论文标题
统一布朗运动的地方法律和刚性
Local law and rigidity for unitary Brownian motion
论文作者
论文摘要
我们在$ n $维统一组上对布朗运动的特征值位置建立了很高的概率估计，并对位于单位圆的任何间隔中的特征值数量进行了估计。这些估计是最佳的，以$ n $中的任意小多项式因素。我们的结果处于光谱边缘（表明极限特征值在$ \ Mathcal {o}（n^{ - 2/3+}）$中，在限制光谱测量的边缘的边缘，在光谱体积中），以及在$ 4 $的时间接近$ 4 $的限制光谱测量的时间。我们的方法是动力学的，基于分析经验光谱测量的孔变换的演变，沿着限制光谱度量满足的PDE的特征，即自由统一的布朗尼运动。
We establish high probability estimates on the eigenvalue locations of Brownian motion on the $N$-dimensional unitary group, as well as estimates on the number of eigenvalues lying in any interval on the unit circle. These estimates are optimal up to arbitrarily small polynomial factors in $N$. Our results hold at the spectral edges (showing that the extremal eigenvalues are within $\mathcal{O} (N^{-2/3+})$ of the edges of the limiting spectral measure), in the spectral bulk, as well as for times near $4$ at which point the limiting spectral measure forms a cusp. Our methods are dynamical and are based on analyzing the evolution of the Borel transform of the empirical spectral measure along the characteristics of the PDE satisfied by the limiting spectral measure, that of the free unitary Brownian motion.
