  This paper establishes universal formulas describing the global asymptotics
of two different models of discrete $\beta$-ensembles in high, low and fixed
temperature regimes. Our results affirmatively answer a question posed by the
second author and \'Sniady.
  We first consider the Jack measures on Young diagrams of arbitrary size,
which depend on the inverse temperature parameter $\beta>0$ and specialize to
Schur measures when $\beta=2$. We introduce a class of Jack measures of
Plancherel-type and prove a law of large numbers and central limit theorem in
the three regimes. In each regime, 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
in terms of weighted lattice paths. We also establish connections between these
measures and the work of Kerov-Okounkov-Olshanski on Jack-positive
specializations and show that this is a rich class of measures parametrized by
the elements in the Thoma cone.
  Second, we show that the formulas from limits of Plancherel-type Jack
measures are universal: they also describe the limit shape and Gaussian
fluctuations for the second model of random Young diagrams of a fixed size
defined by Jack characters with the approximate factorization property (AFP)
studied by the second author and \'Sniady. Finally, we discuss the limit shape
in the high/low-temperature regimes and show that, contrary to the continuous
case of $\beta$-ensembles, there is a phase transition phenomenon in passing
from the fixed temperature regime to the high/low temperature regimes. We note
that the relation we find between the two different models of random Young
diagrams appears to be new, even in the special case of $\beta=2$ that relates
Schur measures to the character measures with the AFP studied by Biane and
\'Sniady.
