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

Abstract The goal of the paper is to introduce a new set of tools for the study of discrete and continuous $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -corners processes. In the continuous setting, our work provides a multi-level extension of the loop equations (also called Schwinger–Dyson equations) for $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -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 $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -ensembles obtained by Borodin, Gorin and Guionnet in (Publications mathématiques de l’IHÉS 125, 1–78, 2017).