Estimating the asymptotics of integer partitions in intermediate dimensions ($d = 3,4,5,6$)

Abstract

It was recently shown by Yeliussizov \cite{Yeliussizov} that integer partitions in dimensions $d \geq 7$ asymptotically grow strictly faster than MacMahon numbers. As MacMahon numbers match with integer partitions in dimensions $d = 1,2$, the comparison of asymptotics of integer partitions with MacMahon numbers in intermediate dimensions ($d = 3,4,5,6$) is an open question. In this work, we perform Markov chain Monte Carlo (MCMC) simulations till $N=15000$ by using adaptive weight learning followed by conventional MCMC steps to numerically estimate the asymptotics of integer partitions in these intermediate dimensions. We numerically establish that in these intermediate dimensions, partitions asymptotically grow faster than MacMahon numbers. More specifically, assuming that the limits exist, we show: $\lim_{n\to\infty}n^{-3/4}\log p_3(n) = 1.8196 \pm 0.0019$, $\lim_{n\to\infty}n^{-4/5}\log p_4(n) = 1.7215 \pm 0.0045$, $\lim_{n\to\infty}n^{-5/6}\log p_5(n) = 1.6521 \pm 0.0059$, and $\lim_{n\to\infty}\log n^{-6/7}p_6(n) = 1.652 \pm 0.021$ for partitions in dimensions $d=3,4,5,$ and $6$ respectively. These numbers are all larger than MacMahon leading order asymptotic coefficients of $1.7898, 1.6614, 1.5737,$ and $1.509$ respectively. Additionally, we also find estimates for some of the sub-leading asymptotic terms in $\log p_d(n)$ in each of the dimensions.