  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. Our proof uses graph limits and information theory. Our
results have consequences for factor of i.i.d.\ processes on the infinite
regular tree.
