Upper bounds for the maximum deviation of the Pearcey process

TL;DR

This paper establishes upper bounds on the maximum deviation of the Pearcey process, a universal point process in random matrix theory, providing insights into its rigidity and fluctuations.

Contribution

It introduces new global rigidity bounds for the Pearcey process and derives a central limit theorem for its fluctuations.

Findings

Proves a global rigidity upper bound for the number of points in the Pearcey process.

Establishes an upper bound for the maximum deviation of points in the process.

Derives a central limit theorem for individual fluctuations.

Abstract

The Pearcey process is a universal point process in random matrix theory and depends on a parameter $\rho \in \mathbb{R}$. Let $N(x)$ be the random variable that counts the number of points in this process that fall in the interval $[-x,x]$. In this note, we establish the following global rigidity upper bound: \begin{align*} \lim_{s \to \infty}\mathbb P\left(\sup_{x> s}\left|\frac{N(x)-\big( \frac{3\sqrt{3}}{4\pi}x^{\frac{4}{3}}-\frac{\sqrt{3}\rho}{2\pi}x^{\frac{2}{3}} \big)}{\log x}\right| \leq \frac{4\sqrt{2}}{3\pi} + \epsilon \right) = 1, \end{align*} where $\epsilon > 0$ is arbitrary. We also obtain a similar upper bound for the maximum deviation of the points, and a central limit theorem for the individual fluctuations. The proof is short and combines a recent result of Dai, Xu and Zhang with another result of Charlier and Claeys.