Asymptotic Analysis of Expectations of Plane Partition Statistics

TL;DR

This paper develops a general asymptotic framework for calculating expectations of various statistics of plane partitions as their size grows large, using generating functions and Hayman's method.

Contribution

It introduces a unified asymptotic scheme for plane partition expectations based on generating functions and Hayman's method, extending previous results to a broader class of statistics.

Findings

Asymptotic formulas for expected largest part and trace of plane partitions.

Method applies to expectations derived from generating functions of the form Q(x)F(x).

Results align with known asymptotics for linear integer partitions.

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 height 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. (2014) related to linear integer partition statistics. We base our study on Hayman's method for admissible power series.