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 $$Q(t,s)=\mathbb{E} \left[ \prod_{i\geq 1} \left(1+\mathrm{e}^{\eta(\lambda_i-i+\frac{1}{2}-s)}\right)^{-1} \right]$$ for fixed $\eta>0$ in the regime $t\to+\infty$ and $s/t=O(1)$. We compute the large-$t$ expansion of $\log Q(t,xt)$ expressing the rate function $\mathcal{F}(x) = -\lim_{t \to \infty}t^{-2}\log Q(t,xt)$ 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.