Abstract
In this paper, we compare the solutions of the Dyson Brownian motion for general $$\beta $$ and potential V and the associated McKean–Vlasov equation near the edge. Under suitable conditions on the initial data and the potential V, we obtain optimal rigidity estimates of particle locations near the edge after a short time $$t={{\,\mathrm{o}\,}}(1)$$ . Our argument uses the method of characteristics along with a careful estimate involving an equation of the edge. With the rigidity estimates as an input, we prove a central limit theorem for mesoscopic statistics near the edge, which, as far as we know, has been done for the first time in this paper. Additionally, combining our results with Landon and Yau (Edge statistics of Dyson Brownian motion. arXiv:1712.03881 , 2017), we give a proof of the local ergodicity of the Dyson Brownian motion for general $$\beta $$ and potential at the edge, i.e., we show the distribution of extreme particles converges to the Tracy–Widom $$\beta $$ distribution in a short time.
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