Global asymptotics for $β$-Krawtchouk corners processes via multi-level loop equations

Abstract: 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.