Estimating the asymptotics of solid partitions

TL;DR

This paper investigates the asymptotic behavior of solid partitions using Monte Carlo simulations, revealing deviations from conjectured values and identifying oscillatory terms that could explain the complexity of higher-dimensional partitions.

Contribution

It provides the first detailed asymptotic analysis of solid partitions, including the discovery of oscillatory terms and deviations from previous conjectures.

Findings

Established the limit of $n^{-3/4} \, \log p_3(n)$ as approximately 1.822

Identified an oscillatory term with period proportional to $n^{1/4}$ in the asymptotics

Found deviations from MacMahon numbers conjecture for solid partitions

Abstract

We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If $p_3(n)$ denotes the number of solid partitions of an integer $n$, we show that $\lim_{n\rightarrow\infty} n^{-3/4} \log p_3(n)\sim 1.822\pm 0.001$. This shows clear deviation from the value $1.7898$, attained by MacMahon numbers $m_3(n)$, that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in $\log p_3(n)$. In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to $n^{1/4}$, the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.