# Uniform point variance bounds in classical beta ensembles

## TL;DR

This paper establishes logarithmic bounds on the variance of point counts in classical beta ensembles, providing insights into their fluctuation behavior in circular and linear settings.

## Contribution

It introduces new variance bounds for point counts in beta ensembles, advancing understanding of their statistical fluctuations.

## Key findings

- Variance bounds are logarithmic in set length.
- Bounds are expected to be optimal up to a constant.
- Results apply to both circular and Gaussian beta ensembles.

## Abstract

In this paper, we give bounds on the variance of the number of points of the circular and the Gaussian $\beta$ ensemble in arcs of the unit circle or intervals of the real line. These bounds are logarithmic with respect to the renormalized length of these sets, which is expected to be optimal up to a multiplicative constant depending only on $\beta$.