In this paper, we compare the solutions of Dyson Brownian motion with general $\beta$ and potential $V$ and the associated McKean-Vlasov equation near the edge. Under suitable conditions on the initial data and potential $V$, we obtain the optimal rigidity estimates of particle locations near the edge for short time $t=\text{o}(1)$. 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, have been done for the first time in this paper. Additionally, combining with \cite{LandonEdge}, our rigidity estimates are used to give a proof of the local ergodicity of Dyson Brownian motion for general $\beta$ and potential at the edge, i.e. the distribution of extreme particles converges to Tracy-Widom $\beta$ distribution in short time.

中文翻译：

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一般 $$\beta $$ 和边缘潜力的戴森布朗运动

在本文中，我们将戴森布朗运动的解与一般 $\beta$ 和潜在 $V$ 以及边缘附近的关联 McKean-Vlasov 方程进行比较。在初始数据和潜在 $V$ 的合适条件下，我们获得了短时间 $t=\text{o}(1)$ 边缘附近粒子位置的最佳刚度估计。我们的论点使用特征方法以及涉及边缘方程的仔细估计。以刚性估计作为输入，我们证明了边缘附近细观统计的中心极限定理，据我们所知，这是本文首次完成的。此外，结合 \cite{LandonEdge}，我们的刚性估计用于证明戴森布朗运动的局部遍历性对于一般 $\beta$ 和边缘的潜力，即