Disclaimer. I only started learning the subject of free probability $1$ day ago, and I'm still trying to absorb the fundamentals, while applying them to my own specific problems arizing in the spectral analysis of certain concrete random matrices.

Let $X_{n,m}$, $Y_{m,k}$, $Z_{m,k}$ be large (large in the sense that $n \to \infty$ such that $m/n,k/n \in (0,\infty)$, say) independent random matrices with entries from $N(0,1)$ and let $A_{m,k}$ be a deterministic matrix. Consider the random psd matrix $R_{m,k} := (X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k})(X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k})^\top$.

Question. How to use tools from free probability (e.g the "linearization trick", etc.) to compute the limiting spectral distribution of $R_{m,k}$ ?

Note. In particular, (when possible) I'd like to have bounds on the extremities of the support of this distribution, as this is my ultimate goal.

Disclaimer. I only started learning the subject of free probability $1$ day ago, and I'm still trying to absorb the fundamentals, while applying them to my own specific problems arizing in the spectral analysis of certain concrete random matrices.

Let $X_{n,m}$, $Y_{m,k}$, $Z_{m,k}$ be large (large in the sense that $n \to \infty$ such that $m/n,k/n \in (0,\infty)$, say) independent random matrices with entries from $N(0,1)$ and let $A_{m,k}$ be a deterministic matrix. Consider the random matrix $R_{n,k} := X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k}$ (or equivalently, of $R_{n,k}R_{n,k}^\top$).

Question. How to use tools from free probability (e.g the "linearization trick", etc.) to compute the limiting singular-value distribution of $R_{n,k}$ ?

Note. In particular, (when possible) I'd like to have bounds on the extremities of the support of this distribution, as this is my ultimate goal.

Disclaimer. I only started learning the subject of free probability $1$ day ago, and I'm still trying to absorb the fundamentals, while applying them to my own specific problems arizing in the spectral analysis of certain concrete random matrices.

Let $X_{n,m}$, $Y_{m,k}$, $Z_{m,k}$ be large independent random matrices with entries from $N(0,1)$ and let $A_{m,k}$ be a deterministic matrix. Consider the random psd matrix $R_{m,k} := (X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k})(X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k})^\top$.

Question. How to use tools from free probability (e.g the "linearization trick", etc.) to compute the limiting spectral distribution of $R_{m,k}$ ?

Disclaimer. I only started learning the subject of free probability $1$ day ago, and I'm still trying to absorb the fundamentals, while applying them to my own specific problems arizing in the spectral analysis of certain concrete random matrices.

Let $X_{n,m}$, $Y_{m,k}$, $Z_{m,k}$ be large (large in the sense that $n \to \infty$ such that $m/n,k/n \in (0,\infty)$, say) independent random matrices with entries from $N(0,1)$ and let $A_{m,k}$ be a deterministic matrix. Consider the random psd matrix $R_{m,k} := (X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k})(X_{n,m}Y_{m,k}+Z_{m,k}+A_{m,k})^\top$.

Question. How to use tools from free probability (e.g the "linearization trick", etc.) to compute the limiting spectral distribution of $R_{m,k}$ ?

Note. In particular, (when possible) I'd like to have bounds on the extremities of the support of this distribution, as this is my ultimate goal.