发表机构
Raman Research Institute（拉曼研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过自适应权重学习结合MCMC模拟，估计中间维度（$d=3,4,5,6$）整数分拆的渐近行为，发现其增长速度快于MacMahon数，给出了主项及部分次项渐近系数的数值估计。
AI 中文摘要
Yeliussizov近期的研究表明，维度$d \geq 7$的整数分拆渐近增长速度严格快于MacMahon数；由于MacMahon数与维度$d=1,2$的整数分拆完全匹配，中间维度（$d = 3,4,5,6$）整数分拆的渐近行为与MacMahon数的对比是一个未解决的问题。本研究中，我们采用自适应权重学习结合常规MCMC步骤的方法，进行直至$N=15000$的马尔可夫链蒙特卡洛（MCMC）模拟，以数值估计这些中间维度整数分拆的渐近行为。我们数值确定，这些中间维度的分拆渐近增长速度快于MacMahon数；具体而言，假设极限存在，对于$d=3,4,5,6$的分拆，我们得到：$\lim_{n\to\infty}n^{-3/4}\log p_3(n) = 1.8196 \pm 0.0019$，$\lim_{n\to\infty}n^{-4/5}\log p_4(n) = 1.7215 \pm 0.0045$，$\lim_{n\to\infty}n^{-5/6}\log p_5(n) = 1.6521 \pm 0.0059$，以及$\lim_{n\to\infty}n^{-6/7}\log p_6(n) = 1.652 \pm 0.021$。这些数值均大于MacMahon数的主项渐近系数，分别为1.7898、1.6614、1.5737和1.509；此外，我们还得到了各维度$\log p_d(n)$中部分次项渐近项的估计值。
英文摘要
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.
Comments24 pages. Comments are welcome
