Gaussian Asymptotics of Jack Measures on Partitions From Weighted Enumeration of Ribbon Paths

Abstract

Abstract In this paper, we determine two asymptotic results for Jack measures $M(v^{\textrm {out}}, v^{\textrm {in}})$, a measure on partitions defined by two specializations $v^{\textrm {out}}, v^{\textrm {in}}$ of Jack polynomials proposed by Borodin–Olshanski in [10]. Assuming $v^{\textrm {out}} = v^{\textrm {in}}$, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to vanishing, fixed, and diverging values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call “ribbon paths,” show for arbitrary $v^{\textrm {out}}, v^{\textrm {in}}$ that certain Jack measure joint cumulants ${\kappa _n}$ are weighted sums of connected ribbon paths on $n$ sites with $n-1+g$ pairings, and derive our two results from the contributions of $(n,g)=(1,0)$ and $(2,0)$, respectively. Our analysis makes use of Nazarov–Sklyanin’s spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack–Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy–Dołęga.