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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.05164 |
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| author | Cafasso, Mattia Mucciconi, Matteo Ruzza, Giulio |
| author_facet | Cafasso, Mattia Mucciconi, Matteo Ruzza, Giulio |
| contents | 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 $$Q(t,s)=\mathbb{E} \left[ \prod_{i\geq 1} \left(1+\mathrm{e}^{η(λ_i-i+\frac{1}{2}-s)}\right)^{-1} \right] $$ 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 initial data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05164 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplicative Averages of Plancherel Random Partitions: Elliptic Functions, Phase Transitions, and Applications Cafasso, Mattia Mucciconi, Matteo Ruzza, Giulio Mathematical Physics Probability 11P82, 33E05, 37K60, 41A60, 60F10, 60K35 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 $$Q(t,s)=\mathbb{E} \left[ \prod_{i\geq 1} \left(1+\mathrm{e}^{η(λ_i-i+\frac{1}{2}-s)}\right)^{-1} \right] $$ 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 initial data. |
| title | Multiplicative Averages of Plancherel Random Partitions: Elliptic Functions, Phase Transitions, and Applications |
| topic | Mathematical Physics Probability 11P82, 33E05, 37K60, 41A60, 60F10, 60K35 |
| url | https://arxiv.org/abs/2601.05164 |