On the almost eigenvectors of random regular graphs

Understand

Let $d\geq 3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices.

• We show that if $n$ is large enough then the entry distribution of every almost eigenvector $v$ of $G$ (with entry sum 0 and normalized to have length $\sqrt{n}$) is close to some Gaussian distribution $N(0,\sigma)$ in the weak topology where $0\leq\sigma\leq 1$.
• Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph.
• Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenvalues are close.