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

Abstract

We introduce and study a class of discrete particle ensembles that naturally arise in connection with classical random matrix ensembles, log-gases and Jack polynomials. Under technical assumptions on a general analytic potential we prove that the global fluctuations of these ensembles are asymptotically Gaussian with a universal covariance that remarkably differs from its counterpart in random matrix theory. Our main tools are certain novel algebraic identities that we have discovered. They play a role of discrete multilevel analogues of loop equations. 1. Introduction 247 2. Nekrasov's equations 257 3. Particular setup 264 4. Application of Nekrasov's equations 271 5. Central limit theorem 283 6. Multilevel extensions and examples 294 7. Continuous limit 305 Appendix A. 321 Appendix B.331 References 340 * Note that we have reversed the order of the indices so that ℓ 1 is now largest and ℓ N is the smallest -this convention is more consistent with the symmetric function origin of (1-3).