Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic random vector in $\mathbb R^d$ with $\mathbb E\|z\|^2=1$. For concreteness, take $z \sim N(0,1/d)$. Consider the random $n \times n$ matrix $W = XX^\top$. It is well-known that the limiting spectral distribution of $W$ is the Marchenko-Pastur distribution with parameter $\gamma$.

Of course if $\gamma > 1$, then the least singular-value of $W$ is $0$, otherwise converges in to $(1-\sqrt{\gamma})^2$ (in some sense...).

Now consider the matrix $A := W\circ W$ with entries given by $A_{i,j} = W_{i,j}^2 = (x_i^\top x_j)^2$.

Question. What is the limiting spectral distribution of $A$ ? In particular does $\lambda_{\min}(A)$ converge (in some sense) to a limit ?

Related: (MO link) Eigenvalues of Hadamard product of two Wishart-type matrices

An experiment

I've noticed that $\lambda_{\min}(A)$ converges to a limit which is bounded away from zero even when $\gamma > 1$. This is inspite of the fact that $\lambda_{\min}(W) = 0$ in this case. See figure below (for $\gamma = 2$).