Multiplicative Averages of Plancherel Random Partitions: Elliptic Functions, Phase Transitions, and Applications

We consider random integer partitions $\lambda$ that follow the Poissonized Plancherel measure of parameter $t^2$. Using Riemann$-$Hilbert techniques, we establish the asymptotics of the multiplicative averages $$Q(t,s)=\mathbb{E}\left[\prod _{i\ge 1}{(1+{\mathrm{e}}^{\eta ({\lambda }_i-i+\frac{1}{2}-s)})}^{-1}\right]$$ for fixed $\eta >0$ in the regime $t\to +\mathrm{\infty }$ and $s\mathrm{/}t=O(1)$. We compute the large-$t$ expansion of $\log Q(t,xt)$ expressing the rate function $\mathcal{F}(x)=-{\lim }_{t\to \mathrm{\infty }}{t}^{-2}\log Q(t,xt)$ and the subsequent divergent and oscillatory contributions explicitly in terms of elliptic theta functions. The associated equilibrium measure presents, in general, nontrivial saturated regions and it undergoes two third-order phase transitions of different nature which we describe. Applications of our results include an explicit characterization of tail probabilities of the height function of the $q$-deformed polynuclear growth model and of the edge of the positive-temperature discrete Bessel process and asymptotics of radially symmetric solutions to the 2D Toda equation with step-like initial data.