Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. Has anyone studied this expectation in asymptotics $$E\_{A}(\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1} A^T A))?$$


Any papers/resources would be helpful, ideal fidnings would be $m,n \mapsto \infty$ as $\frac{m}{n} \mapsto 0$, but anything would be great.



Approximating the expectation of trace inverse of random Gaussian combination
The answer is similar to that of your [earlier question](https://mathoverflow.net/q/424771/11260):


For $m,n\gg 1$, and $m/n\equiv r\in (0,1)$ fixed, an integration over the [Marchenko–Pastur distribution](https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution) gives (with $x\_\pm=(1\pm\sqrt{r})^2$)
$$\lim\_{m,n\rightarrow\infty}\mathbb{E}\bigl[m^{-1}\mathrm{Tr}\,\bigl((n^{-1}AA^\top + \lambda I)^{-1}n^{-1}AA^\top\bigr)\bigr]=\int\_{x\_-}^{x\_+} \frac{x}{x+\lambda}\frac{\sqrt{\left(x\_+-x\right) \left(x-x\_-\right)}}{2 \pi r x}\,dx$$
$$\qquad=\frac{1}{2r}\bigl(\lambda+r+1-\sqrt{\lambda^2+2 \lambda (r+1)+(r-1)^2}\bigr).$$
The rescaling of $AA^\top$ by a factor $1/n$ is needed for a $\lambda$-dependent answer in the large $n$ limit.


In the limit $r\rightarrow 0$ this tends to $1/(1+\lambda)$.


