Overcrowding asymptotics for the Sine_beta process

TL;DR

This paper derives asymptotic probabilities for overcrowding events in the Sine_beta process, revealing how likely it is to find many points in a fixed interval, with precise exponential decay rates.

Contribution

It provides the first detailed overcrowding estimates for the Sine_beta process, including the main exponential decay and the next order term for shrinking intervals.

Findings

Probability of at least n points ~ e^{-rac{eta}{2} n^2 \log(n)}

Identifies next order term when interval size shrinks to zero

Enhances understanding of point process fluctuations in random matrix limits

Abstract

We give overcrowding estimates for the Sine_beta process, the bulk point process limit of the Gaussian beta-ensemble. We show that the probability of having at least n points in a fixed interval is given by $e^{-\frac{\beta}{2} n^2 \log(n)+O(n^2)}$ as $n\to \infty$. We also identify the next order term in the exponent if the size of the interval goes to zero.