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

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