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.
Categories: math.PR math-ph math.MP