The random-vector model conjecture for eigenvectors of sparse random graphs

Let $G(n,p)$ be an Erdős–Rényi random graph. Assume

$$p\geq \frac{(1+\epsilon)\log n}{n}$$

for some constant $\epsilon>0$. Let $v$ be a random unit vector uniformly distributed on the $(n-1)$-dimensional unit sphere, let $u$ be a unit eigenvector of $G(n,p)$, and let $w$ be any fixed $n$-dimensional vector. The random-vector model conjecture. For every $\delta>0$,

$$\mathbf{P}(|w\cdot u-w\cdot v|>\delta)=o(1).$$

This conjecturally says that projections of eigenvectors have the same asymptotic behavior as projections of a uniformly random unit vector above the connectivity threshold; the source presents it as a generalization of preceding eigenvector-delocalization questions.