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Local law and mesoscopic fluctuations of Dyson Brownian motion for general $\beta$ and potential
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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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Universality of the least singular value for the sum of random matrices
The least singular value of R*XT + U*YV, after scaling by N, converges in distribution to 1 - e^{-r^2}, the same limit as for a complex Gaussian matrix.
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