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

TL;DR

This paper introduces multi-level loop equations for $eta$-corners processes, extending existing loop equations to both discrete and continuous settings, providing new analytical tools for these stochastic models.

Contribution

It develops a unified framework of multi-level loop equations for $eta$-corners processes, extending prior single-level equations to multiple levels in both discrete and continuous cases.

Findings

Extended loop equations to multi-level settings.

Unified approach for discrete and continuous $eta$-corners processes.

Facilitates deeper analysis of $eta$-log gases and $eta$-ensembles.

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 math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017).