Multi-level loop equations for $β$-corners processes

Abstract

The goal of the paper is to introduce a new set of tools for the study of discrete and continuous β-corners processes. In the continuous setting, our work provides a multi-level extension of the loop equations (also called Schwinger-Dyson equations) for β-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 β-ensembles obtained by Borodin, Gorin and Guionnet in (Publications mathématiques de l 'IHÉS 125, 1-78, 2017). Contents 1. Introduction and main result 1 2. General setup and the equations 6 3. Application of Nekrasov equations 28 4. Continuous limits 33 References 46

Asymptotics of discrete β-corners processes via two-level discrete loop equations

“…In the paper we consider a two-parameter family of probability measures P θ,K ∞ , which are indexed by β/2 = θ > 0 and K ∈ Z ≥0 . We call these measures β-Krawtchouk corners processes, and they form a special subclass of discrete β-corners processes, which we introduced in [DK21] as integrable discretizations of the β-corners processes from random matrix theory. The goal of this paper is to describe the global behavior of these models as K → ∞.…”

“…The goal of this paper is to describe the global behavior of these models as K → ∞. The tools we use to study P θ,K ∞ are certain multi-level loop equations that were developed in [DK21], which generalize the two-level loop equations from [DK22] and the single-level loop equations from [BGG17], and which originate from the works of Nekrasov and his collaborators [NPS18,NP12,Nek16]. Loop equations (also known as Schwinger-Dyson equations) have proved to be a very efficient tool in the study of global fluctuations of log-gases and random matrices, see [BG24,BG13,Joh98,KS10,Shc13] and the references therein.…”

Dynamical Loop Equation

“…There were hints in the literature, suggesting that the framework of the loop equations might be adaptable to multi-time two-dimensional setting, as in [3,31,33,51,53]. However, these results were mostly isolated and relied on specifics of the stochastic systems they dealt with.…”

“…It is also instructive to compare Theorem 1.1 with two-level discrete loop equations of [31,33]. An important distinction is that in our setup x is deterministic, while in [31,33] it is random (of log-gas type).…”

“…It is also instructive to compare Theorem 1.1 with two-level discrete loop equations of [31,33]. An important distinction is that in our setup x is deterministic, while in [31,33] it is random (of log-gas type). We are going to significantly rely on this feature in the applications of Theorem 1.1: it allows us to iterate the equations and reach joint distributions at arbitrary times, rather than only at adjacent ones.…”