  In this paper we determine two asymptotic results for Jack measures on
partitions, a model defined by two specializations of Jack polynomials proposed
by Borodin-Olshanski in [European J. Combin. 26.6 (2005): 795-834]. Assuming
these two specializations are the same, we derive limit shapes and Gaussian
fluctuations for the anisotropic profiles of these random partitions in three
asymptotic regimes associated to diverging, fixed, and vanishing values of the
Jack parameter. To do so, we introduce a generalization of Motzkin paths we
call "ribbon paths", show for general Jack measures that certain joint
cumulants 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-Dol\k{e}ga.
