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

Abstract

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.