The limit shape of Young diagrams for multiplicative statistics with superpolynomial growth (Q937862)

Multiplicative statistics for permutations were introduced in \textit{A. M. Vershik} [Funct. Anal. Appl. 30, No.\ 2, 90--105 (1996; Zbl 0868.05004)]. Define the \textit{weight} \(a(\lambda)\) of a partition \(\lambda\vdash n\) with \(t_k\) parts equal to \(k\) to be \(\prod a_{t_i(\lambda),i}\), and consider the product \[ F(x) = \prod_{i=1}^{\infty}f_k(x^k) = \sum_n \left(\sum_{\lambda\vdash n}a(\lambda)\right) x^n. \] For any choice of formal power series \(f_k=1+\sum a_{i,k}x^i\), this introduces a probability measure on partitions of \(n\) given by \(\text{Prob}(\lambda)=a(\lambda)/(\sum_{\mu\vdash n}a(\mu))\). The author considers the statistics in the case \(f_k(x)=(1-x)^{-P(k)}\) where \(P(k)=e^{2\alpha\sqrt{n}}(1+o(1))\), with \(\alpha=\sqrt{\zeta(2)}\). These statistics arise in the enumeration of types of conjugacy classes of matrices over finite fields, as studied by \textit{J. A. Green} [Trans.\ Am.\ Math.\ Soc.\ 80, 442--447 (1955; Zbl 0068.25605)] For these statistics, the asymptotics for the part sizes of partitions converge to give a sharp peak at \((\log n)^2/\alpha^2\), producing nearly rectangular Young diagrams.