhn )3
j=k
In particular, set x =
√
b log n/σn,k . We only need to check
∞
(log n)3/2 X j β e−3jhn
= o(1)
3
σn,k
(1 − e−jhn )3
(4.19)
j=k
uniformly in k ≥ 1. In turn, this can be proved in a similar way to (4.5).
(4.18) now easily follows from Chebyschev’s inequality. To prove (4.17), observe
the following relation: for any set B,
Pn,β {(r1 , · · · , rn ) ∈ B} ≤
1
Pβ,q {(r1 , · · · , rn ) ∈ B}
Pβ,q (|λ| = n)
Besides, a similar argument to Lemma 4.2 [10] shows
(β+3)/2
Pβ,q (|λ| = n)
=
hn
p
2πΓ(β + 3)ζ(β + 3, 0)
(1 + o(1)).
These together with (4.18) immediately implies (4.7).
Proof of Theorem 4.1. Since dλ = dλ0 , it is sufficient for us to prove
dλ0
d
(4.20)
h(β+3)/2
log
−
b
−→ N (0, Λ2 ),
n
n
(n!)1/(β+2)
According to the Frobenius formula (4.2),
log
dλ0
1
β+1
= log
+
log n!
Hλ0
β+2
(n!)1/(β+2)
For any partition λ and its dual λ0 , it follows
Q
0
0
1
1≤i<j≤λ1 (λi − λj + j − i)
Q
(4.21)
=
0
Hλ0
1≤i≤λ1 (λi − i + λ1 )!
=:
Mn
,
Nn
and so
log
1
= log Mn − log Nn .
Hλ0
ZHONGGEN SU∗
26
Use (4.14),
(4.22)
log Mn
X
=
log(λ0i − λ0j + j − i)
1≤i<j≤λ1
≤ λ21 log n = Op
log3 n h2n
,
where and in the sequel the notation Xn = Yn + Op (cn ) means P (|Xn − Yn | ≥
M cn ) → 0 for some M > 0.
−(β+3)/2
Note log Mn in (4.22) is negligible compared to the variance hn
when
β > 1. The normal fluctuation in (4.20) mainly comes from the term log Nn . By
Stirling’s approximation for factorial and the upper bound for λ1 , we easily see
X
(4.23)
log(λ0i − i + λ1 )!
log Nn =
1≤i≤λ1
X
=
φ(λ0i − i + λ1 ) + Op
log2 n 1≤i≤λ1
hn
,
where φ(x) = x log x − x.
On the other hand, note the Taylor expansion for the function φ: for any x0
Z x
φ(x) = φ(x0 ) + φ0 (x0 )(x − x0 ) +
φ00 (t)(x − t)dt.
x0
We get
(4.24)
X
φ(λ0i − i + λ1 )
X
=
1≤i≤λ1
φ(µn,i − i + λ1 )
1≤i≤λ1
+
X
(λ0i − µn,i ) log(µn,i − i + λ1 )
1≤i≤λ1
+
X
Rφ,i
1≤i≤λ1
where the error term satisfies
|Rφ,i | ≤
(λ0i − µn,i )2
.
µn,i − i + λ1
Lemma 4.2 can now be used to show the square error term is negligible:
X
1≤i≤λ1
(λ0i − µn,i )2
µn,i − i + λ1
X
= Op log n
1≤i≤λ1
= Op
log2 n hn
σi2 µn,i
.
The second term of the right hand side of (4.24) gives the linear approximation of
log Nn . We proceed to simplify the logarithmic factor. Let
νn,i =
1
hβ+1
n
Ψβ (ihn ),
Normal convergence for random partitions
27
where Ψβ is as in Lemma 1.2, then it is easy to see
X
(λ0i − µn,i ) log(µn,i − i + λ1 )
(4.25)
1≤i≤λ1
=
X
X
(λ0i − µn,i ) log νn,i +
Rlog,i ,
1≤i≤λ1
1≤i≤λ1
where the error term satisfies
|λ0 − µn,i |
|Rlog,i | ≤ i
· |µn,i − νn,i + λ1 − i|.
νn,i ∨ µn,i
Once again, the error term in (4.25) is negligible. Indeed, by Lemma 4.2 it suffices
to control
X
σn,i
1
|µn,i − νn,i | = Op (β+1)/2 .
(4.26)
νn,i ∨ µn,i
hn
1≤i≤λ1
This in turn can be proved as in the proof of Lemma 4.2.
Next let us turn to the first term of the right hand side of (4.25) and (4.24)
respectively. It obviously follows
X
X
X
(λ0i − µn,i ) =
(4.27)
λ0i −
µn,i
1≤i≤λ1
1≤i≤λ1
∞
X
= n−
1≤i≤λ1
X
µn,i +
i=1
µn,i
i>λ1
and
X
µn,i
i>λ1
X k β+1 e−khn
X X k β e−khn
=
1 − e−khn
1 − e−khn
k>λ1
i>λ1 k≥i
Z ∞ β+1 −u
1
u
e
≤
du
−u
1
−
e
hβ+2
λ1 hn
n
1
= Op
(log n)β+1 ,
hn
=
where in the last step we used (4.14). Similarly,
X
X
X
R̄φ,i .
φ(µn,i − i + λ1 ) =
φ(µn,i ) +
1≤i≤λ1
1≤i≤λ1
1≤i≤λ1
where the error term satisfies
|R̄φ,i | ≤ (λ1 − i) log(µn,i − i + λ1 ),
and so by (4.14) and (4.10)
X
|
R̄φ,i | = op
1≤i≤λ1
1
(β+3)/2
hn
.
On the other hand , we have by (4.10) and (4.27)
X
X
φ(µn,i ) =
(µn,i log µn,i − µn,i )
i>λ1
i>λ1
=
Op
1
(log n)β+2 .
hn
ZHONGGEN SU∗
28
Combining the preceding, (4.23) becomes
(4.28)
log Nn
=
X
(λ0i − µn,i ) log Ψβ (ihn ) +
∞
X
φ(µn,i )
i=1
1≤i≤λ1
∞
X
− n−
µn,i log hβ+1
+
o
p
n
1
(β+3)/2
hn
X
β+1
(λ0i − µn,i ) log Ψβ (ihn ) +
=
log n!
β+2
1≤i≤λ1
1
−bn + op (β+3)/2
hn
i=1
where bn is as in (4.4).
It remains to prove the summation term in the right hand side of (4.28) satisfies
the central limit theorem. In fact,
Z ∞
X
λ0[x] − µn,[x] log Ψβ (xhn )dx + Rn ,
(λ0i − µn,i ) log Ψβ (ihn ) =
0
1≤i≤λ1
where Rn is bounded by
∞
X
|λ0i − µn,i |(log Ψβ ((i − 1)hn ) − log Ψβ (ihn )) = op
i=1
Moreover, a change of variable gives
Z ∞
h(β+3)/2
λ0[x] − µn,[x] log Ψβ (xhn )dx
n
0
Z
=
1
(β+3)/2
.
hn
∞
Vn (x) log Ψβ (x)dx
0
Therefore we have by (4.22) and (4.28)
Z ∞
dλ0
(β+3)/2
− bn = −
Vn (x) log Ψβ (x)dx + op (1)
hn
log
(n!)1/(β+2)
0
Now a direct application of Theorem 3.4 concludes the proof of (4.20).
Acknowledgement. The author wishes expresses his gratitude to the anonymous
referee for his careful reading and insightful comments, which substantially improved the paper.
References
[1] G. E. Andrews, The Theory of Partitions, Encyclopedia of Mtahematics and Its Applications,
Vol. 2, Addison-Wesley, Reading, MA, 1976.
[2] P. Billingsley, Weak Convergence of Measures, 2nd edition, John Wiley & Son, Inc. 1999.
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Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, P.R.
China
E-mail address: suzhonggen@zju.eud.cn