Let $a \in \mathbb R$ be a determinstic scalar and let $X$ be and $n \times d$  such that the $n \times n$ psd random matrix $S=XX^T$ has limiting eigenvalue distribution $\mu$, when $n,d \to \infty$ with $n/d = \gamma \in (0,\infty)$. Consider the psd random matrix $T=YY^\top$, where $Y$ is the $n \times d$ random matrix with entries $Y_{ij} = X_{ij} + a$.

>**Question.** What is the limiting eigenvalue distribution $\lambda$ of $T$ ?

**Note.** My guess is that the Stieltjes transform of $\lambda$ would be some how given implicitly via the Stieltjes transform of $\mu$.

The example I have in mind is when $X$ has iid rows drawn from an **isotropic log-concave** distribution in $\mathbb R^d$. For gaussian of uniform on unit-sphere in $\mathbb R^d$.