Macdonald-level extension of beta ensembles and large-N limit transition

TL;DR

This paper introduces a family of deformed beta ensembles linked to Macdonald polynomials and proves a large-N limit transition to infinite-particle processes, advancing understanding of integrable probability models.

Contribution

It presents a novel $(q,t)$-deformed discrete beta ensemble framework and establishes the large-N limit transition to infinite-particle processes, connecting Macdonald polynomials with random matrix theory.

Findings

Existence of a large-N limit transition for the deformed beta ensembles.

Construction of infinite-particle limit processes.

Extension of beta ensemble theory via Macdonald polynomial parameters.

Abstract

We introduce and study a family of $(q,t)$-deformed discrete $N$-particle beta ensembles, where $q$ and $t$ are the parameters of Macdonald polynomials. The main result is the existence of a large-$N$ limit transition leading to random point processes with infinitely many particles.