随机超平面的法向量
Normal vector of a random hyperplane
摘要：
设v_1，...，v_{n-1}是R^n（或C^n）中的n-1个独立向量。研究了v_i所张成的超平面的单位法向量x。我们的主要发现是，在v_i的某些随机性假设下，x类似于从单位球面均匀选择的随机向量。我们的结果在随机矩阵理论中有应用。考虑一个具有iid元素的n乘n随机矩阵。我们首先证明了一个指数界上的最小奇异值，提高早期的线性界的Rudelson和Vershynin。其次，我们推导出对应于小模特征值的特征向量的最优离域。
英文摘要：
Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x, the unit normal vector of the hyperplane spanned by the v_i. Our main finding is that x resembles a random vector chosen uniformly from the unit sphere, under some randomness assumption on the v_i. Our result has applications in random matrix theory. Consider an n by n random matrix with iid entries. We first prove an exponential bound on the upper tail for the least singular value, improving the earlier linear bound by Rudelson and Vershynin. Next, we derive optimal delocalization for the eigenvectors corresponding to eigenvalues of small modulus.
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