Poissonized Plancherel Measure

• Poissonized Plancherel measure is a probability distribution on integer partitions that combines Poisson randomness with classical Plancherel weights, offering a rich model for asymptotic behavior.
• Its determinantal structure, characterized by discrete Bessel kernels and Fredholm determinants, provides explicit results on limit shapes and Tracy–Widom edge fluctuations.
• The measure connects symmetric group representations, random matrix theory, and stochastic growth models, and underpins analyses of phase transitions and advanced combinatorial asymptotics.

The Poissonized Plancherel measure is a probability distribution on the set of all integer partitions, combining the Plancherel measure with Poissonian randomness in the partition size. Distinguished by its deep connections to symmetric group representations, @@@@2@@@@, integrable probability, and combinatorial models, the Poissonized Plancherel measure yields determinantal point processes whose rich asymptotic and structural properties unify aspects of integrable systems, random growth models, and topological recursion. The fine-structure of its associated transforms, limit shapes, and fluctuations reveal connections to Catalan combinatorics, Tracy–Widom distributions, and higher genus enumerative invariants.

1. Definition and Poissonization

Let $\Lambda_n$ denote the set of integer partitions of $n$. The Plancherel measure on $\Lambda_n$ is given by

$$\mathrm{Planch}_n(\lambda) = \frac{(\dim \lambda)^2}{n!},$$

where $\dim \lambda$ is the dimension of the irreducible representation of $S_n$ associated with $\lambda$.

The Poissonized Plancherel measure with parameter $N > 0$ is defined on the set of all partitions $\Lambda = \sqcup_{n \geq 0} \Lambda_n$ by

$$\mathrm{PP}(N)(\lambda) = e^{-N} \frac{N^{|\lambda|}}{|\lambda|!} \cdot \mathrm{Planch}_{|\lambda|}(\lambda) = e^{-N} \frac{N^{|\lambda|} (\dim \lambda)^2}{(|\lambda|!)^2}.$$

Here $|\lambda|$ is the total number of boxes in $\lambda$. This construction "poissonizes" the partition size, so that $|\lambda|$ is distributed as $\mathrm{Poisson}(N)$, after which a partition of that size is drawn from the ordinary Plancherel measure (Waters, 2016, Rostam, 2021, Betea, 2020). As $N \to \infty$, the scaled boundary of the random Young diagram converges to the Vershik–Kerov–Logan–Shepp limit shape.

2. Determinantal Structure and Correlation Kernels

Under the map $$\lambda \mapsto X(\lambda) = \{\lambda_i - i + \frac{1}{2} : i \geq 1\}$$ (the "Maya diagram"), the Poissonized Plancherel measure induces a determinantal point process on $\mathbb{Z} + \frac{1}{2}$ with correlation kernel expressible in multiple forms. The discrete Bessel kernel representation is

$$K_N(x, y) = \frac{J_x(2\sqrt{N}) J_{y+1}(2\sqrt{N}) - J_{x+1}(2\sqrt{N}) J_y(2\sqrt{N})}{x - y},$$

for $x \neq y$, using Bessel functions of the first kind. The process is a specialization of the Schur measure. For any finite configuration $\{x_1, \ldots, x_k\}$,

$$\mathbb{P}\big( \{x_1, \ldots, x_k\} \subset X(\lambda) \big) = \det [K_N(x_i, x_j)]_{i, j = 1}^k$$

(Betea, 2020, Lazag, 2019, Rostam, 2021). Fredholm determinants of $K_N$ encode gap and largest part probabilities, with asymptotics yielding Tracy–Widom (GUE) edge fluctuations. Extensions to "almost symmetric" partitions and symplectic/orthogonal Schur measures yield analogous determinantal laws and edge results.

3. Kerov–Markov–Krein Transform and Fine-Structure Asymptotics

Given a Young diagram $\lambda$, the signed corner measure $\sigma_\lambda$ is supported at the rescaled locations of its inner and outer corners. The Kerov–Markov–Krein (KMK) transform $\mu_\lambda$ is the unique probability measure characterized by the exponential Stieltjes formula

$$\exp \left( -\frac{1}{2} \int \log(x - s)\, d\sigma_\lambda(s) \right) = \int \frac{d\mu_\lambda(s)}{x-s},$$

for $x$ off the support (Waters, 2016).

The averaged $\mu_\lambda$ under $\mathrm{PP}(N)$ behaves analogously to the empirical measure of GUE eigenvalues: as $N \to \infty$, both converge to Wigner's semicircle law $$\rho_0(s) = \frac{1}{2\pi} \sqrt{4 - s^2} \, \mathbf{1}_{[-2,2]}(s)$$. The Stieltjes transform admits a uniform, all-orders $1/N$ expansion: $$\mathbb{E}_{\mathrm{PP}(N)} \left[ \int \frac{d\mu_\lambda(s)}{1 - x s} \right] = \sum_{g \geq 0} N^{-g} \Phi_g(c(x^2)),$$ where $c(x^2) = \sum_{k \ge 0} C_k x^{2k}$ is the Catalan generating function, and $\Phi_g$ are explicit rational functions. The fine-structure theorem asserts

$$[N^{-g}]\,\mathbb{E}_{\mathrm{PP}(N)} \left[ \int \frac{d\mu_\lambda(s)}{1 - x s} \right] = \frac{c}{T^g}\sum_{k = g+1}^{3g-1} \theta_g(k)\left(\frac{\Delta}{T}\right)^k,$$

with $T = 2 - c$, $\Delta = c - 1$, and integer coefficients $\theta_g(k)$. This structure is recursively generated by differential operators acting on the two-variable Catalan kernel $G(x, y) = c / (1 - x y c)$, mirroring the Harer–Zagier expansion for GUE moments, but with genus index $g$ rather than $2g$ and different summation bounds (Waters, 2016).

4. Limit Theorems for Core Sizes and Central Limit Behavior

Let $|\bar\lambda|$ denote the size of the $e$-core of a random partition $\lambda$ under the Poissonized Plancherel measure. As $|\lambda|$ becomes large ($N \to \infty$), suitably normalized $|\bar\lambda|$ converges in distribution to a sum of $(e-1)$ independent gamma random variables with parameters: $$Y_t = \frac{\pi}{4\sqrt{t}} |\bar\lambda| \xrightarrow{\ d \ } \sum_{k = 1}^{e-1} \Gamma\!\left(\frac{1}{2}, \sin \frac{k\pi}{e} \right).$$ This result exhibits full independence for the gamma variables, in contrast to the uniform measure, and relates the fluctuations in $e$-core size to determinantal statistics of the descent set. Explicit mean, variance, and covariance asymptotics for the associated "residuum" vectors $x_i(\lambda)$ are provided, and joint moment generating functions are computed, solidifying the Poissonized Plancherel measure as a determinantal, log-gas-like model with explicit fluctuation laws (Rostam, 2021).

5. Multiplicative Averages, Riemann–Hilbert Analysis, and Phase Transitions

Consider the multiplicative average

$$Q(t, s) = \mathbb{E} \left[ \prod_{i \geq 1} (1 + e^{\eta(\lambda_i - i + \frac{1}{2} - s)})^{-1} \right]$$

for $\eta > 0$, where the underlying random partition is distributed according to the Poissonized Plancherel measure with parameter $t^2$. $Q(t, s)$ is expressible as a Fredholm determinant $\det(I - \mathscr{H}_\eta(t, s))$, with $\mathscr{H}_\eta$ a discrete-integrable operator (built from the Bessel kernel), which lifts to a matrix Riemann–Hilbert problem (Cafasso et al., 8 Jan 2026).

In the regime $t \to \infty$, $s = x t$, the logarithmic rate function

$F(x) = -\lim_{t \to \infty} t^{-2} \log Q(t, x t)$

exhibits distinct analytic behaviors. For $x \leq x_*$ (a critical negative value), $F(x)$ interpolates a quadratic and exponential term; for $x_* < x < 2$, $F(x)$ involves an elliptic integral, and for $x \geq 2$, $F(x) = 0$. The associated equilibrium measure for the underlying log-gas energy displays two third-order phase transitions: Tracy–Widom type ($x \uparrow 2$), and a "birth of a cut" ($x \downarrow x_*$), with explicit dependence on Jacobi $\vartheta_{11}$-functions and elliptic moduli.

Applications link these large deviations to lower-tail probabilities in $q$-deformed polynuclear growth models, spectral behavior at the Bessel-edge for positive-temperature free fermions, and asymptotics in radially symmetric 2D Toda shock solutions (Cafasso et al., 8 Jan 2026).

6. Christoffel Deformations, Palm Measures, and TASEP Applications

The Poissonized Plancherel measure admits "Christoffel deformations"—multiplications of the weight by squared polynomials vanishing at prescribed points—mirroring discrete orthogonal polynomial ensembles. These deformations yield explicit kernels in terms of Wronskians of Bessel functions (via the Charlier limit transition), and link to Palm measures: the latter are the laws of the process conditioned to contain specific points, and can be used to describe exclusion processes with frozen particles (e.g., TASEP with wedge initial data and blocked sites). These constructions provide a precise framework for understanding conditional distributions and the effect of inserting particles (or holes) in the configuration (Lazag, 2019).

7. Connections, Significance, and Structural Analogies

The Poissonized Plancherel measure crystallizes the bridge between symmetric group representation theory, random matrix models (GUE), and integrable combinatorics. Its core probabilistic object is a determinantal point process exhibiting edge universality, fine-structure corrections governed by Catalan combinatorics, and higher-genus expansions paralleling topological recursion structures in random matrix theory. Recent advances encompass rigorous multiplicative averages (linked to elliptic integrals and theta functions), explicit conditioning (Christoffel/Palm theory), and applications to continuous limits, integrable PDEs, and stochastic growth. The full interplay of combinatorics, special function theory, and analysis exemplified by the Poissonized Plancherel measure locates it as a central subject in modern probabilistic and representation-theoretic asymptotics (Waters, 2016, Rostam, 2021, Betea, 2020, Lazag, 2019, Cafasso et al., 8 Jan 2026).