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The goal of the paper is to introduce a new set of tools for the study ofdiscrete and continuous $\beta$-corners processes. In the continuous setting,our work provides a multi-level extension of the loop equations (also calledSchwinger-Dyson equations) for $\beta$-log gases obtained by Borot and Guionnetin (Commun. Math. Phys. 317, 447-483, 2013). In the discrete setting, our workprovides a multi-level extension of the loop equations (also called Nekrasovequations) for discrete $\beta$-ensembles obtained by Borodin, Gorin andGuionnet in (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017).

Multi-level loop equations for $\beta$-corners processes