Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures

TL;DR

This work establishes universal formulas describing the global asymptotics of Jack-deformed discrete β-ensembles across high, low, and fixed temperature regimes. It introduces Jack--Thoma measures and connects them to Jack measures with the approximate factorization property, proving LLN and CLT with explicit limit shapes, transition measures, and Gaussian fluctuations expressed via Łukasiewicz ribbon paths. A key achievement is the universality of these formulas: the same moment and covariance structures govern both Jack--Thoma measures and AFP-character measures, via depoissonization and the Markov–Krein correspondence. The results bridge discrete β-ensembles, free probability, and algebraic combinatorics, and yield concrete edge asymptotics for Jack--Plancherel partitions in extreme temperature regimes, including explicit Bessel-zero descriptions. Overall, the paper provides a robust universal framework for the large-scale geometry of Jack-deformed partitions with wide applicability to representation theory and probabilistic combinatorics.

Abstract

This paper establishes universal formulas describing the global asymptotics of two distinct discrete versions of $β$-ensembles in the high, low and fixed temperature regimes. Our results affirmatively answer a question posed by the second author and Śniady. We first introduce a special class of Jack measures on Young diagrams of arbitrary size, called the ``Jack--Thoma measures'', and prove the LLN and CLT in the three aforementioned limit regimes. In each case, we provide explicit formulas for polynomial observables of the limit shape and Gaussian fluctuations around the limit shape. These formulas have surprising positivity properties and are expressed as sums of weighted lattice paths. Second, we show that the previous formulas are universal: they also describe the limit shape and Gaussian fluctuations for the model of random Young diagrams of a fixed size derived from Jack characters with the approximate factorization property. Finally, in stark contrast with continuous $β$-ensembles, we show that the limit shapes at high and low temperatures of our random Young diagrams are one-sided infinite staircase shapes. For the Jack--Plancherel measure, we describe this shape explicitly by relating its local minima with the zeroes of Bessel functions.

Figures (7)

• Figure 1: The profiles of anisotropically-scaled Young diagrams of size $d=400$ and $d=6400$, sampled by the Jack--Schur--Weyl measure from \ref{['ex:Schur-Weyl']} with parameters $\alpha = 4$, $N = \sqrt{\alpha d}$, where each box is not of size $1\times 1$, but of size $\frac{2}{\sqrt{d}}\times\frac{1}{2\sqrt{d}}$. The blue curve is the limit shape in the fixed temperature regime (fixed $\alpha>0$), discovered by Biane Biane2001.
• Figure 2: The limit shape $\Lambda_{g;(1,0,0,\cdots)}$ with $g\!=\!-\frac{1}{4}$ (low temperature regime), $\mathbf{v}\!=\!(1,0,0,\cdots)$, drawn in the French notation and obtained from \ref{['thm_asymptotic_positions_abbr']}, is shaded in light blue. Juxtaposed is an anisotropically-scaled Young diagram $\lambda^{(d)}$ of size $d = 1600$ sampled by the Jack--Plancherel measure $\mathbb{P}^{(\alpha)}_d$ with $\alpha= \frac{4^2}{d}$, so each box (with gray-colored sides) is not of size $1\!\times\!1$, but of size $\frac{4}{d}\!\times\!\frac{1}{4}$. The three smallest zeroes of the Bessel function $J_{-4z}(8)$ are equal to $l^{(1/4)}_1 = -1.086...$, $l^{(1/4)}_2 = -0.424...$, and $l^{(1/4)}_3 = 0.102...$, therefore \ref{['thm_asymptotic_positions_abbr']} implies $\lim_{d\to\infty}\frac{\lambda^{(d)}_1}{-g\cdot d} = 1.336...$, $\lim_{d\to\infty}\frac{\lambda^{(d)}_2}{-g\cdot d} = 0.924...$, and $\lim_{d\to\infty}\frac{\lambda^{(d)}_3}{-g\cdot d} = 0.647...$ in probability.
• Figure 3: Young diagram $\lambda = (4,3,1,1)$ (left) and the anisotropic Young diagram $T_{2,\frac{1}{2}}\lambda$ (center) are in the French convention. The solid line in the rightmost picture is the profile$\omega_{T_{2,\frac{1}{2}}\lambda}$ of $T_{2,\frac{1}{2}}\lambda$, obtained by switching from the French to the Russian convention. The local minima $x_1,x_2,x_3,x_4$ and maxima $y_1,y_2,y_3$ of the profile are indicated in red and blue, respectively.
• Figure 4: Four excursions $\Gamma_1,\Gamma_2,\Gamma_3$ and $\Gamma_4$. Note that $\Gamma_3$ and $\Gamma_4$ are Łukasiewicz paths, but $\Gamma_1$ and $\Gamma_2$ are not.
• Figure 5: A ribbon path $\vec{\bm{\Gamma}} = (\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4)$ on $4$ sites of lengths $8, 7, 4, 5$, which is also a Łukasiewicz ribbon path belonging to $\mathbf{L}(8, 7, 4, 5)$. The three pairs of dots of different colors (red, green and blue) are examples of pairings, and the colored arcs join the pairs of preceding steps of the same degree. The degrees of the red, green and blue pairings are $4$, $2$ and $2$, respectively.

Theorems & Definitions (104)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Abbreviated version of \ref{['theo:ShapeDS', 'theo:CovDS']}
• Theorem 1.4
• Remark 1
• Definition 2.1: First model of random Young diagrams
• Remark 2
• Remark 3
• Example 1
• Definition 2.2