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

TL;DR

This paper analyzes the asymptotic behavior of multiplicative averages of Plancherel random partitions using advanced techniques, revealing phase transitions and elliptic function connections with applications in growth models and integrable systems.

Contribution

It introduces a novel asymptotic analysis of multiplicative averages for Plancherel partitions, uncovering phase transitions and elliptic function representations with broad applications.

Findings

Explicit large-$t$ expansion of $ ext{log }Q(t,xt)$

Identification of two third-order phase transitions

Connection to elliptic theta functions and applications

Abstract

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.