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The bead process for beta ensembles.

Joseph Najnudel¹, Bálint Virág²

¹School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG UK.

Probability Theory and Related Fields

|November 1, 2021

Summary

Researchers constructed a bead process for general point processes, revealing it as a microscopic scaling limit of the Hermite corner process. This work generalizes random matrix theory, connecting point processes to Markov chains.

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Area of Science:

Probability Theory
Mathematical Physics
Random Matrix Theory

Background:

The bead process, a countable interlacing of point processes, was previously introduced by Boutillier.
Generalizing point processes and their scaling limits is a key area in mathematical physics.
The Hermite beta ensemble and its corner process are significant in random matrix theory.

Purpose of the Study:

To construct the bead process for general point processes.
To establish the bead process as the microscopic scaling limit of the Hermite beta corner process.
To generalize the study of minors in Gaussian Unitary and Orthogonal Ensembles.

Main Methods:

Construction of the bead process as an infinite-dimensional Markov chain with an explicit transition mechanism.

Keywords:

60B20 60F05 60J05 60J55

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Analysis of the microscopic scaling limit in the bulk.

Utilizing variance bounds for point counting in circular and Gaussian beta ensembles.

Main Results:

Successfully constructed the bead process for general point processes.
Demonstrated that this bead process is the microscopic scaling limit of the Hermite beta corner process.
Extended the understanding of minors in Gaussian Unitary and Orthogonal Ensembles.

Conclusions:

The bead process provides a novel framework for studying point processes and their scaling limits.
This research establishes a significant connection between Markov chains and random matrix theory.
The findings generalize existing results in the field and open avenues for further research.