Asymptotic analysis of expectations of plane partition statistics

Abstract

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where $Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}$ is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around $x=1$. The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

Appendix

In the Appendix we deduce Wright’s formula (5), using Hayman’s result (24).

First, by (16) and (18) one has

$$\begin{aligned}&e^{nd_n} =\exp {((2\zeta (3))^{1/3} n^{2/3} -1/36+ O(n^{-\beta }))}, \end{aligned}$$

(A.1)

$$\begin{aligned}&\sqrt{2\pi b(e^{-d_n})} \sim \frac{(6\pi )^{1/2} n^{2/3}}{(2\zeta (3))^{1/6}}. \end{aligned}$$

(A.2)

An asymptotic expression for $Q(e^{-d_n})$ can be obtained using a general lemma due to Meinardus [16] (see also [2, Lemma 6.1]). Since the Dirichlet generating series for the plane partitions is $\zeta (z-1)$ (see also (8)), we get

$$\begin{aligned} Q(e^{-d_n})= & {} \exp {(\zeta (3)d_n^{-2} -\zeta (-1)\log {d_n} +\zeta ^\prime (-1) +O(d_n^{\beta _1}))}\\= & {} \exp {(\zeta (3)d_n^{-2} +\frac{1}{12}\log {d_n} +2\gamma +O(d_n^{\beta _1}))} \end{aligned}$$

where $0<\beta _1<1$ and $\gamma$ is given by (6) (more details on the values of $\zeta (-1)$ and $\zeta ^\prime (-1)$ can be found in [27, Section 13.13] and [6, Section 2.15]). Using (48) and (49), after some algebraic manipulations, we obtain

$$\begin{aligned} Q(e^{-d_n}) =\left( \frac{2\zeta (3)}{n}\right) ^{1/36} \exp {((\zeta (3))^{1/3}(n/2)^{2/3} +1/36 +O(n^{-\beta _1}))}. \end{aligned}$$

(A.3)

Combining (A.1–A.3), we find that

$$\begin{aligned}&q(n) \sim \left( \frac{2\zeta (3)}{n}\right) ^{1/36} \frac{\exp {((2\zeta (3))^{1/3}-1/36+(\zeta (3))^{1/3}(n/2)^{2/3}+1/36+2\gamma )}}{(3\pi )^{1/2} n^{2/3}/(2\zeta (3))^{1/6}}\\&\quad =\frac{(\zeta (3))^{1/6+1/36} n^{-1/36-2/3}}{2^{1/2-1/6-1/36}(3\pi )^{1/2}} \exp {(3(\zeta (3))^{1/3}(n/2)^{2/3}+2\gamma )} \\&\quad =\frac{(\zeta (3))^{7/36}}{2^{11/36}(3\pi )^{1/2}} n^{-25/36} \exp {(3(\zeta (3))^{1/3}(n/2)^{2/3}+2\gamma )}. \end{aligned}$$

$\square$