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

Mattia Cafasso, Matteo Mucciconi, Giulio Ruzza

TL;DR

This work analyzes multiplicative averages $Q(t,s)$ under the Poissonized Plancherel measure, revealing large-$t$ asymptotics governed by a rate function $\mathcal{F}(x)$ and elliptic corrections. The authors develop a discrete Riemann–Hilbert framework and execute a nonlinear steepest descent analysis, with a variational problem linking $Q$ to a log-gas energy and a g-function tied to equilibrium measures. They uncover two third-order phase transitions in the equilibrium density, including a discreteness-induced birth of a cut, and provide explicit density formulas in one- and two-cut regimes expressed via elliptic data. The results have broad implications: they yield tail characterizations for the $q$-PNG height, describe the edge behavior of the positive-temperature discrete Bessel process, and inform step-like shock solutions to the cylindrical Toda equation, illustrating a deep connection between random partitions, integrable systems, and log-gas phenomena.

Abstract

We consider random integer partitions~$λ$ that follow the Poissonized Plancherel measure of parameter~$t^2$. Using Riemann--Hilbert techniques, we establish the asymptotics of the multiplicative averages \[ Q(t,s)=\mathbb{E} \left[ \prod_{i\geq 1} \left(1+\e^{η(λ_i-i+\frac{1}{2}-s)}\right)^{-1} \right] \] for fixed $η>0$ 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 shock initial data.

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

TL;DR

This work analyzes multiplicative averages under the Poissonized Plancherel measure, revealing large- asymptotics governed by a rate function and elliptic corrections. The authors develop a discrete Riemann–Hilbert framework and execute a nonlinear steepest descent analysis, with a variational problem linking to a log-gas energy and a g-function tied to equilibrium measures. They uncover two third-order phase transitions in the equilibrium density, including a discreteness-induced birth of a cut, and provide explicit density formulas in one- and two-cut regimes expressed via elliptic data. The results have broad implications: they yield tail characterizations for the -PNG height, describe the edge behavior of the positive-temperature discrete Bessel process, and inform step-like shock solutions to the cylindrical Toda equation, illustrating a deep connection between random partitions, integrable systems, and log-gas phenomena.

Abstract

We consider random integer partitions~ that follow the Poissonized Plancherel measure of parameter~. Using Riemann--Hilbert techniques, we establish the asymptotics of the multiplicative averages \[ Q(t,s)=\mathbb{E} \left[ \prod_{i\geq 1} \left(1+\e^{η(λ_i-i+\frac{1}{2}-s)}\right)^{-1} \right] \] for fixed in the regime and . We compute the large- expansion of expressing the rate function 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 -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 shock initial data.
Paper Structure (52 sections, 65 theorems, 585 equations, 23 figures)

This paper contains 52 sections, 65 theorems, 585 equations, 23 figures.

Table of Contents

  1. Introduction and results
  2. Overview
  3. Results
  4. Minimization of logarithmic energy
  5. Large-$t$ asymptotic expansion
  6. Methods
  7. Open directions
  8. Open directions
  9. Applications
  10. $q$-deformed polynuclear growth
  11. Positive-temperature discrete Bessel process
  12. Solutions to the cylindrical Toda equation with step-like shock initial conditions
  13. Minimization of logarithmic energy
  14. Poissonized Plancherel measure and logarithmic energy
  15. Variational problem
  16. ...and 37 more sections

Key Result

Theorem 1.2

Let $\eta>0$ and $x\in \mathbb{R}$. The minimizer $\rho_{\eta,x}$ of the logarithmic energy $\mathcal{E}_{\eta,x}$ is given explicitly as follows. (1) If $x \le x_*$, $\rho_{\eta,x}(\mu)=\rho_{\mathrm{VKLS}}\bigl(\mathrm{e}^{\eta/2}\mu\bigr)$. (2) If $x_*<x<2$, where The endpoints $a=a(\eta,x)$, $b=b(\eta,x)$, $c=c(\eta,x)$, and $d=d(\eta,x)$ are given by where $\mathcal{K}=\mathcal{K}(x)$ is g

Figures (23)

  • Figure 1: In orange, the Young diagram (in Russian notation) of the partition $\lambda = (11,8,8,7,5,3,2,2,1,1,1)$. In blue, the Maya diagram $\mathscr{D}(\lambda)$. The darker shaded cells represent the hook of the cell $(3,2)$, whose length is $12$.
  • Figure 2: The various phases of the equilibrium measure $\rho_{\eta,x}$ (in blue) and the corresponding limiting partition (in orange) in Russian notation. Here $\eta=\log 5$. In the top-left panel, when $x \le x_*$, the density $\rho_{\eta,x}$ is a rescaling of the Vershik--Kerov--Logan--Shepp density by a factor $\mathrm{e}^{-\eta/2}$. In the top-right and bottom-left panels the cases when $\rho_{\eta,x}$ possesses two saturated regions and its explicit form is given in \ref{['eq:maintheorem:densitymiddle']}. In the bottom-right panel, where $x \ge 2$, $\rho_{\eta,x}$ coincides with the Vershik--Kerov--Logan--Shepp density.
  • Figure 3: A plot of the endpoints $a$, $b$, $c$, and $d$ (in blue, orange, green, and red, respectively) as functions of $x\in(x_*,2)$. The dot-dashed thin curve is the graph of $x$ and lies between $b$ and $c$. We take $\eta=\log 20$, such that $x_*=-\frac{19}{10\log 20}=-0.634236\ldots$
  • Figure 4: Top: plot of the rate function $\mathcal{F}(x)$ (the different colors correspond to $x\leq x_*$, $x_*<x<2$, and $x\geq 2$; the thin black dashed part is the analytic continuation of the parabola defining $\mathcal{F}(x)$ for $x\leq x_*$). Bottom: plots of the functions $\mathcal{K}(x)$ and $\mathcal{L}(x)$, respectively, for $x\in(x_*,2)$. We take $\eta=\log 5$, such that $x_* = -\frac{8}{5\, \log 5} = -0.994136\ldots$
  • Figure 5: A depiction of the $q$-PNG dynamics. Red segments represent the collision interface between two islands. Green segments correspond to nucleations of new islands of infinitesimal width.
  • ...and 18 more figures

Theorems & Definitions (142)

  • Definition 1.1: Implicit half-period
  • Theorem 1.2: Equilibrium measure
  • Definition 1.3: Rate function
  • Remark 1.4
  • Theorem 1.5: Asymptotic expansion
  • Theorem 1.6: Phase transitions
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 132 more