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