Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. Finally, defined $\alpha := w^\top G^2 w$.

Question. In the limit $n,d \to \infty$ with $n/d \to \rho \in (0,\infty)$, what is the limitting value of $\alpha$ as a function of $\lambda$ and $\rho$ ?

A useful subcase is when $\lambda \to 0^+$.

Question. What is the value of $\lim_{\lambda \to 0^+}\lim_{n,d \to \infty \\ n/d \to \rho}\alpha$ as a function of $\rho$ ?