Multi-level loop equations for \(\beta \)-corners processes

The goal of the paper is to introduce a new set of tools for the study of discrete and continuous \(\beta \) $\beta$ -corners processes. In the continuous setting, our work provides a multi-level extension of the loop equations (also called Schwinger–Dyson equations) for \(\beta \) $\beta$ -log gases obtained by Borot and Guionnet in (Commun. Math. Phys. 317, 447–483, 2013). In the discrete setting, our work provides a multi-level extension of the loop equations (also called Nekrasov equations) for discrete \(\beta \) $\beta$ -ensembles obtained by Borodin, Gorin and Guionnet in (Publications mathématiques de l’IHÉS 125, 1–78, 2017).