Normal vector of a random hyperplane

TL;DR

This paper investigates the distribution of the normal vector to a hyperplane spanned by random vectors, showing it behaves like a uniform sphere vector, and applies this to improve bounds in random matrix theory.

Contribution

It establishes that the hyperplane normal vector is approximately uniformly distributed on the sphere under certain conditions, and applies this to improve bounds on singular values and eigenvector delocalization in random matrices.

Findings

Normal vector resembles a uniform sphere vector.

Provides exponential tail bounds for the least singular value.

Achieves optimal delocalization results for eigenvectors.

Abstract

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.