A Gaussian distribution for refined DT invariants and 3D partitions
If \(p_n\) is the number of partitions of \(n\) (non-increasing sequences of integers summing to \(n\)), the celebrated Hardy-Ramanujan asymptotic expansion implies that as \(n\to\infty\), \[ p_n \sim \frac{1}{4n\sqrt{3}} \exp\biggl(\pi\sqrt{\frac{2n}{3}}\biggr) \] If, instead, \(p_n\) is the number of \textit{plane} partitions of \(n\) (non-increasing double sequences of integers summing to \(n\)), a result of \textit{E. M. Wright} [Q. J. Math., Oxf. Ser. 2, 177--189 (1931; Zbl 0002.38202)] is that as \(n\to\infty\), \[ p_n \sim \frac{\zeta(3)^{7/36}}{2^{11/36}\sqrt{3\pi} \, n^{25/36}} e^{3\bigl(\frac{\zeta(3)}{4}\bigr)^{1/3}n^{2/3}+\zeta'(-1)} \] If, then, \(w(\pi)\) is the weight of such a plane partition \(\pi=(\pi_{i,j})\) given by \( w(\pi) = \sum_{i,j} \text{sgn}(i-j)\pi_{i,j}\,, \) and one defines the Laurent polynomial \( p_n(q) = \sum_{\pi} q^{w(\pi)} \) by a sum over all plane partitions of \(n\), then clearly \( p_n = p_n(q)|_{q=1}\,. \) The main result of the present paper is that the statistics of \(w/n^{2/3}\) over plane partitions is Gaussian with variance \((2\zeta(3))^{-1/3}\) in the limit \(n\to\infty\). This statement is equivalent to the asymptotics \[ \bigl(q\frac{d}{dq}\bigr)^k p_n(q)|_{q=1} \sim n^{2k/3}\cdot(k-1)!!(2\zeta(3))^{-k/6} \cdot \frac{\zeta(3)^{7/36}}{2^{11/36}\sqrt{3\pi} \, n^{25/36}} e^{3\bigl(\frac{\zeta(3)}{4}\bigr)^{1/3}n^{2/3}+\zeta'(-1)} \] for even \(k\), which is proven by an application of the circle method. The result is of interest in the Donaldson-Thomas theory of certain Calabi-Yau threefolds.
scientific article; zbMATH DE number 6341598
scientific article