Does the expectation of eigenvectors of resolvents of random matrices vary with the constants?

Suppose X is a real $p\times n$ random matrix and $Y=XX^T/n $. The resolvent of Y is$$R(z)=(Y-z)^{-1},$$ where $z$ is a complex number. Since the Stieltjes transform of a random matrix can be written as $m(z)=\frac{1}{p}\mathrm{tr}R(z)$, it's very useful in random matrix theory. I know that $R(z)$ shares the same eigenvectors as $Y$. I want to know whether $\mathbb{E}[R(z_1)]$ and $\mathbb{E}[R(z_2)], z_1\neq z_2$ also share the same eigenvectors. In other words, I guess $z$ has no effect on the eigenvectors of $\mathbb{E}R(z)$. Will the correlation between eigenvalue and eigenvector lead to a negative answer?

Many thanks.