INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A13

THE SIZE OF THE LARGEST PART OF RANDOM PLANE PARTITIONS OF LARGE INTEGERS

Ljuben Mutafchiev^{1}

American University in Bulgaria, 2700 Blagoevgrad, Bulgaria and Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences

                              ljuben@aubg.bg

Received: 11/25/05, Revised: 3/20/06, Accepted: 4/12/06, Published: 4/26/06

Abstract

We study the asymptotic behavior of the largest part size of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that this characteristic, appropriately normalized, tends weakly, as n → ∞, to a random variable having an extreme value probability distribution with distribution function, equal to e^{−e^{−z}}, −∞ < z < ∞. The representation of a plane partition as a solid diagram shows that the same limit theorem holds for the numbers of rows and columns of a random plane partition of n.

1. Introduction

A plane partition ω of the positive integer n is an array of non-negative integers

\begin{matrix}ω_{1,1} & ω_{1,2} & ω_{1,3} & ... \\ ω_{2,1} & ω_{2,2} & ω_{2,3} & ... \\ ... & ... & ... & ... \end{matrix}  (1.1)

for which ∑_{i,j} ω_{i,j} = n and the rows and columns are arranged in decreasing order:

ω_{i,j} ≥ ω_{i+1,j}, ω_{i,j} ≥ ω_{i,j+1}

1A significant part of this work was done during author’s visit at the Department of Mathematics of the University of South Florida. The author wishes to thank for the support, hospitality and helpful discussions.

The author is also grateful to the referee for the careful reading of the manuscript and for his or her valuable comments and suggestions.

for all i, j ≥ 1. The non-zero entries ω_{i,j} > 0 are called parts of ω. If there are λ_{i} parts in the ith row of ω, so that for some r

λ_{1} ≥ λ_{2} ≥ ... ≥ λ_{r} > λ_{r+1} = 0,

then the partition λ = (λ_{1}, λ_{2}, ..., λ_{r}) of the integer p = λ_{1} + λ_{2} + ... + λ_{r} is called the shape of ω, denoted by λ. We also say that ω has r rows and p parts. Sometimes, for the sake of brevity, the zeroes in the array (1.1) are deleted. For instance, the abbreviation

3 2 1 1 1

is assumed to present a plane partition of n = 8 having r = 2 rows and p = 5 parts.

It seems that MacMahon was the first who introduced the idea of a plane partition; see [9]. He deals with the general problem of such partitions, enumerating them by the size of each part, number of rows and number of columns. These problems have been subsequently reconsidered by other authors who have developed methods, entirely different from those of MacMahon. For important references and more details in this direction, we refer the reader to the monographs of Andrews [4; Chap. 11] and Stanley [15; Chap. 7], as well as to the survey paper [14; Chap. V].

Let q(n) denote the total number of plane partitions of the integer n ≥ 1. It turns out that

Q(x) = 1 + ∑_{n=1}^{∞} q(n)x^{n} = ∏_{j=1}^{∞}(1 − x^{j})^{−j}  (1.2)

(see [4; Corollary 11.3] or [14; Corollary 18.2]). The asymptotic of q(n) has been obtained by Wright [17]. It is given by the following formula:

q(n) ∼ \frac{[ζ(3)]^{7/36}}{2^{11/36}3^{1/2}π^{1/2}} n^{−25/36} exp \{3[ζ(3)]^{1/3}(n/2)^{2/3} + 2γ\},  (1.3)

where

ζ(s) = ∑_{j=1}^{∞} j^{−s}  (1.4)

is the Riemann’s zeta function and

γ = ∫_{0}^{∞} \frac{y log y}{e^{2πy} − 1}dy.

The asymptotic of the coefficients in the power series expansion of ∏_{j=1}^{∞}(1−x^{j})^{−a_{j}} under a general scheme of assumptions on the sequence of non-negative numbers {a_{j}}_{j≥1} was obtained by Meinardus [10]. For plane partitions, we have a_{j} = j, j = 1, 2, .... Meinardus’ theorem confirms Wright’s formula (1.3). (In the original statement of (1.3) [17; p. 179] the constant 3^{1/2} in the denominator is missing. However, it appears at the end of the proof on p. 189. Further results on the asymptotic expansion of q(n) may be found in [2,3].)

We introduce the uniform probability measure P on the set of all plane partitions of n, assuming that the probability 1/q(n) is assigned to each plane partition.

Let L_{n}, C_{n} and R_{n} denote the size of the largest part, the number of columns and number of rows in a plane partition of n, respectively. With respect to the probability measure P, L_{n}, C_{n} and R_{n} become random variables defined on the set of plane partitions of n. Our aim in this paper is to study their limiting distributions as n → ∞.

We notice that any plane partition ω, defined by array (1.1), has an associated diagram D(ω) = {(i, j, k) ∈ N^{3} : 1 ≤ k ≤ ω_{i,j}}. Here N denotes the set of the positive integers. Any permutation σ of the three coordinate axis (i, j, k), different from the identical one, transforms D(ω) in a diagram that uniquely determines another plane partition σ ∘ ω. The permutation σ also permutes the three statistics (L_{n}, C_{n}, R_{n}). Then, if one of these statistics is restricted by certain inequality, the same restriction occurs on the statistics permuted by σ. The one to one correspondence between ω and σ ∘ ω implies that L_{n}, C_{n} and R_{n} have one and the same probability distribution (for more details see also [15; p. 371]).

The starting point in our asymptotic analysis is the following generating function identity, which follows from a stronger result of MacMahon [9; Section 495]:

1 + ∑_{n=1}^{∞} P (L_{n} ≤ m, R_{n} ≤ r)q(n)x^{n} = ∏_{k=1}^{m} ∏_{j=1}^{r}(1 − x^{j+k−1})^{−1}, m, r = 1, 2, ....

(For more details and other proofs of this result we refer the reader to [14; Chap. V]). If we keep either of the parameters m and r fixed, setting the other one := ∞, we obtain

1 + ∑_{n=1}^{∞} P (X_{n} ≤ m)q(n)x^{n} = ∏_{k=1}^{m}(1 − x^{k})^{−k} ∏_{j=m+1}^{∞}(1 − x^{j})^{−m}

= Q(x) ∏_{j=m+1}^{∞}(1 − x^{j})^{j−m}, X_{n} = L_{n}, C_{n}, R_{n}.  (1.5)

Here Q(x) was defined by (1.2). Note that C_{n} is also included in (1.5) since its distribution coincides with those of L_{n} and R_{n}.

The object of this paper is to provide an asymptotic analysis for the Cauchy integrals stemming from (1.5) in order to prove the following theorem.

Theorem 1. For any real and finite z, we have

lim_{n→∞} P \{[\frac{2ζ(3)}{n}]^{1/3} X_{n} − log [\frac{n}{2ζ(3)}]^{2/3} − log [\frac{1}{2} log^{2} (n^{2/3})] ≤ z\} = e^{−e^{−z}},  (1.6)

where ζ(s) denotes Riemann’s zeta function (1.4) and X_{n} = L_{n}, C_{n}, R_{n}.

The proof of this result is based on a method developed by Hayman [7]. The idea comes from the fact that Q(x), the generating function enumerating the numbers q(n) of plane partitions of n, satisfies Hayman’s admissibilty properties in a neighborhood of its main singularity x = 1 and outside it (see Lemmas 1 and 2 of the next section, respectively). Hence, Hayman’s asymptotic result for the coefficients q(n) compensates, roughly speaking, their contribution to the general asymptotic of the coefficients in (1.5). A careful analysis of the product

∏_{j=m+1}^{∞}(1 − x^{j})^{j−m},

around x = 1 then yields the weak convergence to the extreme value distribution (1.6). A relevant analytic approach to problems concerning random integer partitions is presented in [11,12]. A different probabilistic method was previously suggested by Fristedt [6], who used a conditioning device to transfer the problems related to integer partition statistics to problems dealing with functionals of independent and geometrically distributed random variables.

We organize the paper as follows. Section 2 contains auxiliary facts on the admissibility of Q(x) and on the asymptotic behavior of the coefficients in its power series expansion. In Section 3 we prove our main result.

We conclude this section noticing that plane partitions have many applications to such diverse topics as ballot problems (see [14; Chap. IV] and [5]), symmetric functions (see [14; Chap. II - IV], [15; Chap. 7]) and the references therein), the representation theory of the symmetric group (see [14; Chap III] and [15; Chap. 7]), and analysis of algorithms (see [8; Section 5.2.4] and [13; Chap. 12]).

2. Preliminary Asymptotics

In order to get an asymptotic estimate for Q(x) around its main singularity x = 1, we will use a general result due to Meinardus [10] (see also [4; Chap. 6]) on the asymptotic behavior of the nth coefficient in the expansion of

G(x) = ∏_{j=1}^{∞}(1 − x^{j})^{−a_{j}},

where {a_{j}}_{j=1}^{∞} is a sequence of given non-negative numbers. He introduced a set of assumptions on the Dirichlet’s series

D(s) = ∑_{j=1}^{∞} a_{j}j^{−s},  (2.1)

generated by the sequence {a_{j}}_{j=1}^{∞}. A basic restriction in his scheme states that D(s) has to converge in the half-plane ℜe(s) > α > 0 and can be analytically continued into the half-plane ℜe(s) ≥ −α_{0} for some α_{0} ∈ (0, 1). Moreover, in ℜe(s) ≥ −α_{0}, D(s) is analytic except for a simple pole at s = α with residue A. Under these and certain other restrictions on D(s), for v = y + 2πiw, Meinardus [10] (see also [4; Lemma 6.1]) proved that

G(e^{−v}) = exp {AΓ(α)ζ(α + 1)v^{−α} − D(0) log v + D′(0) + O(y^{α_{0}})}  (2.2)

as y → 0 uniformly for | arg v |≤ π/4 and | w |≤ 1/2. (Here Γ(α) denotes Euler’s gamma function and log (.) presents the main branch of the logarithmic function satisfying log v < 0 for 0 < v < 1.)

Let us take now a sequence {r_{n}}_{n=1}^{∞} which, as n → ∞, satisfies

r_{n} = 1 − \frac{[2ζ(3)]^{1/3}}{n^{1/3}} + \frac{[2ζ(3)]^{2/3}}{2n^{2/3}} − \frac{ζ(3)}{n} + O(n^{−4/3}).  (2.3)

For the sake of brevity, for 0 < r < 1, we also set

b(r) = \frac{6ζ(3)}{(1 − r)^{4}} .  (2.4)

It is easy to check that (2.3) and (2.4) imply

b(r_{n}) = \frac{3n^{4/3}}{[2ζ(3)]^{1/3}} + O(n)  (2.5)

as n → ∞.

The next lemma suggests a tool that we will subsequently use in Section 3 to obtain the main terms in our asymptotic.

Lemma 1. If r_{n} satisfies (2.3) for large n, then

Q(r_{n}e^{iθ})e^{−iθn} = Q(r_{n})e^{−θ^{2}b(r_{n})/2}[1 + O(1/ log^{3} n)]

as n → ∞ uniformly for | θ |≤ δ_{n}, where

δ_{n} = \frac{n^{−5/9}}{log n}  (2.6)

and b(r_{n}) is determined by (2.4).

Proof. We apply first Meinardus’ asymptotic formula (2.2) to the plane partition generating function Q(x) (see (1.2)). Since in this case we have a_{j} = j, j = 1, 2, ... and D(s) = ζ(s − 1) (see (2.1)), we find that α = 2, A = 1 and D(0) = −1/12. Classical results on the ζ function (see [16; Section 13.51]) imply that (2.2) is valid with v = y + 2πiw. Therefore, we can write

Q(e^{−v}) = exp \{ζ(3)v^{−2} + \frac{1}{12} log v + D′(0) + O(y^{α_{0}})\}  (2.7)

as y → 0 uniformly for | w |≤ 1/2 and | arg (v) |≤ π/4 (here α_{0} ∈ (0, 1) is the constant specified in Meinardus assumption [10] for the analytical continuation of D(s) = ζ(s − 1)). Setting

e^{−v} = r_{n}e^{iθ},  (2.8)

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A13 6

we see that y = y_{n} = − log r_{n} and w = −θ/2π. An asymptotic expansion for − log r_{n}, as n → ∞, can be found using (2.3) as follows:

y_{n} = − log r_{n} = \frac{[2ζ(3)]^{1/3}}{n^{1/3}} − \frac{[2ζ(3)]^{2/3}}{2n^{2/3}} + \frac{ζ(3)}{n}

+\frac{[2ζ(3)]^{2/3}}{n^{2/3}} − \frac{2^{4/3}[2ζ(3)]^{2/3}[ζ(3)]^{1/3}}{4n} + O(n^{−4/3})

= \frac{[2ζ(3)]^{1/3}}{n^{1/3}} + O(n^{−4/3}).  (2.9)

Combining (2.7) - (2.9), we observe that

\frac{Q(r_{n}e^{iθ})}{Q(r_{n})}e^{−iθn} = (\frac{y_{n} − iθ}{y_{n}})^{1/12} exp \{ζ(3)[(y_{n} − iθ)^{−2} − y_{n}^{−2}] − iθn + O(y_{n}^{α_{0}})\}. (2.10)

A Taylor’s formula expansion for | θ |≤ δ_{n} yields

(y_{n} − iθ)^{−2} − y_{n}^{−2} = 2iθy_{n}^{−3} − 6\frac{θ^{2}}{2} y_{n}^{−4} + O(| θ |^{3} y_{n}^{−5})

= 2iθy_{n}^{−3} − 3θ^{2}y_{n}^{−4} + O(δ_{n}^{3}y_{n}^{−5}).  (2.11)

Using (2.9), we also get the following estimate for the factor outside the exponent in (2.10):

(\frac{y_{n} − iθ}{y_{n}})^{1/12} = \{\frac{[2ζ(3)]^{1/3} − iθn^{1/3} + O(n^{−1})}{[2ζ(3)]^{1/3} + O(n^{−1})}\}^{1/12} = 1 + O(δ_{n}n^{1/3}), n → ∞. (2.12)

Finally we notice that (2.9) implies the bound

y_{n}^{α_{0}} = O(n^{−α_{0}/3}), n → ∞,  (2.13)

where α_{0} ∈ (0, 1). Hence, inserting (2.5), (2.9), (2.11) - (2.13) into (2.10), we obtain

\frac{Q(r_{n}e^{iθ})}{Q(r_{n})}e^{−iθn} = [1 + O(δ_{n}n^{1/3})] exp\{ζ(3)[\frac{2iθn}{2ζ(3)[1 + O(n^{−1})]^{3}}

−\frac{θ^{2}}{2} \frac{6n^{4/3}}{2^{4/3}[ζ(3)]^{4/3}[1 + O(n^{−1})]^{4}} + O(δ_{n}^{3}y_{n}^{−5})] − iθn + O(n^{−α_{0}/3})\}

= [1 + O(n^{−2/9}/ log n)] exp\{ζ(3)[\frac{iθn}{ζ(3)}(1 + O(n^{−1}))

−\frac{θ^{2}}{2} [b(r_{n}) + O(n)](1 + O(n^{−1})) + O(δ_{n}n^{5/3})] − iθn + O(n^{−α_{0}/3})\}

= [1 + O(n^{−2/9}/ log n)] exp{iθn + O(δ_{n}) − θ^{2}b(r_{n})/2 + O(nδ_{n}^{2}) +O(δ_{n}^{2}b(r_{n})n^{−1}) + O(δ_{n}^{3}n^{5/3}) − iθn + O(n^{−α_{0}/3})} = [1 + O(n^{−2/9}/ log n)] exp {−θ^{2}b(r_{n})/2 + O(1/ log^{3} n)} = e^{−θ^{2}b(r_{n})/2}[1 + O(1/ log^{3} n)], n → ∞.

This completes the proof. □

We also need another lemma that will establish a uniform estimate for

Q_{m}(x) = ∏_{j=1}^{m}(1 − x^{j})^{−j}, m = 1, 2, ...,  (2.14)

where | x |= r_{n} and arg (x) is outside the range (−δ_{n}, δ_{n}). For the sake of convenience, we also let

f_{m,n}(θ) = ∑_{j>m} jr_{n}^{j}[cos (jθ) − 1].  (2.15)

Lemma 2. If r_{n} and δ_{n} satisfy (2.3) and (2.6), respectively, then there exist two positive constants ϵ and n_{0} such that

| Q_{m}(r_{n}e^{iθ}) |≤ Q_{m}(r_{n}) exp {ϵ − (2.5)n^{2/9}/[2ζ(3)]^{4/3} log^{2} n − f_{m,n}(θ)}

uniformly for m = 1, 2, ..., π ≥| θ |≥ δ_{n} and n ≥ n_{0}.

Proof. By taking logarithms in (2.14), for | x |< 1, we get

log Q_{m}(x) = log [∏_{j=1}^{m}(1 − x^{j})^{−j}] = − ∑_{j=1}^{m} j log (1 − x^{j})

= ∑_{j=1}^{m} j ∑_{l=1}^{∞} \frac{x^{jl}}{l} = ∑_{l=1}^{∞} \frac{1}{l} ∑_{j=1}^{m} j(x^{l})^{j}.

Thus, substituting x = r_{n}e^{iθ}, we obtain

| Q_{m}(r_{n}e^{iθ}) |=| exp \{log Q_{m}(r_{n}e^{iθ})\} |=| exp\{∑_{l=1}^{∞} \frac{1}{l} ∑_{j=1}^{m} jr_{n}^{lj}e^{ijlθ}\} |

=| exp\{∑_{j=1}^{m} jr_{n}^{j}e^{ijθ} + ∑_{l=2}^{∞} \frac{1}{l} ∑_{j=1}^{m} jr_{n}^{lj}e^{ijlθ}\} |

= exp\{∑_{j=1}^{m} jr_{n}^{j}ℜe(e^{ijθ}) + ∑_{l=2}^{∞} \frac{1}{l} ∑_{j=1}^{m} jr_{n}^{lj}ℜe(e^{ijlθ})\}

= exp\{∑_{j=1}^{m} jr_{n}^{j} cos (jθ) + ∑_{l=2}^{∞} \frac{1}{l} ∑_{j=1}^{m} jr_{n}^{lj} cos (ljθ)\}

≤ exp\{∑_{j=1}^{m} jr_{n}^{j} cos (jθ) + ∑_{l=2}^{∞} \frac{1}{l} ∑_{j=1}^{m} jr_{n}^{lj}\}

= exp\{∑_{j=1}^{m} jr_{n}^{j}[cos (jθ) − 1] + ∑_{l=1}^{∞} \frac{1}{l} ∑_{j=1}^{m} jr_{n}^{lj}\}

= Q_{m}(r_{n}) exp {H_{m,n}(θ)},  (2.16)

where

H_{m,n}(θ) = ∑_{j=1}^{m} jr_{n}^{j}[cos (jθ) − 1]

= ℜe [\frac{r_{n}e^{iθ}}{(1 − r_{n}e^{iθ})^{2}} ] − \frac{r_{n}}{(1 − r_{n})^{2}} − f_{m,n}(θ)

= \frac{r_{n}(cos θ + r_{n}^{2} cos θ − 2r_{n})}{(1 − 2r_{n} cos θ + r_{n}^{2})^{2}} − \frac{r_{n}}{(1 − r_{n})^{2}} − f_{m,n}(θ).  (2.17)

The function cos θ + r_{n}^{2} cos θ attains its maximum in the range δ_{n} ≤| θ |≤ π at θ = δ_{n} and θ = −δ_{n}. Note that

cos δ_{n} = 1 − \frac{δ_{n}^{2}}{2} + O(δ_{n}^{4}) = 1 − \frac{n^{−10/9}}{2 log^{2} n} + O(n^{−20/9}/ log^{4} n)

and

(1 − r_{n})^{−2} = [\frac{n}{2ζ(3)}]^{2/3} + O(n^{1/3}).

Substituting these estimates into the right-hand side of (2.17), after some manipulations, we obtain

H_{m,n}(θ) ≤ \frac{r_{n}[1 − n^{−4/9}/2[2ζ(3)]^{2/3} log^{2} n + O(n^{−7/9}/ log^{2} n)]}{(1 − r_{n})^{2} [1 + \frac{r_{n}n^{−10/9}}{(1−r_{n})^{2} log^{2} n} + O (\frac{n^{−20/9}}{(1−r_{n})^{2} log^{4} n})]^{2}}

−\frac{r_{n}}{(1 − r_{n})^{2}} − f_{m,n}(θ)

= \frac{r_{n}[1 − n^{−4/9}/2[2ζ(3)]^{2/3} log^{2} n + O(n^{−7/9}/ log^{2} n)]}{(1 − r_{n})^{2}[1 + 2n^{−4/9}/[2ζ(3)]^{2/3} log^{2} n + O(n^{−7/9}/ log^{2} n)}

−\frac{r_{n}}{(1 − r_{n})^{2}} − f_{m,n}(θ)

= \frac{r_{n}[1 − n^{−4/9}/2[2ζ(3)]^{2/3} log^{2} n + O(n^{−7/9}/ log^{2} n)]}{(1 − r_{n})^{2}}

× \{1 − \frac{2n^{−4/9}}{[2ζ(3)]^{2/3} log^{2} n} + O (\frac{n^{−7/9}}{log^{2} n})\} − \frac{r_{n}}{(1 − r_{n})^{2}} − f_{m,n}(θ)

= −r_{n} \{[\frac{n}{2ζ(3)}]^{2/3} + O(n^{1/3})\} \frac{5n^{−4/9}}{2[2ζ(3)]^{2/3} log^{2} n} + o(1) − f_{m,n}(θ)

= −\frac{(2.5)n^{2/9}}{[2ζ(3)]^{4/3} log^{2} n} + o(1) − f_{m,n}(θ).

Inserting this estimate into (2.16), we obtain the required result. □

Further, we will essentially use the asymptotic form of the numbers q(n). Although this is given by Wright’s formula (1.3), we need this result in a slightly different form. Since Lemmas 1 and 2, together with (2.3) - (2.5), provide Hayman’s admissiblity [7] for the generating function Q(x) (see (1.2)), we can apply his general asymptotic formula to the coefficients q(n). The next lemma encompasses these results. We only sketch its proof and insert a remark explaining the role of the asymptotic expansion (2.3) there.

Lemma 3. For Q(x), the generating function of the numbers q(n) of plane partitions of n defined by (1.2), we have

q(n) ∼ Q(r_{n})r_{n}^{−n}/\sqrt{2πb(r_{n})}  (2.18)

as n → ∞, where r_{n} satisfies the equation

rQ′(r)/Q(r) = n  (2.19)

for sufficiently large n and b(r_{n}) is defined by (2.4).

Sketch of the proof. It is clear that | x |= 1 is a natural boundary for Q(x). Lemma 1 shows the behavior of Q(x) around its main singularity x = 1 (condition I of Hayman’s definition [7]); Lemma 2 establishes that the growth of Q(x) as x → x_{0}, | x_{0} |= 1 and x_{0} ≠ 1 is negligible (condition II of [7]). It is then easily seen that

\frac{rQ′(r)}{Q(r)} = 2 ∑_{j=1}^{∞} \frac{r^{2j}}{(1 − r^{j})^{3}} + ∑_{j=1}^{∞} \frac{r^{j}}{(1 − r^{j})^{2}}

= \frac{2ζ(3)}{(1 − r)^{3}} + o((1 − r)^{−1})

as r → 1^{−}. This enables one to conclude that r_{n}, determined for sufficiently large n by (2.19), can be substituted by the asymptotic expansion (2.3). Thus, one can obtain (2.18) after direct application of Hayman’s theorem [7]. □

Finally, notice that (2.3) - (2.5) and (2.7) imply the coincidence of the right sides of (1.3) and (2.18).

3. Proof of the Main Result

We apply first Cauchy’s coefficient formula to (1.5) using the circle x = r_{n}e^{iθ}, −π < θ ≤ π as a contour of integration, where r_{n} is determined by (2.3). Thus, for X_{n} = L_{n}, we obtain

P (L_{n} ≤ m)q(n) = \frac{r_{n}^{−n}}{2π} ∫_{−π}^{π} Q(r_{n}e^{iθ})e^{−iθn} ∏_{j=m+1}^{∞}(1 − r_{n}e^{ijθ})^{j−m}dθ.

Then, we break up the range of integration as follows:

P (L_{n} ≤ m)q(n) = J_{1}(m, n) + J_{2}(m, n),  (3.1)

where

J_{1}(m, n) = \frac{r_{n}^{−n}}{2π} ∫_{−δ_{n}}^{δ_{n}} Q(r_{n}e^{iθ})e^{−iθn} ∏_{j=m+1}^{∞}(1 − r_{n}^{j}e^{ijθ})^{j−m}dθ,  (3.2)

J_{2}(m, n) = \frac{r_{n}^{−n}}{2π} ∫_{δ_{n}≤|θ|≤π} Q(r_{n}e^{iθ})e^{−iθn} ∏_{j=m+1}^{∞}(1 − r_{n}^{j}e^{ijθ})^{j−m}dθ.  (3.3)

We next let

e^{F_{m}(x)} = ∏_{j=m+1}^{∞}(1 − x^{j})^{j−m},

that is

F_{m}(x) = ∑_{j=m+1}^{∞}(j − m) log (1 − x^{j}).  (3.4)

3.1. An Asymptotic Estimate for J_{1}(m, n)

We will employ a more compact form for J_{1}(m, n). It is based on the notation given in (3.4). We can write

J_{1}(m, n) = \frac{r_{n}^{−n}}{2π} ∫_{−δ_{n}}^{δ_{n}} Q(r_{n}e^{iθ}) exp \{F_{m}(r_{n}e^{iθ}) − iθn\}dθ.  (3.5)

Using Taylor’s formula expansion, we obtain

F_{m}(r_{n}e^{iθ}) = F_{m}(r_{n}) + r_{n}(e^{iθ} − 1)F′_{m}(r_{n}) + \frac{r_{n}^{2}}{2} (e^{iθ} − 1)^{2}F′′_{m}(r_{n})

+O(| θ |^{3}| F′′′_{m}(r_{n}) |).

Substituting this expression into (3.5) and applying the result of Lemma 1 to Q(r_{n}e^{iθ}), we get a new expression for J_{1}(m, n), namely:

J_{1}(m, n) = \frac{r_{n}^{−n}}{2π} Q(r_{n})e^{F_{m}(r_{n})}[1 + O(1/ log^{3} n)]

× ∫_{−δ_{n}}^{δ_{n}} exp\{(e^{iθ} − 1)r_{n}F′_{m}(r_{n}) + \frac{1}{2}(e^{iθ} − 1)^{2}r_{n}^{2}F′′_{m}(r_{n})

+O(δ_{n}^{3} | F′′′_{m}(r_{n}) |)}dθ.  (3.6)

Next, we will obtain asymptotic expressions for F_{m}(r_{n}), F′_{m}(r_{n}), F′′_{m}(r_{n}) and F′′′_{m}(r_{n}) in terms of the functions

ψ_{j}(v) = ∫_{0}^{∞} \frac{u^{j}du}{e^{u+v} − 1}, j = 0, 1, 2, 3,  (3.7)

where the parameter v > 0 will be further substituted by

v = v_{n} = my_{n}  (3.8)

(see again (2.9) for the asymptotic expansion of y_{n}). The functions ψ_{j}(v) are closely related to the Debye functions (see, e.g., [1; Section 27.1]). We will interpret the sums that we will obtain as Riemann ones with step sizes equal to y_{n} = − log r_{n}.

Starting with F_{m}(r_{n}), we find, as n → ∞, that

F_{m}(r_{n}) = ∑_{j>m}(j − m) log (1 − r_{n}^{j})

= y_{n}^{−2} ∑_{jy_{n}>v_{n}}(jy_{n})y_{n} log (1 − e^{−jy_{n}}) − my_{n}^{−1} ∑_{jy_{n}>v_{n}} y_{n} log (1 − e^{−jy_{n}})

∼ y_{n}^{−2} ∫_{v_{n}}^{∞}(u − v_{n}) log (1 − e^{−u})du  (3.9)

with v_{n} defined by (3.8). Integrating by parts the integral in (3.9) and using (3.7) with j = 2, we find that

e^{F_{m}(r_{n})} = [1 + o(1)] exp \{−\frac{y_{n}^{−2}}{2} ψ_{2}(v_{n})\}  (3.10)

as n → ∞. In the same way, we show that

r_{n}F′_{m}(r_{n}) = ∑_{j>m} \frac{(j − m)jr_{n}^{j}}{1 − r_{n}^{j}}

= −y_{n}^{−3} ∑_{jy_{n}>v_{n}} \frac{(jy_{n})^{2}y_{n}e^{−jy_{n}}}{1 − e^{−jy_{n}}} + my_{n}^{−2} ∑_{jy_{n}>v_{n}} \frac{(jy_{n})y_{n}e^{−jy_{n}}}{1 − e^{−jy_{n}}}

= [1 + o(1)](−y_{n}^{−3}) ∫_{v_{n}}^{∞} \frac{u(u − v_{n})}{e^{u} − 1}du = −y_{n}^{−3}[ψ_{2}(v_{n}) + v_{n}ψ_{1}(v_{n})].  (3.11)

For the asymptotic of the second derivative of F_{m} we have

r_{n}^{2}F′′_{m}(r_{n}) = − ∑_{j>m}(j − m)j [\frac{jr_{n}^{j}}{1 − r_{n}^{j}} − \frac{r_{n}^{j}}{1 − r_{n}^{j}} + \frac{jr_{n}^{2j}}{(1 − r_{n}^{j})^{2}} ]

= − ∑_{j>m} j^{3} [\frac{r_{n}^{j}}{1 − r_{n}^{j}} + \frac{r_{n}^{2j}}{(1 − r_{n}^{j})^{2}} ] + m ∑_{j>m} j^{2} [\frac{r_{n}^{j}}{1 − r_{n}^{j}} + \frac{r_{n}^{2j}}{(1 − r_{n}^{j})^{2}} ]

+ ∑_{j>m} j^{2} \frac{r_{n}^{j}}{1 − r_{n}^{j}} − m ∑_{j>m} j \frac{r_{n}^{j}}{1 − r_{n}^{j}}

= [1 + o(1)] \{−y_{n}^{−4} ∫_{v_{n}}^{∞} u^{2}(u − v_{n}) [\frac{1}{e^{u} − 1} + \frac{1}{(e^{u} − 1)^{2}} ] du + y_{n}^{−3} ∫_{v_{n}}^{∞} u(u − v_{n})\frac{du}{e^{u} − 1}\}

= −[1 + o(1)]y_{n}^{−4} ∫_{v_{n}}^{∞} u^{2}(u − v_{n}) [\frac{1}{e^{u} − 1} + \frac{1}{(e^{u} − 1)^{2}} ] du + O(y_{n}^{−3})  (3.12)

as n → ∞. Since the integrand in (3.12) includes (e^{u} − 1)^{2} in its denominator, we need to make an auxiliary calculation. Thus, for any a ≥ 1, we have

I_{a} = ∫_{v_{n}}^{∞} \frac{u^{a}}{(e^{u} − 1)^{2}} du = \frac{v_{n}^{a}}{e^{v_{n}} − 1} + a ∫_{0}^{∞} \frac{(z + v_{n})^{a−1}}{e^{z+v_{n}} − 1} dz − ∫_{0}^{∞} \frac{(z + v_{n})^{a}}{e^{z+v_{n}} − 1}dz.

Setting a = 3 and a = 2, after simple manipulations, we easily obtain

∫_{v_{n}}^{∞} \frac{u^{2}(u − v_{n})}{(e^{u} − 1)^{2}}du = I_{3} − v_{n}I_{2} = ∫_{0}^{∞} \frac{(3 − 2v_{n})z^{2} + v_{n}z + v_{n}^{2} − z^{3}}{e^{z+v_{n}} − 1} dz

= −ψ_{3}(v_{n}) + (3 − 2v_{n})ψ_{2}(v_{n}) + v_{n}ψ_{1}(v_{n}) + v_{n}^{2}ψ_{0}(v_{n})
= −ψ_{3}(v_{n}) + (3 − 2v_{n})ψ_{2}(v_{n}) + v_{n}ψ_{1}(v_{n}) − log (1 − e^{−v_{n}}).  (3.13)

On the other hand,

∫_{v_{n}}^{∞} \frac{u^{2}(u − v_{n})}{e^{u} − 1}du = ∫_{0}^{∞} \frac{(z^{3} + 2v_{n}z^{2} + v_{n}^{2}z)dz}{e^{z+v_{n}} − 1} = ψ_{3}(v_{n}) + 2v_{n}ψ_{2}(v_{n}) + v_{n}^{2}ψ_{1}(v_{n}). (3.14)

Upon replacing (3.13) and (3.14) into (3.12), we get the required asymptotic of the second derivative of F_{m}:

r_{n}^{2}F′′_{m}(r_{n}) = −[1 + o(1)]y_{n}^{−4}[3ψ_{2}(v_{n}) + (v_{n}^{2} + v_{n})ψ_{1}(v_{n}) + v_{n}^{2}ψ_{0}(v_{n})], n → ∞.  (3.15)

Finally, following similar but simpler analysis, we obtain the following O-bound for the third derivative of F_{m}:

F′′′_{m}(r_{n}) = O (y_{n}^{−5} ∫_{v_{n}}^{∞} u^{4} [\frac{1}{e^{u} − 1} + \frac{1}{(e^{u} − 1)^{2}} + \frac{1}{(e^{u} − 1)^{3}} ] du) = O(y_{n}^{−5}), n → ∞. (3.16)

To find a complete asymptotic estimate for the integrand in (3.6), we approximate e^{iθ} − 1 and (e^{iθ} − 1)^{2} by iθ + O(δ_{n}^{2}) and −θ^{2} + O(δ_{n}^{3}), respectively. Hence, for | θ |≤ δ_{n}, by (2.6) we get

e^{iθ} − 1 = iθ + O(n^{−10/9}/ log^{2} n),  (3.17)
(e^{iθ} − 1)^{2} = −θ^{2} + O(n^{−5/3}/ log^{3} n)  (3.18)

as n → ∞. Furthermore, we notice that (2.9) implies, for large n, that

y_{n}^{−k} = [\frac{n}{2ζ(3)}]^{k/3} [1 + O(1/n)] = [\frac{n}{2ζ(3)}]^{k/3} + O(n^{k/3−1}), k = 2, 3, 4, 5.

We substitute these estimates in the right sides of (3.10), (3.11), (3.15) and (3.16). Combining the results obtained after these substitutions with those of (3.17) and (3.18), we see now that (3.6) becomes

J_{1}(m, n) = \frac{r_{n}^{−n}}{2π} Q(r_{n})[1 + o(1)] exp \{−\frac{1}{2} [\frac{n}{2ζ(3)}]^{2/3} ψ_{2}(v_{n})\}

× ∫_{−δ_{n}}^{δ_{n}} exp\{−\frac{iθn}{2ζ(3)}[ψ_{2}(v_{n}) + v_{n}ψ_{1}(v_{n})] − \frac{θ^{2}b(r_{n})}{2}

+\frac{θ^{2}}{2} [\frac{n}{2ζ(3)}]^{4/3} [3ψ_{2}(v_{n}) + (v_{n}^{2} + v_{n})ψ_{1}(v_{n}) − v_{n}^{2} log (1 − e^{−v_{n}})]\}dθ, n → ∞.

By letting θ = t/b^{1/2}(r_{n}), −∞ < t < ∞, in this integral with b(r_{n}) defined by (2.4) (see also (2.5)), we get

J_{1}(m, n) = \frac{r_{n}^{−n}}{2πb^{1/2}(r_{n})}Q(r_{n})[1 + o(1)]exp \{−\frac{1}{2} [\frac{n}{2ζ(3)}]^{2/3} ψ_{2}(v_{n})\}

× ∫_{−δ_{n}b^{1/2}(r_{n})}^{δ_{n}b^{1/2}(r_{n})} exp {−itA_{n} − B_{n}t^{2}/2}dt, n → ∞,  (3.19)

where

A_{n} = \frac{n}{2ζ(3)b^{1/2}(r_{n})}[ψ_{2}(v_{n}) + v_{n}ψ_{1}(v_{n})],  (3.20)

B_{n} = 1 − [\frac{n}{2ζ(3)}]^{4/3} [3ψ_{2}(v_{n}) + (v_{n}^{2} + v_{n})ψ_{1}(v_{n}) − v_{n}^{2} log (1 − e^{−v_{n}})]/b(r_{n}).  (3.21)

On the other hand, (2.5) and (2.6) imply that

δ_{n}b^{1/2}(r_{n}) ∼ dn^{1/9} log n, d = \sqrt{3}/[2ζ(3)]^{1/6}.  (3.22)

Further, we also need some asymptotic estimates for ψ_{j}(v_{n}), j = 1, 2, as v_{n} → ∞, presented in a slightly different form. Integrating by parts the right side of ψ_{2}(v_{n}) (see (3.7)), we get

ψ_{2}(v_{n}) = −v_{n}^{2} log (1 − e^{−v_{n}}) − 2 ∫_{v_{n}}^{∞} u log (1 − e^{−u})du.  (3.23)

For u ∈ [v_{n}, ∞), we have

− log (1 − e^{−u}) = e^{−u} + O(e^{−2u}) = e^{−u}[1 + O(e^{−v_{n}})],

and therefore,

− ∫_{v_{n}}^{∞} u log (1 − e^{−u})du = [1 + O(e^{−v_{n}})] ∫_{v_{n}}^{∞} ue^{−u}du
= [1 + O(e^{−v_{n}})](v_{n}e^{−v_{n}} + e^{−v_{n}}) = O(v_{n}e^{−v_{n}}), v_{n} → ∞.

Substituting this into (3.23), we conclude that

ψ_{2}(v_{n}) = −v_{n}^{2} log (1 − e^{−v_{n}}) + O(v_{n}e^{−v_{n}}), v_{n} → ∞.  (3.24)

In the same way for ψ_{1}(v_{n}) we obtain the following asymptotic:

ψ_{1}(v_{n}) = −v_{n} log (1 − e^{−v_{n}}) + O(e^{−v_{n}}), v_{n} → ∞.  (3.25)

We are now ready to specify the sequence {v_{n}}_{n=1}^{∞} that will produce a non-degenerate limiting distribution. We set

v_{n} = \frac{2}{3} log n + log log n + z, −∞ < z < ∞, n = 1, 2, ....  (3.26)

Obviously, assumption (3.26) implies that

− log (1 − e^{−v_{n}}) = e^{−v_{n}} + O(e^{−2v_{n}}) = \frac{e^{−z}}{n^{2/3} log^{2} n} + O(n^{−4/3}/ log^{4} n),

and therefore,

−v_{n}^{2} log (1 − e^{−v_{n}}) = \frac{4e^{−z}}{9n^{2/3}} [1 + O (\frac{log log n}{log n})]  (3.27)

as v_{n} → ∞. Therefore, the exponent outside the integral of the right side of (3.19) becomes

exp \{−\frac{1}{2} [\frac{n}{2ζ(3)}]^{2/3} ψ_{2}(v_{n})\} = exp \{−\frac{2e^{−z}}{9[2ζ(3)]^{2/3}} \} + O (\frac{log log n}{log n})  (3.28)

as n → ∞. Turning to the integrand of (3.19), note that (3.24) - (3.27) imply somewhat less precise but shorter estimates for ψ_{1} and ψ_{2}. We have

ψ_{2}(v_{n}) = O(n^{−2/3}), v_{n}ψ_{1}(v_{n}) = O(n^{−2/3}), n → ∞.

Inserting them into (3.20) and taking into account the asymptotic form of b(r_{n}) from (2.5), we get

A_{n} = O(n^{−2/3}), n → ∞.  (3.29)

A similar analysis of (3.21) shows that

B_{n} = 1 + O(n^{−2/3} log n).  (3.30)

Therefore the integrand of (3.19) tends to e^{−t^{2}/2} as n → ∞.

Our final remark concerns the bounds of the integral in (3.19). They can be substituted by −∞ and ∞ with an error term that is estimated, e.g., by using the asymptotic expansion of the function (2π)^{−1/2} ∫_{v}^{∞} e^{−t^{2}}dt as v → ∞ (see [1; Chap. 7]). Hence, by (3.22) we find that its order, as n → ∞, is at most O(n^{−1/9}(log n) exp {−d^{2}n^{2/9}/ log^{2} n}) = o(1). This, in combination with (3.28) - (3.30) and the Lebesgue dominated convergence theorem, shows that

J_{1}(m, n) = \frac{r_{n}^{−n}}{2πb^{1/2}(r_{n})}Q(r_{n})[1 + o(1)] exp \{−\frac{2e^{−z}}{9[2ζ(3)]^{2/3}} \}

× ∫_{−∞}^{∞} e^{−t^{2}/2}dt = \frac{r_{n}^{−n}}{\sqrt{2πb(r_{n})}}Q(r_{n})[1 + o(1)] exp \{−\frac{2e^{−z}}{9[2ζ(3)]^{2/3}} \}  (3.31)

as n → ∞. Thus the estimate for J_{1}(m, n) is completed.

3.2. An Asymptotic estimate for J_{2}(m, n)

First, we recall (2.14) and (3.3). It is also clear that

Q(r_{n}e^{iθ}) ∏_{j=m+1}^{∞}(1 − r_{n}^{j}e^{ijθ})^{j−m} = Q_{m}(r_{n}e^{iθ}) ∏_{j=m+1}^{∞} \frac{1}{(1 − r_{n}^{j}e^{ijθ})^{m}} .  (3.32)

The infinite product in the right side of (3.32) can be estimated as follows:

| ∏_{j=m+1}^{∞} \frac{1}{(1 − r_{n}^{j}e^{ijθ})^{m}} |≤ ∏_{j=m+1}^{∞} \frac{1}{(1 − r_{n}^{j})^{m}} < ∏_{j=m+1}^{∞} \frac{1}{(1 − r_{n}^{j})^{j}} .  (3.33)

This bound is uniform for m = 1, 2, .... Now, to estimate the integrand in J_{2}(m, n), we apply successively Lemma 2 and (3.33). We obtain

| Q(r_{n}e^{iθ})e^{−iθn} ∏_{j=m+1}^{∞}(1 − r_{n}^{j}e^{ijθ})^{j−m} |

≤ Q_{m}(r_{n}) exp \{ϵ − (2.5)n^{2/9}/[2ζ(3)]^{4/3} log^{2} n − f_{m,n}(θ)\} ∏_{j=m+1}^{∞} \frac{1}{(1 − r_{n}^{j})^{j}}

= Q(r_{n}) exp {ϵ − (2.5)n^{2/9}/[2ζ(3)]^{4/3} log^{2} n − f_{m,n}(θ)},  (3.34)

since by (2.14)

Q(r_{n}) = Q_{m}(r_{n}) ∏_{j=m+1}^{∞} \frac{1}{(1 − r_{n}^{j})^{j}} .

Using (2.15), (2.9) and (3.26), we find that

| f_{m,n}(θ) |≤ 2 ∑_{j>m} jr_{n}^{j} = 2y_{n}^{−2} ∑_{jy_{n}>my_{n}} y_{n}(jy_{n})e^{−jy_{n}}

= O (n^{2/3} ∫_{v_{n}}^{∞} ue^{−u}du) = O(1).

In order to get an estimate which contains a term that is asymptotically equivalent to q(n), we apply first the bound from (3.34), and then multiply and divide the whole expression by \sqrt{2πb(r_{n})}. Thus, using (2.4) again, we finally get

| J_{2}(m, n) |≤ r_{n}^{−n}Q(r_{n}) exp {−(2.5)n^{2/9}/[2ζ(3)]^{4/3} log^{2} n + O(1)}

= \frac{r_{n}^{−n}Q(r_{n})}{\sqrt{2πb(r_{n})}}exp\{−(2.5)n^{2/9}/[2ζ(3)]^{4/3} log^{2} n + log \sqrt{2πb(r_{n})} + O(1)\}

= \frac{r_{n}^{−n}Q(r_{n})}{\sqrt{2πb(r_{n})}} exp \{−(2.5)n^{2/9}/[2ζ(3)]^{4/3} log^{2} n + \frac{4}{3} log n + O(1)\}  (3.35)

as n → ∞.

3.3. End of the Proof

Applying the result of Lemma 3 to (3.31) and (3.35), we obtain

J_{1}(m, n) = exp \{−\frac{2e^{−z}}{9[2ζ(3)]^{2/3}} \}q(n)[1 + o(1)]

and

J_{2}(m, n) = o(q(n)).

Replacing these estimates into (3.1) - (3.3), after appropriate cancellations we find that

P (L_{n} ≤ m) = exp \{−\frac{2e^{−z}}{9[2ζ(3)]^{2/3}} \} + o(1).  (3.36)

Here m = v_{n}/y_{n} is determined by (3.8). Using estimate (2.9) for y_{n} and expression (3.26) for v_{n}, we conclude that

m = \frac{\frac{2}{3} log n + 2 log log n + z}{y_{n}} = [\frac{n}{2ζ(3)}]^{1/3} (\frac{2}{3} log n + 2 log log n + z) [1 + O(1/n)]

= [\frac{n}{2ζ(3)}]^{1/3} (\frac{2}{3} log n + 2 log log n + z) + O(n^{−2/3} log n)

as n → ∞. Substituting this value into (3.36), we obtain

P \{[\frac{2ζ(3)}{n}]^{1/3} L_{n} − log (n^{2/3}) − log (log^{2} n) ≤ z + o(1)\}

= exp \{−\frac{2e^{−z}}{9[2ζ(3)]^{2/3}} \} + o(1).  (3.37)

Now, it remains to replace z with z + log {2/9[2ζ(3)]^{2/3}} and apply simple algebra to see that

− log (n^{2/3}) − log (log^{2} n) − log {2/9[2ζ(3)]^{2/3}}
= − log [\frac{n}{2ζ(3)}]^{2/3} − log [\frac{1}{2} log^{2} (n^{2/3})].  (3.38)

The limiting distribution result (1.6) now follows from (3.37), (3.38) and the fact that L_{n}, C_{n} and R_{n} are equidistributed. □

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