  We introduce a two-parameter family of probability distributions, indexed by
$\beta/2 = \theta > 0$ and $K \in \mathbb{Z}_{\geq 0}$, that are called
$\beta$-Krawtchouk corners processes. These measures are related to Jack
symmetric functions, and can be thought of as integrable discretizations of
$\beta$-corners processes from random matrix theory, or alternatively as
non-determinantal measures on lozenge tilings of infinite domains. We show that
as $K$ tends to infinity the height function of these models concentrates
around an explicit limit shape, and prove that its fluctuations are
asymptotically described by a pull-back of the Gaussian free field, which
agrees with the one for Wigner matrices. The main tools we use to establish our
results are certain multi-level loop equations introduced in our earlier work
arXiv:2108.07710.
