Local law and mesoscopic fluctuations of Dyson Brownian motion for general $\beta$ and potential

Jiaoyang Huang; Benjamin Landon

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arXiv · 1612.06306

Local law and mesoscopic fluctuations of Dyson Brownian motion for general $\beta$ and potential

Jiaoyang Huang·Benjamin Landon

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Abstract

We study Dyson Brownian motion with general potential $V$ and for general $\beta \geq 1$. For short times $t = o (1)$ and under suitable conditions on $V$ we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale $\eta_* \ll t \ll 1$, we prove a mesoscopic central limit theorem for linear statistics at all scales $N^{-1}\ll\eta\ll t$

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Jiaoyang Huang, Benjamin Landon. 2016-12-19. Local law and mesoscopic fluctuations of Dyson Brownian motion for general $\beta$ and potential. https://arxiv.org/abs/1612.06306

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