# The bead process for beta ensembles

## TL;DR

This paper constructs a general bead process for sine beta ensembles as an infinite-dimensional Markov chain, linking it to the bulk scaling limit of Hermite beta corner processes and generalizing classical random matrix minors.

## Contribution

It introduces a new construction of the bead process for sine beta ensembles and establishes its connection to Hermite beta corner processes as a microscopic limit.

## Key findings

- Constructed the bead process for general sine beta processes.
- Proved the process as the bulk scaling limit of Hermite beta corner processes.
- Extended classical results for Gaussian and orthogonal ensembles.

## Abstract

The bead process introduced by Boutillier is a countable interlacing of the determinantal sine-kernel point processes. We construct the bead process for general sine beta processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite beta corner process introduced by Gorin and Shkolnikov, generalizing the process of the minors of the Gaussian unitary and orthogonal ensembles. In order to prove our results, we use bounds on the variance of the point counting of the circular and the Gaussian beta ensembles, proven in a companion paper.