Upper bounds for the maximum deviation of the Pearcey process

2020/09/28 by Charlier, Christophe
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2009.13225

Abstract

The Pearcey process is a universal point process in random matrix theory and depends on a parameter ρ∈ 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: lim_{s → ∞}\mathbb P(sup_{xgt; s}|\fracN(x)-( (3√(3))/(4π)x^(4)/(3)-(√(3)ρ)/(2π)x^(2)/(3) )log x| ≤ (4√(2))/(3π) + ε) = 1, where ε> 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.

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