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

Abstract

This paper establishes universal formulas describing the global asymptotics of two distinct discrete versions of $\beta$-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 $\beta$-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.