Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts

Leonid V. Bogachev

Published 2011-11-14, updated 2014-05-02Version 4

We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form $\mathcal{F}(z)=\prod_{\ell=1}^\infty \mathcal{F}_0(z^\ell)$ (which entails equal weighting among possible parts $\ell\in\mathbb{N}$). Under mild technical assumptions on the function $H_0(u)=\ln(\mathcal{F}_0(u))$, we show that the limit shape $\omega^*(x)$ exists and is given by the equation $y=\gamma^{-1}H_0(\mathrm{e}^{-\gamma x})$, where $\gamma^2=\int_0^1 u^{-1}H_0(u)\,\mathrm{d}u$. The wide class of partition measures covered by this result includes (but is not limited to) representatives of the three meta-types of decomposable combinatorial structures --- assemblies, multisets and selections. Our method is based on the usual randomization and conditioning; to this end, a suitable local limit theorem is proved. The proofs are greatly facilitated by working with the cumulants of sums of the part counts rather than with their moments.